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\begin{document} \begin{CJK*}{GBK}{song} \title{Quantum state transfer between three ring-connected atoms}
\author{Guo Yan-Qing$^{*}$, Deng Yao, Pei Pei\footnotetext{ $^{*}$ Corresponding author. Email: [email protected]}, Wang Dian-Fu} \affiliation{Department of Physics, Dalian Maritime University, Dalian, 116026} \pacs{03.67.Mn, 42.50.Pq}
\keywords{Quantum State Transfer; Distant Atoms; Ising Model} \begin{abstract} A robust quantum state transfer scheme is discussed for three atoms that are trapped by separated cavities linked via optical fibers in ring-connection. It is shown that, under the effective three-atom Ising model, arbitrary quantum state can be transferred from one atom to another deterministically via an auxiliary atom with maximum unit fidelity. The only required operation for this scheme is replicating turning on/off the local laser fields applied to the atoms for two steps with time cost $\frac{\sqrt{2}\pi}{\Gamma_{0}}$. The scheme is insensitive to cavity leakage and atomic position due to the condition $\Delta \approx \kappa\gg g$. Another advantage of this scheme is that the cooperative influence of spontaneous emission and operating time error can reduce the time cost for maximum fidelity and thus speed up the implementation of quantum state transfer. \end{abstract} \maketitle
Long-range communication channels between distant qubits are essential for practical quantum information processing. One of the most important goal for constructing the channels is the implementation of quantum state transfer (QST) from one qubit to another in a deterministic way, especially for unknown quantum state $^{[1-11]}$. Many schemes that based on spin systems that including Heisenberg model or Ising Model, or atom-photon systems that including cavity QED systems have been proposed to implement QST between spins in quantum dots$^{[12-14]}$, atoms or photons in cavities$^{[15-17]}$. The advantage of spin systems is the spins can be easily controlled through magnetic field due to the simple and regular interaction between neighboring spin sites. While, differing from the short-range communication channels of spin systems, QED systems that including intra-cavity atoms connected via optical fibers extend QST to macroscopic length scale that is necessary for long range quantum communications. We believe a quantum model, such as the model suggested by Zhong et al $^{[18]}$, combined with these two kinds of systems should be of much importance for QST process.
However, there is a disadvantage in the schemes using QED systems, that the schemes usually work in a probabilistic way. One of the ways to improve the success probability and fidelity is constructing precisely controlled coherent evolutions of the global system and weaken the affect of probing impulse detection inefficiency. One kind of the controlled evolutions are dominated based on global control of the system. For example, in the scheme considered by Serafini et al $^{[1]}$, the technic turning off the interaction between atoms in separated cavities is used to implement quantum swap gate and C-phase gate. In the scheme proposed by Yin and Li$^{[9]}$, deterministic QST can be achieved through turning off the interaction between distant atomic groups. In the scheme proposed by Bevilacqua and Renzoni, laser pulses are used to implement QST $^{[16]}$. Another kind of the controlled evolutions are dominated based on local control of the system. For example, in the scheme proposed by Mancini and Bose $^{[2]}$, the only required control to obtain maximally entangled states is synchronously turning on and off of the locally applied laser fields applied on individual intro-cavity atoms.
In the present paper, we propose an alternative QST scheme based on a simple quantum network consists of three distant atoms trapped in distinct cavities. Such a system is treated as an effective spin-spin interacting Ising model for distant atoms. A two-step operation consisting of simply replicating turning on/off of the local laser fields is put forward to implement QST between two atoms. We demonstrate that the scheme works in deterministic way with high fidelity. We also investigate the affect of atomic spontaneous emission on the fidelity of the scheme.
We firstly recall the model put forward in Ref. [19]. The schematic setup of the model is shown in Fig. 1. Three two-level atoms 1, 2 and 3 are trapped in spatially separated optical cavities which are assumed to be single-sided. Atoms interact with cavity field in a dispersive way. Three off-resonant driving external fields are added on cavities. Two neighboring cavities are connected via optical fiber. The global system is located in vacuum. Using the input-output theory, taking the adiabatic approximation $^{[20]}$ and applying the methods developed in Refs. [2], we obtain the effective Hamiltonian of the global system as $H_{eff}=H_{zz}$, where $H_{zz}=J(\sigma _{1}^{z}\sigma _{2}^{z}+\sigma _{2}^{z}\sigma _{3}^{z}+\sigma _{3}^{z}\sigma _{1}^{z})$, $\sigma_{i}^{z}$ is spin operators of atom i. And
$J=2\kappa\chi ^{2}Im\left\{ |\alpha|^2(Me^{i\phi }+\kappa e^{i2\phi})/(M^{3}-W^{3})\right\}$, where $\kappa$ is the cavity leaking rate, $\chi=\frac{g^2}{\Delta}$, $g$ is the coupling strength between atom and cavity field, $\Delta$ is the detuning. In deducing $H_{eff}$, the condition $\Delta\approx\kappa\gg g$ is assumed, $M=i\Delta+\kappa$, $W^{3}=\kappa ^{3}e^{i3\phi}$. $\phi _{21}$, $\phi$ is the phase delay caused by the photon transmission along optical fiber. And $\alpha=\varepsilon\frac{M^2+M\kappa e^{i\phi}+\kappa^{2}e^{i2\phi}}{M^{3}-W^3}$. Such a system is undoubtedly an Ising ring model with uniform coupling strengthes.
It has been approved that, in an isotropic Heisenberg model, the arbitrarily perfect QST can be achieved only by applying a magnetic field along the spin chain $^{[13]}$. Thus, we assume local weak laser fields are applied to resonantly interact with the atoms. Without losing of generality, we allow a simple spatial variation of the laser fields so that the Rabi frequencies are different for individual atoms. The effective Hamiltonian is now written as $H_{eff}=H_{zz}+H_{x}$, where $H_{x}=\sum\limits_{i}\Gamma_{i}\sigma_{x},\sigma_{x}=(\sigma _{i}^{-}+\sigma _{i}^{+})$, $\sigma_{i}^{+} (\sigma_{i}^{-})$ is raising (lowering) operator of atom $i$. This can be interpreted as an Ising ring with electromagnetic fields applied on individual spins in perpendicular direction. The system plays an important role in quantum information process since two-atom entangled stated can be generated in such system by synchronously turning off the local laser fields $^{[21-23]}$. This paper aims to study the QST governed by the Hamiltonian. Under the condition $\Gamma_{i}\ll J$, the secular part of the effective Hamiltonian can be obtained through the transformation $UH_{x}U^{-1}$, $U=e^{-iH_{zz}t}$, as $^{[24]}$ \begin{eqnarray} \tilde{H}=\sum\limits_{ijk}\frac{\Gamma_{i}}{2}\sigma_{i}^{x}(1-\sigma_{j}^{z}\sigma_{k}^{z}) . \end{eqnarray} where the subscripts $ijk$ are permutations of $1, 2, 3$. The straight forward interpretation of this Hamiltonian is: the spin of an atom in the Ising ring flips \textbf{if and only if} its two neighbors have opposite spins.
The task of arbitrary unknown quantum state transfer (QST) between two two-level systems a and b is to accomplish the implementation
$(\alpha|e\rangle_{a}+\beta|g\rangle_{a})\otimes|g\rangle_{b}\longrightarrow
|g\rangle_{a}\otimes(\alpha|e\rangle_{b}+\beta|g\rangle_{b})$ deterministically, where $\alpha$ and $\beta$ are unknown complex number and meet the condition of normalization, and the former in above equation is the inputting initial state while the latter is the outputting target state.
To this end, we assume atom 1 is inputting qubit and initially in coherent state $\alpha|e\rangle_{1}+\beta|g\rangle_{1}$, atom 2 is outputting qubit and initially in ground state, atom 3 is an auxiliary qubit and initially in ground state, and suppose the local laser field applied on atom 3 is kept zero, which leads to an unchanged state of atom 3. The secular part of the effective Hamiltonian can be written as $\tilde{H}=\Gamma_{1}\sigma_{1}^{x}(1-\frac{1}{2}\sigma_{2}^{z}\sigma_{3}^{z})+\Gamma_{2}\sigma_{2}^{x}(1-\frac{1}{2}\sigma_{1}^{z}\sigma_{3}^{z})$. The evolution of the first term of initial state
$(\alpha|e\rangle_{1}+\beta|g\rangle_{1})\otimes|g\rangle_{2}\otimes|g\rangle_{3}$ is restricted within the subspace spanned by the following basis vectors \begin{eqnarray}
|\phi_{1}\rangle=|e\rangle_{1}|e\rangle_{2}|g\rangle_{3},
|\phi_{2}\rangle=|e\rangle_{1}|g\rangle_{2}|g\rangle_{3},
|\phi_{3}\rangle=|g\rangle_{1}|e\rangle_{2}|g\rangle_{3}, \end{eqnarray} while the second term remains unchanged. The Hamiltonian in Eq. (7) is now written as \begin{eqnarray} \tilde{H}=\left(\begin{array}{ccc} 0 & \Gamma_{2} & \Gamma_{1}\\ \Gamma_{2} & 0 & 0\\ \Gamma_{1} & 0 & 0\\ \end{array}\right). \end{eqnarray}
The eigenvalues of the Hamiltonian can be obtained as $E_{12}=\pm\sqrt{\Gamma_{1}^{2}+\Gamma_{2}^{2}}$,$E_{3}=0$, and the corresponding eigenvectors are \begin{eqnarray}
|\psi\rangle_{i}=\sum\limits_{j}S_{ij}|\phi_{j}\rangle \end{eqnarray} where \begin{eqnarray} S=\left(\begin{array}{ccc} \frac{1}{\sqrt{2}} & \frac{\Gamma_{2}}{\sqrt{2(\Gamma_{1}^{2}+\Gamma_{2}^{2})}} & \frac{\Gamma_{1}}{\sqrt{2(\Gamma_{1}^{2}+\Gamma_{2}^{2})}} \\ -\frac{1}{\sqrt{2}} & \frac{\Gamma_{2}}{\sqrt{2(B_{1}^{2}+\Gamma_{2}^{2})}} & \frac{\Gamma_{1}}{\sqrt{2(\Gamma_{1}^{2}+\Gamma_{2}^{2})}} \\ 0 & -\frac{\Gamma_{1}}{\sqrt{\Gamma_{1}^{2}+\Gamma_{2}^{2}}} & \frac{\Gamma_{2}}{\sqrt{\Gamma_{1}^{2}+\Gamma_{2}^{2}}} \\ \end{array}\right) \end{eqnarray}, which represents unitary transformation matrix between eigenvectors and basis vectors. For initial system state
$|\Psi(0)\rangle=\sum\limits_{i}c_{i}(0)|\phi_{i}\rangle$, the evolving global system state can be written as
$|\Psi(t)\rangle=\sum\limits_{i}c_{i}(t)|\phi_{i}\rangle$ and is governed by the Schr\"{o}dinger equation $i\frac{\partial
|\Psi(t)\rangle}{\partial t}=\tilde{H}|\Psi(t)\rangle$. The coefficients $c_{i}(t)$ are then given by $^{[9]}$ $c_{i}(t)=\sum\limits_{j}[S^{-1}]_{ij}[Sc(0)]_{j}e^{-iE_{j}t}$, where $c(0)=[c_{1}(0),c_{2}(0),c_{3}(0)]^{T}$.
For initial coefficients $c(0)=[0,\alpha,0]^{T}$, the coefficients can be obtained as \begin{eqnarray} c_{1}(t)&=&-i\frac{\alpha\Gamma_{2}}{\Omega}\textrm{sin}\Omega t,\nonumber\\ c_{2}(t)&=&\frac{\alpha\Gamma_{1}^{2}}{\Omega^{2}}+\frac{\alpha\Gamma_{2}^{2}}{\Omega^{2}}\textrm{cos}\Omega t\nonumber \\ c_{3}(t)&=&\frac{-\alpha\Gamma_{1}\Gamma_{2}}{\Omega^{2}}+\frac{\alpha\Gamma_{1}\Gamma_{2}}{\Omega^{2}}\textrm{cos}\Omega t. \end{eqnarray} where $\Omega=\sqrt{\Gamma_{1}^{2}+\Gamma_{2}^{2}}$.
It is easily shown that, one can take $\Gamma_{1}=\Gamma_{2}=\Gamma_{0}$ and turn off the local laser fields applied to atom 1 and atom 2 synchronously at $t_{p}=\frac{(2k-1)\pi}{\Omega}, k=1,2,3, ...$ and obtain the system state as \begin{eqnarray}
|\Psi(t_{p_{0}})\rangle=|g\rangle_{1}\otimes(-\alpha|e\rangle_{2}+\beta|g\rangle_{2})|g\rangle_{3}. \end{eqnarray}
The above state differs from the target state
$|g\rangle_{1}\otimes(\alpha|e\rangle_{2}+\beta|g\rangle_{2})\otimes|g\rangle_{3}$ due to a minus sign. To obtain the target state exactly, one may program the operating process as in Table. 1 (the term '$\pi$ pulse' in the table is used to denote an equivalent evaluating time $t_{p_{0}}=\frac{\pi}{\Omega}$):
\begin{tabular}{l|l} \multicolumn{2}{l}{Table. 1 Operation sequence for implementing QST}\\ \hline operation sequence & system state\\ \hline $\Gamma_{1}=\Gamma_{3}=\Gamma_{0},\Gamma_{2}=0$ & initial state
$(\alpha|e\rangle_{1}+\beta|g\rangle_{1})|g\rangle_{2}|g\rangle_{3}$\\ \hline $\pi$ pulse on atoms 1 and 3 &
$|g\rangle_{1}|g\rangle_{2}(-\alpha|e\rangle_{3}+\beta|g\rangle_{3})$\\ \hline $\Gamma_{2}=\Gamma_{3}=\Gamma_{0},\Gamma_{2}=0$ &
$|g\rangle_{1}|g\rangle_{2}(-\alpha|e\rangle_{3}+\beta|g\rangle_{3})$\\ \hline $\pi$ pulse on atoms 2 and 3 &
target state $|g\rangle_{1}(\alpha|e\rangle_{2}+\beta|g\rangle_{2})|g\rangle_{3}$\\ \hline \end{tabular}
The above operating process can be interpreted as two steps:
Firstly, turning on the laser field acting on atom 1 and 3 while keeping the laser field acting atom 3 zero. At the specific time $t_{p_{1}}=\frac{(2k-1)\pi}{\Omega}, k=1,2,3, ...$, turning off the laser fields synchronously.
Secondly, turning on the laser field acting on atom 2 and 3 while keeping the laser field acting atom 1 zero. At the specific time $t_{p_{2}}=\frac{(2k-1)\pi}{\Omega}, k=1,2,3, ...$, turning off the laser fields synchronously.
In this procedure, one do not need to know the values of coefficients $\alpha$ and $\beta$, and do not require any methods of quantum coincidence measurement on atoms. An unknown QST is implemented deterministically with $100\%$ success probability.
To illustrate the the efficiency of the QST, not only at specific times, but also in overall view of time scales, we plot the fidelity of QST between atoms 1 and 2 with respect to operating times $t_{1}$, which represents the operating time of the first step, and $t_{2}$, which denotes that of the second step. The average fidelity is defined as $^{[9]}$ \begin{eqnarray}
F=\frac{1}{2\pi}\int_{0}^{2\pi}|\langle\Psi_{f}|\Psi(t)\rangle|^{2} d\theta \end{eqnarray}
where, $|\Psi_{f}\rangle$ is the target state. In Fig. 2 (a), it can be seen that the overall quantity of fidelity is governed by operating times $t_{p_{1}}$ and $t_{p_{2}}$ almost equally. The fidelity periodically reaches the maximum 1 at specific times $t_{p_{1}}=t_{p_{2}}=\frac{\pi}{\Omega}$. The time cost of the scheme can be estimated as $t_{p}\approx \frac{2\pi}{\Omega}$.
The above results explicitly demonstrate a deterministic two-step QST scheme between atoms 1 and 2 which is accomplished by only turning on two identical local laser fields applied on atoms 1 and 3 and turning them off at typical times synchronously, and duplicate the step for atoms 2 and 3. Similarly, QST between atoms 2 and 3 or between atoms 1 and 3 can be accomplished similarly.
So, the total procedure of the scheme implementation consists only two steps: a step turning on/off the laser fields synchronously for inputting atom and auxiliary atom and repeat the step for auxiliary atom and outputting atom.
In this model, the leakage of cavity fields is assumed to be large enough to keep the validity of the adiabatic approximation for obtaining effective Hamiltonian. While, the inevitable atomic spontaneous emission still challenges the efficiency of the scheme and results in a dissipative effect, which can be estimated by adding a non-Hermitian conditional term to the Hamiltonian in Eqn. (1)$^{[25]}$. The global Hamiltonian can be written as
$H_{s}=-i\gamma\sum\limits_{i}|e\rangle_{i}\langle e|+\tilde{H}$, where $\gamma$ represents the atomic spontaneous emission rate. In the subspace spanned by
$|\phi_{1}\rangle=|e\rangle_{1}|g\rangle_{2}|e\rangle_{3}$,
$|\phi_{2}\rangle=|e\rangle_{1}|g\rangle_{2}|g\rangle_{3}$,
$|\phi_{3}\rangle=|g\rangle_{1}|g\rangle_{2}|e\rangle_{3}$, for initial state $|e\rangle_{1}|g\rangle_{2}|g\rangle_{3}$, the evolved coefficients can be obtained as \begin{eqnarray} c_{1}(t)&=&-\frac{i\alpha\Gamma_{3}}{\Lambda}e^{-\frac{3}{2}\gamma t}\textrm{sin}\Lambda t,\nonumber \\ c_{2}(t)&=&\frac{\alpha\Gamma_{1}^{2}}{\Omega^{2}}e^{-\gamma t}(1+\frac{\Gamma_{3}^{2}}{\Gamma_{1}^{2}}e^{-\frac{\gamma t}{2}}\textrm{cos}\Lambda t +\frac{\gamma}{\Lambda}\frac{\Gamma_{3}^{2}}{\Gamma_{1}^{2}}e^{-\frac{\gamma t}{2}}\textrm{sin}\Lambda t),\nonumber \\ c_{3}(t)&=&\frac{\alpha\Gamma_{1}\Gamma_{3}}{\Omega^{2}}e^{-\gamma t}(-1+e^{-\frac{\gamma t}{2}}\textrm{cos}\Lambda t +\frac{\gamma}{\Lambda}e^{-\frac{\gamma t}{2}}\textrm{sin}\Lambda t), \end{eqnarray} where $\Lambda=\sqrt{\Gamma_{1}^{2}+\Gamma_{3}^{2}-\frac{\gamma^{2}}{4}}$.
Taking $\Gamma_{1}=\Gamma_{3}=\Gamma_{0}$ and shutting down the laser fields applied to atom 1 and atom 3 synchronously at specific time $t_{p_{1}}=\frac{(2k-1)\pi}{\Lambda}, k=1,2,3, ...$ , one can obtain the system state as
\begin{eqnarray}\Psi(t_{p_{1}})&=& \alpha A_{1}e^{-\gamma t_{p_{1}}}|e\rangle_{1}|g\rangle_{2}|g\rangle_{3}\nonumber\\
&-&\alpha B_{1}e^{-\gamma t_{p_{1}}}|g\rangle_{1}|g\rangle_{2}|e\rangle_{3}+\eta\beta
|g\rangle_{1}|g\rangle_{2}|g\rangle_{3}, \end{eqnarray} where $\eta$ is an additional normalized factor, $A_{1}=\frac{(1-e^{-\frac{\gamma t_{p_{1}}}{2}})}{2}, B_{1}=\frac{(1+e^{-\frac{\gamma t_{p_{1}}}{2}})}{2}$. Now, we let the above state be new inputting initial state without delay and take $\Gamma_{2}=\Gamma_{3}=\Gamma_{0}$, and shutting down the laser fields applied to atom 2 and atom 3 synchronously at specific time $t_{p_{2}}=\frac{(2k-1)\pi}{\Lambda}, k=1,2,3, ...$. It can be proved that, under this condition, there is no transition between the first term in Eqn. (10) and other three-atom excited states such as
$|e\rangle_{1}|e\rangle_{2}|g\rangle_{3}$,
$|e\rangle_{1}|g\rangle_{2}|e\rangle_{3}$. After some complicated calculation, the system state can be obtained analytically as \begin{eqnarray}
|\Psi(t_{p})&=&\alpha e^{-\gamma(t_{p_{1}}+t_{p_{2}})}B_{1}B_{2}|g\rangle_{1}|e\rangle_{2}|g\rangle_{3}\\
\nonumber &-&\alpha e^{-\gamma(t_{p_{1}}+t_{p_{2}})}B_{1}A_{2}|g\rangle_{1}|g\rangle_{2}|e\rangle_{3}\\
\nonumber &-&\alpha e^{-\gamma(t_{p_{1}}+3t_{p_{2}})}A_{1}|e\rangle_{1}|g\rangle_{2}|g\rangle_{3}+\lambda
\beta |g\rangle_{1}|g\rangle_{2}|g\rangle_{3}, \end{eqnarray} where $A_{2}=\frac{(1-e^{-\frac{\gamma t_{p_{2}}}{2}})}{2}, B_{2}=\frac{(1+e^{-\frac{\gamma t_{p_{2}}}{2}})}{2}$, $\lambda$ is an additional normalized factor. The time cost of the scheme is now estimated as $t_{p}=t_{p_{1}}+t_{p_{2}}\approx \frac{2\pi}{\Lambda}$.
In Fig. 2 (b), we plot the fidelity of QST with respect to operating time $t_{p_{1}}$ and $t_{p_{2}}$ for atomic spontaneous emission rate $\gamma=0.1\Gamma_{0}$. Obviously, the atomic spontaneous emission reduces the overall quantity of fidelity and smoothing the oscillation of the fidelity. In Fig. 3 (a), we plot fidelity of QST between atoms 1 and 2 with respect to atomic spontaneous emission. Obviously, the spontaneous emission monotonically decreases the maximum quantity of fidelity of QST, which corresponds to the strict operating time condition
$t_{p_{1}}=t_{p_{2}}=\frac{\pi}{\Lambda}$. While, in practical case, operating time error emerges inevitably. The existence of operating time error can lead to a more complicated system state includes $|e\rangle_{1}|e\rangle_{2}|g\rangle_{3}$,
$|e\rangle_{1}|g\rangle_{2}|e\rangle_{3}$,
$|g\rangle_{1}|e\rangle_{2}|e\rangle_{3}$,
$|e\rangle_{1}|g\rangle_{2}|g\rangle_{3}$,
$|g\rangle_{1}|g\rangle_{2}|e\rangle_{3}$,
$|g\rangle_{1}|e\rangle_{2}|g\rangle_{3}$. Only the last term contributes to the fidelity of QST. In Fig. 3 (b), we plot fidelity of QST between atoms 1 and 2 with respect to operating time error $(t-t_{p})\Lambda/ \pi$ for different atomic spontaneous emission rates. It is interesting that operating time error, for larger spontaneous emission, on one hand decreases the maximum quantity of fidelity, on the other hand reduces the time cost for achieving maximum quantity of fidelity and, in other words, speeds up the implementation of QST, which is the cooperative influence of spontaneous emission and operating time error. Further more, the sensitivity of fidelity to operating time error is decreased for larger spontaneous emission.
In summary, we have discussed an arbitrary QST scheme in a system contains three distant atoms by simply replicating the operation of synchronously turning on/off the locally applied laser fields for individual atoms. The auxiliary atom is used to avoid additional single qubit phase shift operation and the resulting QST is deterministic and in 100\% fidelity. We discuss the affect of atomic spontaneous emission on QST. It is shown that the atomic spontaneous emission decreases the quantity of fidelity, while the cooperative influence of spontaneous emission and operating time error reduces the time cost $\frac{2\pi}{\Lambda}$ for maximum fidelity and thus speeds up the implementation of QST. It has been demonstrated that the dissipation of the photon leakage along optical fibers can be included in the spin-spin coupling coefficients by replacing the phase factor $e^{i\phi}$ in Eq. (3) with $e^{i\phi-\nu L}$ $^{[2]}$ , where $\nu$ is the fiber decay per meter and $L$ is the length of the fiber between atoms $i$ and $j$. For typical fibers $^{[26]}$, the decay per meter is $\nu\thickapprox 0.08$. The spin-spin coupling coefficient $J_{0}^{'}$ is now about $90\%$ of $J_{0}$. The rotating wave approximation in deriving secular part of effective Hamiltonian is still kept valid under the condition $\Gamma_{i} \ll J_{0}^{'}\thickapprox 0.9 J_{0}$. So the QST gate still works with high fidelity. Furthermore, we have assumed $\kappa \gg g$ in the calculation of deriving effective Ising model, which ensures the scheme is insensitive to the slight variation of strong leakage rate. As is concluded, the scheme works in a robust way since both the affected aspects of fiber lossy and cavity dissipation can be neglected. It should also be noticed that to avoid the inevitable time-delay affect caused by mismatch of practical and theoretical controlling times $^{[27]}$, remedial methods such as Lyapunov control can be used in the extended scheme. Many of the present schemes only contain two atoms trapped in separated cavities. From a realistic point of view, a robust quantum network must contains many distant quantum nodes. QST must be implemented between any two quantum nodes in high fidelity. The model used and the results obtained in this scheme may act as a possible candidate.
We thank Professor Dian Min Tong and Professor Dong Mi for helpful discussions and their encouragement. This work is supported by NSF of China under Grant No. 11305021 and the Fundamental Research Funds for the Central Universities.
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\begin{document}
\title[Approximation of norms on Banach spaces]{Approximation of norms on Banach spaces} \begin{abstract} Relatively recently it was proved that if $\Gamma$ is an arbitrary set, then any equivalent norm on $c_0(\Gamma)$ can be approximated uniformly on bounded sets by polyhedral norms and $C^\infty$ smooth norms, with arbitrary precision. We extend this result to more classes of spaces having uncountable symmetric bases, such as preduals of the `discrete' Lorentz spaces $d(w,1,\Gamma)$, and certain symmetric Nakano spaces and Orlicz spaces. We also show that, given an arbitrary ordinal number $\alpha$, there exists a scattered compact space $K$ having Cantor-Bendixson height at least $\alpha$, such that every equivalent norm on $C(K)$ can be approximated as above. \end{abstract}
\author{Richard J.~Smith} \address{School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland} \email{[email protected]} \urladdr{http://mathsci.ucd.ie/~rsmith}
\author{Stanimir Troyanski} \address{Institute of Mathematics and Informatics, Bulgarian Academy of Science, bl.8, acad. G. Bonchev str. 1113 Sofia, Bulgaria, and Departamento de Matem\'aticas, Universidad de Murcia, Campus de Espinardo. 30100 Murcia, Spain} \email{[email protected]}
\thanks{The first author thanks the Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, for its hospitality, when he visited in April 2017. Both authors were supported financially by Science Foundation Ireland under Grant Number `SFI 11/RFP.1/MTH/3112'. The second author was partially supported by MTM2014-54182-P (MINECO/FEDER), MTM2017-86182-P (AEI/FEDER, UE) and Bulgarian National Scientific Fund under Grant DFNI/Russia,01/9/23.06.2017. Both authors are grateful to D.~Leung and G.~Lancien for very helpful remarks.}
\subjclass[2010]{46B03, 46B20, 46B26} \keywords{Polyhedrality, smoothness, approximation, renorming} \date{\today} \maketitle
\section{Introduction}
Let $(X,\ensuremath{\left\|\cdot\right\|})$ be a Banach space and let \textbf{P} denote some geometric property of norms, such as strict convexity, polyhedrality or $C^k$-smoothness. We shall say that $\ensuremath{\left\|\cdot\right\|}$ can be \emph{approximated} by norms having \textbf{P} if, given $\ensuremath{\varepsilon}>0$, there exists a norm $\ensuremath{{\displaystyle |\kern-.9pt|\kern-.9pt|}\cdot{\displaystyle |\kern-.9pt|\kern-.9pt|}}$ on $X$ having \textbf{P}, such that \[ \n{x} \;\leqslant\; \tn{x} \;\leqslant\; (1+\ensuremath{\varepsilon})\n{x}, \] for all $x \in X$.
The question of whether all equivalent norms on a given Banach space can be approximated by norms having \textbf{P} has been the subject of a number of papers. A norm on $X$ is called \emph{polyhedral} if, given a finite-dimensional subspace $E \subseteq X$, the restriction of the unit ball to $E$ is a polytope, that is, it has only finitely many extreme points. Given $k \in \mathbb{N}$, a norm is called \emph{$C^k$-smooth} if its $k$th Fr\'echet derivative exists and is continuous at every non-zero point. The norm said to be \emph{$C^{\infty}$-smooth} if this holds for all $k \in \mathbb{N}$. It has been shown that if $X$ is separable and admits a single equivalent polyhedral or $C^k$-smooth norm, where $k \in \mathbb{N}\cup\{\infty\}$, then every equivalent norm on $X$ can be approximated by polyhedral norms or $C^k$-smooth norms, respectively \cite{dfh:98,ht:14}. Other approximation results can be found in \cite{dfh:96,fhz:97,fzz:81,hp:14,pwz:81}.
In the context of approximation by polyhedral and $C^k$-smooth norms, very little was known about the non-separable case until relatively recently. The natural norm of the Banach space $c_0(\Gamma)$, where $\Gamma$ is an arbitrary set, is easily seen to be polyhedral. Moreover, the fact that it admits an equivalent $C^\infty$-smooth norm, at least in the separable case, has been known for some time:~an example of Kuiper was given in \cite{bf:66}. Some results on the approximation of norms on $c_0(\Gamma)$ by $C^\infty$-smooth norms in restricted cases can be found in \cite{fhz:97,pwz:81}. The next result answers the question in the case of $c_0(\Gamma)$ in full generality.
\begin{thm}[{\cite[Theorem 1.7]{bs:16}}]\label{c0_approx} Let $\Gamma$ be an arbitrary set. Then every equivalent norm on $c_0(\Gamma)$ can be approximated by both polyhedral norms and $C^\infty$-smooth norms. \end{thm}
The proof of this theorem makes use of a number of geometric and topological techniques. It is worth pointing out that this result extends to all subspaces of $c_0(\Gamma)$, because any equivalent norm on such a subspace $X$ can be extended to an equivalent norm on the whole space \cite[Lemma II.8.1]{dgz:93}, and by inspection of the definitions it is clear that the restriction to $X$ of any approximation of the extended norm will inherit the desired properties.
The purpose of this paper is to find new examples of spaces with the property that every equivalent norm on the space can be approximated by both polyhedral norms and $C^\infty$-smooth norms (given the above remark, such examples cannot be isomorphic to subspaces of $c_0(\Gamma)$). In Section \ref{sect_approx_frame} we present the general framework and preparatory lemmas that will be used in Section \ref{sect_main_tools}, in some new approximation theorems are presented. The subsequent sections are devoted to applying these theorems to find new examples of spaces whose norms can be approximated in the manner described above.
\section{A framework for approximation}\label{sect_approx_frame}
In this section we build the technical tools required in Section \ref{sect_main_tools}. Let $\Gamma$ be an infinite set and let $X$ be a Banach space supporting a system of non-zero projections $(P_\gamma)_{\gamma \in \Gamma}$, satisfying four properties: \begin{enumerate} \item $P_\alpha P_\beta = 0$ whenever $\alpha \neq \beta$, \item $\sup (\n{P_\gamma})_{\gamma \in \Gamma} < \infty$, \item $X = \cl{\lspan}^{\ensuremath{\left\|\cdot\right\|}}(P_\gamma X)_{\gamma \in \Gamma}$, and \item $X^* = \cl{\lspan}^{\ensuremath{\left\|\cdot\right\|}}(P^*_\gamma X^*)_{\gamma \in \Gamma}$. \end{enumerate} This concept generalizes the well-known idea of a (shrinking, bounded) \emph{Markushevich basis} (M-basis). If the space $X$ admits a shrinking bounded Markushevich basis (M-basis) $(e_\gamma,e^*_\gamma)_{\gamma \in \Gamma}$, then defining $P_\gamma x = e^*_\gamma(x)e_\gamma$ yields such a system, where $\dim P_\gamma X = 1$ for all $\gamma \in \Gamma$. Conversely, if each $P_\gamma$ has 1-dimensional range then we obtain such a basis. In view of (4), it makes sense to call any such system of projections shrinking.
To simplify notation, given $x \in X$ and $f \in X^*$, we will denote $P_\gamma x$ and $P^*_\gamma f$ by $x_\gamma$ and $f_\gamma$, respectively. We define the \emph{support} and the \emph{range} of $f \in X^*$ (with respect to the system of projections) to be the sets \[ \supp(f) \;=\; \set{\gamma \in \Gamma}{f_\gamma \neq 0} \qquad\text{and}\qquad \ran(f) \;=\; \set{\n{f_\gamma}}{\gamma \in \Gamma}, \] respectively. Likewise, we define the support of $x \in X$ to be \[ \supp(x) \;=\; \set{\gamma \in \Gamma}{x_\gamma \neq 0}. \]
\begin{lem}\label{lem_ran(f)_structure} Given $f \in X^*$ and $\ensuremath{\varepsilon}>0$, the set $\set{\gamma \in \Gamma}{\n{f_\gamma}\geqslant \ensuremath{\varepsilon}}$ is finite. \end{lem}
\begin{proof} According to properties (1), (2) and (4) above, given $f \in X^*$ and $\ensuremath{\varepsilon}>0$, there exists a finite set $F \subseteq \Gamma$ and $g \in \lspan(P_\gamma^* X^*)_{\gamma \in F} $, such that $\n{f-g} < \ensuremath{\varepsilon}/D$, where $D:=\sup (\n{f_\gamma})_{\gamma \in \Gamma}$. It follows that $\n{f_\alpha} < \ensuremath{\varepsilon}$ whenever $\alpha \in \Gamma\setminus F$. \end{proof}
We proceed to define a series of numerical quantities, sets and `approximating functionals' associated with subsets of $\Gamma$ and elements of $X^*$. First, given finite $F \subseteq \Gamma$, define \begin{equation}\label{defn_rho} \rho(F) \;=\; \sup\set{\n{\sum_{\gamma \in F}f_\gamma}}{f \in X^*\text{ and }\n{f_\gamma}\leqslant 1 \text{ whenever }\gamma \in F}. \end{equation} Evidently, $\rho$ is increasing, in the sense that $\rho(F)\leqslant \rho(G)$ whenever $F \subseteq G \subseteq \Gamma$ and $G$ is finite.
\begin{lem}\label{lem_lim_inf} Let $(F^\alpha)$ be a net of finite subsets of $\Gamma$, such that \[ A \;:=\; \lim \inf F_\alpha \;=\; \bigcup_\alpha \bigcap_{\beta \geqslant \alpha} F_\beta, \] is infinite. Then $\rho(F^\alpha) \to \infty$. \end{lem}
\begin{proof} Suppose that $\rho(F^\alpha) \not\to \infty$. By taking a subnet if necessary, we can assume that there exists some $L>0$ such that $\rho(F^\alpha) \leqslant L$ for all $\alpha$ (note that $A$ remains infinite). Since the $P_\gamma$ are non-zero, for each $\gamma \in \bigcup_\alpha F^\alpha$, we can select $g_\gamma \in P_\gamma^* X^*$ such that $\n{g_\gamma}=1$. Given $\alpha$, define $f^\alpha = \sum_{\gamma \in F^\alpha} g_\gamma$. By property (1) of our system of projections, we see that $f^\alpha_\gamma = P_\gamma^* f^\alpha= g_\gamma$ if $\gamma \in F^\alpha$, and $f^\alpha_\gamma=0$ otherwise. We have $\n{f^\alpha} \leqslant \rho(F^\alpha) \leqslant L$ for all $n$, and thus the net $(f^\alpha)$ admits a $w^*$-accumulation point $f$. Given the definition of $A$ and the $w^*$-$w^*$-continuity of each $P_\gamma^*$, we conclude that $f_\gamma=g_\gamma$ for all $\gamma \in A$. However, the fact that $\n{f_\gamma}=1$ for infinitely many $\gamma \in \Gamma$ violates Lemma \ref{lem_ran(f)_structure}. \end{proof}
Next, we define a series of numerical quantities, subsets and approximating functionals associated with elements of $X^*$. Let $f \in X^*$ and $k \in \mathbb{N}$. It follows from Lemma \ref{lem_ran(f)_structure} that if $\ran(f)$ is infinite then it has a single accumulation point at $0$. Therefore, if $|\ran(f)| \geqslant k$, it is legitimate to define $p_k(f)$ to be the $k$th largest value of $\ran(f)$. If $|\ran(f)| < k$, set $p_k(f)=0$. It is clear that $p_k(f) \to 0$ as $k \to \infty$.
We continue by defining \[ q_k(f) \;=\; p_k(f)-p_{k+1}(f), \qquad G_k(f) \;=\; \set{\gamma \in \supp(f)}{\n{f_\gamma} \geqslant p_k(f)}, \] and \[ H_k(f) \;=\; \set{\gamma \in \supp(f)}{\n{f_\gamma} = p_k(f)} \;=\; G_k(f)\setminus G_{k-1}(f), \] where we set $G_0(f)=\varnothing$ for convenience. Evidently, $q_k(f)>0$ if and only if $p_k(f)>0$. Also, $G_k(f)$ is always finite. If $p_k(f)>0$ then $G_k(f)$ is finite by Lemma \ref{lem_ran(f)_structure}. If $p_k(f)=0$ then $G_k(f)=\supp(f)$, which must be finite in this case. It is obvious that the $G_k(f)$ form an increasing sequence of subsets of $\Gamma$.
Now define a function $\map{\theta}{X^*}{[0,\infty]}$ by \[ \theta(f) \;=\; \sum_{k=1}^\infty q_k(f)\rho(G_k(f)). \]
Given $t \neq 0$, it is clear that $q_k(t f)=|t|q_k(f)$ and $G_k(t f)=G_k(f)$, and thus $\theta$ is absolutely homogeneous.
\begin{lem}\label{lem_w^*-lsc} The function $\theta$ is $w^*$-lower semicontinuous. \end{lem}
\begin{proof} Let $f \in X^*$, $\lambda \geqslant 0$ and assume $\theta(f) > \lambda$. Fix minimal $n\in\mathbb{N}$ such that \[ \alpha:=\sum_{k=1}^n q_k(f)\rho(G_k(f)) > \lambda. \] By the minimality of $n$, $p_n(f) \geqslant q_n(f)>0$. Set \[ \ensuremath{\varepsilon} \;=\; \min\left\{\frac{\alpha-\lambda}{\rho(G_n(f))},\, p_n(f)\right\}>0. \] By the $w^*$-$w^*$-continuity of each $P_\gamma^*$ and the natural lower semicontinuity of dual norms, the functions $g \mapsto \n{g_\gamma}$, $\gamma \in \Gamma$, are $w^*$-lower semicontinuous. Therefore, as $G_n(f)$ is finite, the set \[ U:=\set{g \in X^*}{\n{g_\gamma} \;>\; \n{f_\gamma} - \ensuremath{\varepsilon} \text{ whenever }\gamma \in G_n(f)}, \] is $w^*$-open (and clearly contains $f$).
Let $g \in U$. Given $1 \leqslant k \leqslant n$, let $\beta_k = \min\set{\n{g_\gamma}}{\gamma \in G_k(f)}$. From the definition of $\ensuremath{\varepsilon}$ and $U$, we know that \[ \beta_k \;>\; p_k(f) - \ensuremath{\varepsilon} \;\geqslant\; 0, \] for all such $k$. Clearly the $\beta_k$ are non-increasing. Now find $j_k \in \mathbb{N}$, $1 \leqslant k \leqslant n$, such that $p_{j_k}(g) = \beta_k$. Observe that $G_k(f)\subseteq G_{j_k}(g)$ whenever $1 \leqslant k \leqslant n$. Since the $\beta_k$ are non-increasing, the integers $j_k$ must be non-decreasing. Because $\rho$ is an increasing function, as observed above, it follows that \begin{align*} \theta(g) &\;>\; \sum_{k=1}^{n-1}\sum_{i=j_k}^{j_{k+1}-1} q_i(g)\rho(G_i(g)) + \sum_{i=j_n}^\infty q_i(g)\rho(G_i(g)) \\ &\geqslant\; \sum_{k=1}^{n-1} \rho(G_{j_k}(g))\sum_{i=j_k}^{j_{k+1}-1}(p_i(g)-p_{i+1}(g)) + \rho(G_{j_n}(g))\sum_{i=j_n}^\infty (p_i(g)-p_{i+1}(g))\\ &=\; \sum_{k=1}^{n-1} \rho(G_{j_k}(g))(\beta_k-\beta_{k+1}) + \rho(G_{j_n}(g))\beta_n\\ &\geqslant\; \sum_{k=1}^{n-1} \rho(G_k(f))(\beta_k-\beta_{k+1}) + \rho(G_n(f))\beta_n\\ &=\; \beta_1 \rho(G_1(f)) + \sum_{k=2}^n \beta_k(\rho(G_k(f))-\rho(G_{k-1}(f)))\\ &>\; (p_1(f)-\ensuremath{\varepsilon})\rho(G_1(f)) + \sum_{k=2}^n (p_k(f)-\ensuremath{\varepsilon})(\rho(G_k(f))-\rho(G_{k-1}(f)))\\ &=\; \sum_{k=1}^{n-1} (p_k(f)-p_{k+1}(f))\rho(G_k(f)) + p_n(f)\rho(G_n(f)) - \ensuremath{\varepsilon}\rho(G_n(f))\\ &\geqslant\; \sum_{k=1}^n q_k(f)\rho(G_k(f)) + \lambda - \alpha \;=\; \lambda. \qedhere \end{align*} \end{proof}
Given $f \in X^*$ and $m,n \in \mathbb{N}$, $m<n$, we define associated functionals $h_m(f)$, $g_{m,n}(f)$ and $j_{m,n}(f)$. The $j_{m,n}(f)$ will be the functionals that we use to approximate $f$ in Section \ref{sect_main_tools}. First, given $k \in \mathbb{N}$, define the auxiliary functionals \[ \omega_k(f) \;=\; \sum_{\gamma \in G_k(f)} \frac{f_\gamma}{\n{f_\gamma}}. \] By the definition of $\rho$, $\n{\omega_k(f)}\leqslant\rho(G_k(f))$ for all $k \in \mathbb{N}$. Now define \[ h_m(f) \;=\; \sum_{k=1}^m q_k(f)\omega_k(f). \] The following straightforward observation will be used in the next lemma: \begin{equation}\label{eqn_h_less_than_theta} \n{h_m(f)} \;\leqslant\; \sum_{k=1}^m q_k(f)\n{\omega_k(f)} \;\leqslant\; \sum_{k=1}^m q_k(f)\rho(G_k(f)) \;\leqslant\; \theta(f), \end{equation} for all $m$.
\begin{lem}\label{lem_theta_dominate} We have $\n{f} \leqslant \theta(f)$ for all $f \in X^*$, and $\n{f - h_m(f)} \to 0$ whenever $\theta(f) < \infty$. \end{lem}
\begin{proof} Assume that $f \in X^*$ is non-zero and that $\theta(f)<\infty$. Given $\gamma \in G_m(f)$, let $k_\gamma$ denote the unique index $k \leqslant m$ for which $\gamma \in H_k(f)$. Assuming $p_m(f)>0$, we have \begin{align}\label{eqn_h} h_m(f) &\;=\; \sum_{k=1}^m q_k(f)\left(\sum_{\gamma \in G_k(f)} \frac{f_\gamma}{\n{f_\gamma}} \right)\nonumber \\ &\;=\; \sum_{\gamma \in G_m(f)} \frac{f_\gamma}{\n{f_\gamma}} \left(\sum_{k = k_\gamma}^m q_k(f) \right)\nonumber \\ &\;=\; \sum_{\gamma \in G_m(f)} \frac{f_\gamma}{\n{f_\gamma}}(p_{k_\gamma}(f)-p_{m+1}(f)) \;=\; \left(\sum_{\gamma \in G_m(f)} f_\gamma\right) - p_{m+1}(f)\omega_m(f). \end{align}
If $\ran(f)$ is finite, then the definition of $h_m(f)$, together with (\ref{eqn_h}), shows that $h_m(f)=f$ whenever $m \geqslant |\ran(f)|-1$. Coupling this with (\ref{eqn_h_less_than_theta}) yields the conclusion.
For the remainder of the proof, we assume that $\ran(f)$ is infinite and thus that (\ref{eqn_h}) applies for all $m \in \mathbb{N}$. According to properties (1) and (3) of our system of projections above, given $\ensuremath{\varepsilon}>0$, there exists a finite set $F\subseteq \Gamma$ and $x \in X$ such that $\n{x}=1$, $x \;=\; \sum_{\gamma \in F} x_\gamma$ and $\n{f} \leqslant f(x) + \ensuremath{\varepsilon}$. Choose $M\in \mathbb{N}$ large enough so that $F \cap G_m(f) = F \cap G_M(f)$ whenever $m \geqslant M$. Observe that if $\gamma \in F\setminus G_M(f)$, then $f_\gamma=0$. If not, then as $p_m(f)\to 0$, there must exist $m\geqslant M$ such that $\gamma \in G_m(f)$, giving $\gamma \in F \cap G_m(f)=F \cap G_M(f)$, which is false. Hence, using the above facts, given $m \geqslant M$, we have \begin{align*} \n{f}-\ensuremath{\varepsilon} \;\leqslant\; f(x) &\;=\; \sum_{\gamma \in F} f_\gamma(x)\\ &\;=\; \sum_{\gamma \in F \cap G_m(f)} f_\gamma(x) + \sum_{F\setminus G_m(f)} f_\gamma(x) + \sum_{G_m(f)\setminus F} f_\gamma(x) \\ &\;=\; \sum_{\gamma \in G_m(f)} f_\gamma(x) \\ &\;=\; h_m(f)(x) + p_{m+1}(f)\omega_n(f)(x)\\ &\;\leqslant\; \theta(f) + p_{m+1}(f)\omega_M(f)(x) \;\to\; \theta(f), \end{align*} using equation (\ref{eqn_h_less_than_theta}) above, the fact that $\omega_m(f)(x)=\omega_M(f)(x)$ (by the choice of $M$), and as $p_m(f) \to 0$ as $m\to\infty$. Since this holds for all $\ensuremath{\varepsilon} > 0$, it follows that $\n{f} \leqslant \theta(f)$.
Now assume that $\theta(f) < \infty$. Fix $m\in\mathbb{N}$ and set $g=f-h_m(f)$. We observe that $p_k(g)=p_{m+k}(f)$ and $G_k(g)=G_{m+k}(f)$ for all $k \in \mathbb{N}$. Therefore, using the above applied to $g$ yields \begin{equation}\label{eqn_f_h_m(f)} \n{f-h_m(f)} \;\leqslant\; \theta(g) \;=\; \sum_{k=1}^\infty q_k(g)\rho(G_k(g)) \;=\; \sum_{k=m+1}^\infty q_k(f)\rho(G_k(f)). \end{equation} Because $\theta(f)<\infty$, the quantity on the right hand side tends to $0$ as $m\to\infty$. \end{proof}
We continue by defining, for $m,n \in \mathbb{N}$, $m<n$, \[ g_{m,n}(f) \;=\; \begin{cases} {\displaystyle \frac{\theta(f-h_m(f))}{\rho(G_n(f))}}\omega_n(f) & \text{if $p_n(f) > 0$}\\ 0 & \text{otherwise,}\end{cases} \] and the approximating functionals \[ j_{m,n}(f) \;=\; h_m(f) + g_{m,n}(f). \] One of the principal reasons for considering the $j_{m,n}(f)$ is brought to light in the next lemma.
\begin{lem}\label{lem_convex_combination} Let $f \in X^*$, $m \in \mathbb{N}$, and suppose that $\theta(f)<\infty$. Then there exist constants $\lambda_n \geqslant 0$, $n>m$, such that \[ \sum_{n=m+1}^\infty \lambda_n \;=\; 1 \qquad\text{and}\qquad f \;=\; \sum_{n=m+1}^\infty \lambda_n j_{m,n}(f). \] \end{lem}
\begin{proof} Fix $m \in \mathbb{N}$ and assume $\theta(f)<\infty$. We know, from Lemma \ref{lem_theta_dominate} and (\ref{eqn_f_h_m(f)}), that \begin{equation} f \;=\; \sum_{n=1}^\infty q_n(f)\omega_n(f) \qquad\text{and}\qquad \theta(f-h_m(f)) \;=\; \sum_{n=m+1}^\infty q_n(f)\rho(G_n(f)). \label{eqn_infinite_sums} \end{equation} There are two cases. If $f=h_m(f)$, then $p_n(f)=0$ whenever $n>m$, which implies that $j_{m,n}(f)=h_m(f)=f$ for all such $n$. In this case, it is clearly sufficient to let $\lambda_{m+1}=1$ and $\lambda_n=0$ for $n>m+1$. Instead, if $f\neq h_m(f)$ then $\theta(f-h_m(f)) > 0$. This time, given $n>m$, set \[ \lambda_n \;=\; \frac{q_n(f)\rho(G_n(f))}{\theta(f-h_m(f))}. \] By (\ref{eqn_infinite_sums}) and Lemma \ref{lem_theta_dominate}, we have $\sum_{n=m+1}^\infty \lambda_n = 1$ and \begin{align*} \sum_{n=m+1}^\infty \lambda_n j_{m,n}(f) \;&=\; \sum_{n=m+1}^\infty \lambda_n h_m(f) + q_n(f)\omega_n(f)\\ &=\; h_m(f) + \sum_{n=m+1}^\infty q_n(f)\omega_n(f) \;=\; f. \tag*{\qedhere} \end{align*} \end{proof}
In our final lemma, we consider what happens when we have a net $j_{m^\alpha,n^\alpha}(f^\alpha)$ of the approximating functionals converging to an infinitely supported element.
\begin{lem}\label{lem_net_1} Let $d \in X^*$ have infinite support, and consider a net $j^\alpha := j_{m^\alpha,n^\alpha}(f^\alpha)$, such that $j^\alpha \stackrel{w^*}{\to} d$. Suppose moreover that $\sup_\alpha \theta(f^\alpha) <\infty$. Then $f^\alpha \stackrel{w^*}{\to} d$. \end{lem}
\begin{proof} Let $L > 0$ such that $\theta(f^\alpha) \leqslant L$ for all $\alpha$. The proof will be comprised of a number of steps. In the first step, we show that $\rho(G_{n^\alpha}(f^\alpha)) \to \infty$. Let $\gamma \in \supp(d)$. As $P^*_\gamma$ is $w^*$-$w^*$-continuous, $j^\alpha_\gamma \stackrel{w^*}{\to} d_\gamma \neq 0$. In particular, there exists $\alpha$ such that $j^\beta_\gamma \neq 0$ for all $\beta \geqslant \alpha$. It follows that \[ \supp(d) \;\subseteq\; \bigcup_\alpha \bigcap_{\beta \geqslant \alpha} \supp(j^\beta) \;\subseteq\; \bigcup_\alpha \bigcap_{\beta \geqslant \alpha} G_{n^\beta}(f^\beta). \] Consequently, $\rho(G_{n^\alpha}(f^\alpha)) \to \infty$ by Lemma \ref{lem_lim_inf}.
In the second step, we use the first step to prove that $g_{m^\alpha,n^\alpha}(f^\alpha) \stackrel{w^*}{\to} 0$. Fix $\gamma \in \Gamma$. Since $\n{\omega_n(f)_\gamma} \leqslant 1$, we observe that if $p_{n^\alpha}(f^\alpha)>0$ then \[ \n{g_{m^\alpha,n^\alpha}(f^\alpha)_\gamma} \;\leqslant\; \frac{\theta(f^\alpha-h_{m^\alpha}(f^\alpha))}{\rho(G_{n^\alpha}(f^\alpha))} \;\leqslant\; \frac{\theta(f^\alpha)}{\rho(G_{n^\alpha}(f^\alpha))} \;\leqslant\; \frac{L}{\rho(G_{n^\alpha}(f^\alpha))}, \] and $\n{g_{m^\alpha,n^\alpha}(f^\alpha)_\gamma}=0$ otherwise. In any case, it follows that \begin{equation}\label{eqn_g_1} \n{g_{m^\alpha,n^\alpha}(f^\alpha)_\gamma} \to 0. \end{equation}
Moreover, by the definition of $\rho$, \begin{equation}\label{eqn_g_2} \n{g_{m^\alpha,n^\alpha}(f^\alpha)} \;=\; \frac{\theta(f^\alpha-h_{m^\alpha}(f^\alpha))\n{\omega_{n^\alpha}(f^\alpha)}}{\rho(G_{n^\alpha}(f^\alpha))} \;\leqslant\; \theta(f^\alpha-h_{m^\alpha}(f^\alpha)) \;\leqslant\; L, \end{equation} if $p_{n^\alpha}(f^\alpha)>0$, and if not then of course this inequality still holds. Hence, by considering (\ref{eqn_g_1}), (\ref{eqn_g_2}) and property (3) of our system of projections, we deduce that $g_{m^\alpha,n^\alpha}(f^\alpha) \stackrel{w^*}{\to} 0$.
Of course, this implies that $h_{m^\alpha}(f^\alpha) \stackrel{w^*}{\to} d$. We complete the proof by showing, in this final step, that \begin{equation}\label{eqn_h_limit} f^\alpha - h_{m^\alpha}(f^\alpha) \;\stackrel{w^*}{\to}\; 0. \end{equation} We repeat the first step, substituting $G_{m^\alpha}(f^\alpha)$ for $G_{n^\alpha}(f^\alpha)$ and noting that $\supp (h_{m^\alpha}(f^\alpha))$ $\subseteq G_{m^\alpha}(f^\alpha)$, to obtain $\rho(G_{m^\alpha}(f^\alpha)) \to \infty$. Since $\rho$ is increasing, given arbitrary $f\in X^*$ and $m\in\mathbb{N}$ we have \begin{equation}\label{eqn_f_1} p_m(f)\rho(G_m(f)) \;=\; \sum_{n=m}^\infty q_n(f)\rho(G_m(f)) \;\leqslant\; \sum_{n=m}^\infty q_n(f)\rho(G_n(f)) \;\leqslant\; \theta(f). \end{equation} If we fix $\gamma \in \Gamma$, then by (\ref{eqn_h}) and (\ref{eqn_f_1}), whenever $p_{m^\alpha}(f^\alpha)>0$ we have \[ \n{f^\alpha_\gamma - h_{m^\alpha}(f^\alpha)_\gamma} \;=\; \n{p_{m^\alpha+1}(f^\alpha)\omega_{m^\alpha}(f^\alpha)_\gamma} \;\leqslant\; p_{m^\alpha+1}(f^\alpha) \;\leqslant\; \frac{L}{\rho(G_{m^\alpha}(f^\alpha))}, \] and $f^\alpha = h_{m^\alpha}(f^\alpha)$ otherwise. Therefore \begin{equation}\label{eqn_f_2} \n{f^\alpha_\gamma - h_{m^\alpha}(f^\alpha)_\gamma} \;\to\; 0, \end{equation} for all $\gamma \in \Gamma$. Moreover, \begin{equation}\label{eqn_f_3} \n{f^\alpha - h_{m^\alpha}(f^\alpha)} \;\leqslant\; \theta(f^\alpha-h_{m^\alpha}(f^\alpha)) \;\leqslant\; \theta(f^\alpha) \;\leqslant\; L. \end{equation} Therefore, by considering (\ref{eqn_f_2}), (\ref{eqn_f_3}) and again property (3) of our projections, we obtain (\ref{eqn_h_limit}) as required. \end{proof}
\section{Approximation of norms}\label{sect_main_tools}
We begin this section with a few more preliminaries. Given a Banach space $(X,\ensuremath{\left\|\cdot\right\|})$ and a bounded set $C \subseteq X^*$, recall that $B \subseteq C$ is called a \emph{James boundary} of $C$ if, given $x \in X$, there exists $b \in B$ such that \[ b(x) \;=\; \sup\set{f(x)}{f \in C}. \] Hereafter, we will simply refer to a James boundary of a given set as a \emph{boundary}. Given an equivalent norm $\ensuremath{{\displaystyle |\kern-.9pt|\kern-.9pt|}\cdot{\displaystyle |\kern-.9pt|\kern-.9pt|}}$ on $X$, we say that $B$ is a boundary of $(X,\ensuremath{{\displaystyle |\kern-.9pt|\kern-.9pt|}\cdot{\displaystyle |\kern-.9pt|\kern-.9pt|}})$, or simply $\ensuremath{{\displaystyle |\kern-.9pt|\kern-.9pt|}\cdot{\displaystyle |\kern-.9pt|\kern-.9pt|}}$, if it is a boundary of $B_{(X,\ttri\cdot\ttri)^*}$, i.e.~$\tn{b}\leqslant 1$ for all $b \in B$ and, given $x \in X$, there exists $b \in B$ such that $b(x)=\tn{x}$. For example, by the Hahn-Banach and Krein-Milman Theorems, the set of extreme points $\ext(B_{(X,\ttri\cdot\ttri)^*})$ is always a boundary of $\ensuremath{{\displaystyle |\kern-.9pt|\kern-.9pt|}\cdot{\displaystyle |\kern-.9pt|\kern-.9pt|}}$.
As well as boundaries, we recall the notion of norming sets. Let $V$ be a subspace of $X$. A bounded set $E\subseteq X^*$ is called \emph{norming for $V$} if there exists $r>0$ such that \[ \sup\set{f(x)}{f \in E} \;\geqslant\; r\n{x}, \] for all $x \in V$. When $V=X$ we just say that $E$ is norming.
We will require the following notion, which has been used to provide a sufficient condition for the existence of equivalent polyhedral and $C^\infty$-smooth norms.
\begin{defn}[{\cite[Definition 11]{fpst:14}}]\label{defn_lrc} Let $X$ be a Banach space. We say that $E \subseteq X^*$ is \emph{$w^*$-locally relatively norm-compact} ($w^*$-LRC for short) if, given $f \in E$, there exists a $w^*$-open set $U \subseteq X^*$, such that $f \in U$ and $E \cap U$ is relatively norm-compact. We say that $E$ is \emph{$\sigma$-$w^*$-LRC} if it can be expressed as the union of countably many $w^*$-LRC sets. \end{defn}
The following result motivates their introduction. We will call a subset $E \subseteq X^*$ $w^*$-$K_\sigma$ if it is the union of countably many $w^*$-compact sets.
\begin{thm}[{\cite[Theorem 2.1]{b:14}} and {\cite[Theorem 7]{fpst:14}}]\label{thm_tb} Let $(X,\ensuremath{\left\|\cdot\right\|})$ be a Banach space having a boundary that is both $\sigma$-$w^*$-LRC and $w^*$-$K_\sigma$. Then $\ensuremath{\left\|\cdot\right\|}$ can be approximated by both $C^\infty$-smooth norms and polyhedral norms. \end{thm}
If $(e_\gamma,e^*_\gamma)_{\gamma \in \Gamma}$ is a bounded M-basis, then $E:=\set{e^*_\gamma}{\gamma \in \Gamma} \cup \{0\}$ is both $\sigma$-$w^*$-LRC and $w^*$-$K_\sigma$, as is $\lspan(E)$. More information concerning the topological behaviour of $w^*$-LRC sets can be found in \cite{smith:17}. We quote one result from that study here.
\begin{thm}[{\cite[Theorem 2.3]{smith:17}}]\label{thm_linspan} If $E$ is a $\sigma$-$w^*$-LRC and $w^*$-$K_\sigma$ subset of a dual Banach space $X^*$, then so is the subspace $\lspan(E)$. \end{thm}
Now let the Banach space $X$ support a system of projections $(P_\gamma)_{\gamma \in \Gamma}$ as in Section \ref{sect_approx_frame}. Suppose that, for each $\gamma \in \Gamma$, we are given a linear (but not necessarily closed) subspace $A_\gamma \subseteq P^*_\gamma X^*$. Given finite $F \subseteq \Gamma$, we define additional subspaces \begin{align*} V_F \;&=\; \lspan(P_\gamma X)_{\gamma \in F} \;=\; \set{x \in X}{\supp(x) \subseteq F},\\ W_F \;&=\; \lspan(P^*_\gamma X^*)_{\gamma \in F} \;=\; \set{f \in X^*}{\supp(f)\subseteq F}, \text{ and}\\ A_F \;&=\; \lspan(A_\gamma)_{\gamma \in F}. \end{align*} Bearing in mind property (1) of our system of projections $(P_\gamma)_{\gamma \in \Gamma}$, by standard methods we see that $B_{W_F}$ is norming for $V_F$.
\begin{defn}\label{defn_admissible_set} We will call the family $(A_\gamma)_{\gamma \in \Gamma}$ \emph{admissible} if, given $\ensuremath{\varepsilon}>0$, a finite set $F\subseteq \Gamma$, and a $w^*$-compact subset $C \subseteq W_F$ that is norming for $V_F$, there exists a set $D \subseteq W_F$ such that \[ C \;\subseteq\; \cl{D}^{w^*} \;\subseteq\; C + \ensuremath{\varepsilon} B_{W_F} \qquad\text{and}\qquad \cl{D}^{w^*} \cap A_F \text{ is a boundary of }\cl{D}^{w^*}. \] \end{defn}
It is obvious that the family $(P_\gamma^* X^*)_{\gamma \in \Gamma}$ is admissible. We will make use of this fact, together with other examples, later on.
Now we can state and prove our main tool.
\begin{thm}\label{thm_main_tool} Let $(A_\gamma)_{\gamma \in \Gamma}$ be an admissible family as above, and let $B \subseteq C \subseteq X^*$ be sets such that $C$ is $w^*$-compact and norming, and $B$ is a boundary of $C$ with the property that $\theta(f) < \infty$ whenever $f \in B$. Then, given $\ensuremath{\varepsilon}>0$, there exists $D \subseteq X^*$ such that \[ B \;\subseteq\; \cl{D}^{w^*} \;\subseteq\; C + \ensuremath{\varepsilon} B_{X^*} \qquad\text{and}\qquad \cl{D}^{w^*} \cap A \text{ is a boundary of }\cl{D}^{w^*}, \] where $A:=\lspan(A_\gamma)_{\gamma \in \Gamma}$. \end{thm}
\begin{proof} By rescaling $C$ if necessary, we can assume that $C \subseteq B_{X^*}$. Since $C$ is norming and a subset of $B_{X^*}$, there exists $r \in (0,1]$ such that $\sup\set{f(x)}{f \in C} \geqslant r\n{x}$ for all $x \in X$. Let $\ensuremath{\varepsilon} \in (0,1)$. Given $k \in \mathbb{N}$, define \[ C_k \;=\; \set{f \in C}{\theta(f) \leqslant k}. \] By Lemma \ref{lem_w^*-lsc}, this set is $w^*$-compact. Since $\theta(f)$ is finite for all $f \in B$, we know that $B\subseteq \bigcup_{k=1}^\infty C_k$. Define sets of approximating functionals \begin{equation}\label{defn_J_k} J_k \;=\; \set{j_{m,n}(f)}{f \in C_k,\; m,n \in \mathbb{N},\;m<n\text{ and } \theta(f-h_m(f))<2^{-k-2}r\ensuremath{\varepsilon}}. \end{equation} Following Lemma \ref{lem_theta_dominate} and (\ref{defn_rho}), \begin{equation}\label{est_j_mn} \n{f-j_{m,n}(f)} \;\leqslant\; \n{f-h_m(f)} + \n{g_{m,n}(f)} \;\leqslant\; 2\theta(f-h_m(f)) \;\to\; 0, \end{equation} as $m \to \infty$. Thus we see that \begin{equation}\label{est_J_k} C_k \;\subseteq\; \cl{J_k}^{\ensuremath{\left\|\cdot\right\|}} \;\subseteq\; \cl{J_k}^{w^*} \;\subseteq\; C_k + 2^{-k-2}r\ensuremath{\varepsilon} B_{X^*}. \end{equation} Given non-empty finite $F \subseteq \Gamma$, define the $w^*$-compact set \begin{equation}\label{defn_C_kF} C_{k,F} \;=\; \set{j \in \cl{J_k}^{w^*}}{\supp(j) \subseteq F} \;\subseteq \; W_F. \end{equation}
We apply Definition \ref{defn_admissible_set} to $\frac{1}{2}\cdot 3^{-k-|F|}\ensuremath{\varepsilon}$ and $C_{k,F} + \frac{1}{2}\cdot 3^{-k-|F|}\ensuremath{\varepsilon} B_{W_F}$ (which is norming for $V_F$) to obtain a set $D_{k,F} \subseteq W_F$ such that \begin{equation}\label{est_D_kF}
{\textstyle C_{k,F} + \frac{1}{2}\cdot 3^{-k-|F|}\ensuremath{\varepsilon} B_{W_F}} \;\subseteq\; \cl{D_{k,F}}^{w^*} \;\subseteq\; C_{k,F} + 3^{-k-|F|}\ensuremath{\varepsilon} B_{W_F}, \end{equation} and \begin{equation}\label{D_kF_boundary} \cl{D_{k,F}}^{w^*} \cap A_F \text{ is a boundary of }\cl{D_{k,F}}^{w^*}. \end{equation} Now define \[
D \;=\; \bigcup\set{(1+ 2^{-k}\ensuremath{\varepsilon})D_{k,F}}{F \subseteq \Gamma,\;k, |F| \in \mathbb{N}}. \]
First, we show that $B \subseteq \cl{D}^{w^*} \subseteq C + \ensuremath{\varepsilon} B_{X^*}$. By (\ref{est_J_k}), (\ref{defn_C_kF}), (\ref{est_D_kF}) and the fact that $C \subseteq B_{X^*}$, \[
D_{k,F} \;\subseteq\; C_{k,F} + 3^{-k-|F|}\ensuremath{\varepsilon} B_{W_F} \;\subseteq\; \cl{J_k}^{w^*} + {\textstyle\frac{1}{9}}\ensuremath{\varepsilon} B_{X^*} \;\subseteq\; C + {\textstyle\frac{1}{4}}\ensuremath{\varepsilon} B_{X^*} \;\subseteq\; {\textstyle\frac{5}{4}}B_{X^*}, \] giving \[ (1 + 2^{-k}\ensuremath{\varepsilon})D_{k,F} \;\subseteq\; C + {\textstyle\frac{1}{4}}\ensuremath{\varepsilon} B_{X^*} + {\textstyle\frac{5}{8}}\ensuremath{\varepsilon} B_{X^*} \;\subseteq\; C+\ensuremath{\varepsilon} B_{X^*}. \] As $C+\ensuremath{\varepsilon} B_{X^*}$ is $w^*$-closed, it follows that $\cl{D}^{w^*} \subseteq C + \ensuremath{\varepsilon} B_{X^*}$. Now let $f \in B$ and fix $k \in \mathbb{N}$ large enough so that $f \in C_k$. Given $\ell \geqslant k$, it follows that \begin{align*}
f \in C_\ell \;\subseteq\; \cl{J_\ell}^{w^*} \;&\subseteq\; \cl{\bigcup\set{C_{\ell,|F|}}{F\subseteq \Gamma,\;|F| \in \mathbb{N}}}^{w^*}\\
&\subseteq\; \cl{\bigcup\set{D_{\ell,|F|}}{F\subseteq \Gamma,\;|F| \in \mathbb{N}}}^{w^*}, \end{align*} and thus $(1+2^{-\ell}\ensuremath{\varepsilon})f \in \cl{D}^{w^*}$. Consequently, $f \in \cl{D}^{w^*}$, and we have $B \subseteq \cl{D}^{w^*}$ as required.
Now we show that $\cl{D}^{w^*} \cap A$ is a boundary of $\cl{D}^{w^*}$. Fix $x \in X$, $\n{x}=1$, and set \[ \eta \;=\; \sup\set{f(x)}{f \in \cl{D}^{w^*}}. \] Evidently, as $B$ is a boundary of $C$ and $B \subseteq \cl{D}^{w^*}$, we have $\eta \geqslant r > 0$.
We need to find an element $e \in \cl{D}^{w^*} \cap A$ satisfying $e(x)=\eta$. By $w^*$-compactness, there exists $d \in \cl{D}^{w^*}$ such that $d(x)=\eta$. Our first task is to show that there exists $k \in \mathbb{N}$ having the property that \begin{equation}\label{d_fixed_k}
d \in \cl{\bigcup\set{(1+ 2^{-k}\ensuremath{\varepsilon})D_{k,F}}{F\subseteq \Gamma,\;|F| \in \mathbb{N}}}^{w^*}. \end{equation} Observe that if there is no $k \in \mathbb{N}$ for which (\ref{d_fixed_k}) holds, then \begin{equation}\label{est_d_intersection}
d \in \bigcap_{k=1}^\infty \left(\cl{\bigcup\set{(1+ 2^{-\ell}\ensuremath{\varepsilon})D_{k,F}}{F\subseteq \Gamma,\; \ell, |F| \in \mathbb{N},\; \ell\geqslant k}}^{w^*}\right). \end{equation} We show that the hypotheses preclude this possibility from taking place. We start by setting $\xi = \sup\set{f(x)}{f \in C} \geqslant r >0$. Given $k \in \mathbb{N}$, (\ref{est_D_kF}) and (\ref{est_d_intersection}) imply \[ d(x) \;\leqslant\; (1+ 2^{-k}\ensuremath{\varepsilon})(\xi + 3^{-k-1}\ensuremath{\varepsilon}). \] As this holds for all $k\in\mathbb{N}$, we have $d(x)\leqslant \xi$. On the other hand, as $B$ is a boundary of $C$, there exists $k \in \mathbb{N}$ and $b \in C_k$, such that $b(x)=\xi$. Using the definition of $J_k$ and (\ref{est_j_mn}), there exists $m \in \mathbb{N}$ such that $j_{m,m+1}(b) \in J_k$ and \[ j_{m,m+1}(b)(x) \;\geqslant\; b(x) - \n{b-j_{m,n}(b)} \;>\; \xi - 2^{-k-2}r\ensuremath{\varepsilon}. \] Since $(1+2^{-k}\ensuremath{\varepsilon})J_k \subseteq \cl{D}^{w^*}$, and recalling that $\ensuremath{\varepsilon}<1$, we have \begin{align*} \eta \;\geqslant\; (1+2^{-k}\ensuremath{\varepsilon})j_{m,m+1}(b)(x) \;&>\; (1+2^{-k}\ensuremath{\varepsilon})(\xi - 2^{-k-2}r\ensuremath{\varepsilon})\\ &\geqslant\; (1+2^{-k}\ensuremath{\varepsilon})(1-2^{-k-2}\ensuremath{\varepsilon})\xi\\ &=\; (1+{\textstyle\frac{3}{4}}\cdot 2^{-k}\ensuremath{\varepsilon} - 2^{-2k-2}\ensuremath{\varepsilon}^2)\xi\\ &>\; (1+{\textstyle\frac{5}{8}}\cdot 2^{-k}\ensuremath{\varepsilon})\xi \;>\; d(x), \end{align*} which is a contradiction. Therefore (\ref{est_d_intersection}) cannot hold.
Hence (\ref{d_fixed_k}) holds for some $k \in \mathbb{N}$ that we fix for the remainder of the proof. For convenience, set $d' = (1+2^{-k}\ensuremath{\varepsilon})^{-1}d$. There are two cases to consider. First of all, it is possible that \begin{equation}\label{eqn_case_1}
d' \in \bigcap_{n=1}^\infty \left(\cl{\bigcup\set{D_{k,F}}{F\subseteq \Gamma,\;|F| \in \mathbb{N},\; |F|\geqslant n}}^{w^*}\right). \end{equation} In this case, by (\ref{defn_C_kF}) and (\ref{est_D_kF}), we have \begin{equation}\label{d_prime_in_J_k}
d' \in \bigcap_{n=1}^\infty \left(\cl{\bigcup\set{C_{k,F}}{F\subseteq \Gamma,\;|F| \in \mathbb{N},\; |F|\geqslant n}}^{w^*} + 3^{-k-n}\ensuremath{\varepsilon} B_{X^*}\right) \;\subseteq\; \cl{J_k}^{w^*}.
\end{equation} Now we need to consider whether $d$ (and hence $d'$) has infinite support or not. If $\supp(d)$ is infinite, then according to (\ref{defn_J_k}), Lemma \ref{lem_net_1} and the $w^*$-compactness of $C_k$, we have $d' \in C_k$. Then, using (\ref{defn_J_k}) and Lemma \ref{lem_convex_combination}, there exists $j \in J_k$ such that $d'(x) \leqslant j(x)$. As $F:=\supp(j)$ is finite, using (\ref{defn_C_kF}) and (\ref{est_D_kF}) \[ j \in C_{k,F} \;\subseteq\; \cl{D_{k,F}}^{w^*}, \] and therefore, by (\ref{D_kF_boundary}), there exists $e' \in \cl{D_{k,F}}^{w^*} \cap A_F$ such that $j(x) \leqslant e'(x)$. Define $e=(1+2^{-k}\ensuremath{\varepsilon})e' \in \cl{D}^{w^*} \cap A_F \subseteq \cl{D}^{w^*} \cap A$. Evidently, \[ \eta \;=\; d(x) \;\leqslant\; e(x) \;\leqslant\; \eta, \] meaning that we have found what we wanted. This concludes the case where $\supp(d)$ is infinite. If $G:=\supp(d)$ is finite, then from (\ref{defn_C_kF}) and (\ref{d_prime_in_J_k}) we see straightaway that $d' \in C_{k,G}$. Then we repeat what we have just done, but without $j$ (this time, by (\ref{D_kF_boundary}), there exists $e' \in \cl{D_{k,G}}^{w^*} \cap A_G$ such that $d'(x) \leqslant e'(x)$).
We have dealt with the first case, where (\ref{eqn_case_1}) holds. If (\ref{eqn_case_1}) does not hold then there exists some $n \in \mathbb{N}$ such that \[
d' \in \cl{\bigcup\set{D_{k,F}}{F\subseteq \Gamma,\;|F|= n}}^{w^*}. \]
We take nets $F^\alpha \subseteq \Gamma$, $|F^\alpha|=n$ and $d^\alpha \in D_{k,F^\alpha}$, such that $d^\alpha \stackrel{w^*}{\to} d'$. The set of subsets of $\Gamma$ having cardinality at most $n$ is compact in the topology of pointwise convergence on $\Gamma$. Among the accumulation points of the net $(F^\alpha)$, let $G$ be one having maximal cardinality. By taking a subnet if necessary, we can assume that $F^\alpha \to G$ and $G \subseteq F^\alpha$ for all $\alpha$. That $G$ is an accumulation point the $F^\alpha$ implies $\supp(d') \subseteq G$, so $d' \in W_G$. Either $|G|=n$ or $|G|<n$. If $|G|=n$, then $G=F^\alpha$ for all $\alpha$, giving $d' \in \cl{D_{k,G}}^{w^*}$. In this case we can finish the proof as above.
In the final part of the proof, we show that it is impossible for the strict inequality $|G|<n$ to hold. For a contradiction, suppose otherwise, and let $H^\alpha = F^\alpha \setminus G$. If $H$ is an accumulation point of $(H^\alpha)$ then $G \cup H$ is an accumulation point of $(F^\alpha)$. This forces $H=\varnothing$, by maximality of the cardinality of $G$. Hence, by compactness, $H^\alpha \to \varnothing$ in the topology of pointwise convergence. According to (\ref{est_D_kF}), there exists $c^\alpha \in C_{k,F^\alpha}$ such that $\n{d^\alpha - c^\alpha} \leqslant 3^{-k-n}\ensuremath{\varepsilon}$. By taking a further subnet if necessary, we can assume that $c^\alpha \to c'$, where $c' \in \cl{J_k}^{w^*}$. Since $H^\alpha \to \varnothing$, it follows that $\supp(c') \subseteq G$, and thus $c' \in C_{k,G}$. Moreover, by $w^*$-lower semicontinuity of the dual norm, $\n{d'-c'} \leqslant 3^{-k-n}\ensuremath{\varepsilon}$. Therefore, by this and (\ref{est_D_kF}), \[
d' + {\textstyle \frac{1}{2}}\cdot 3^{-k-n}\ensuremath{\varepsilon} B_{W_G} \;\subseteq\; C_{k,G} + {\textstyle \frac{1}{2}}\cdot 3^{-k-n+1}\ensuremath{\varepsilon} B_{W_G} \;\subseteq\; C_{k,G} + {\textstyle \frac{1}{2}}\cdot 3^{-k-|G|}\ensuremath{\varepsilon} B_{W_G} \;\subseteq\; \cl{D_{k,G}}^{w^*}. \] It follows that \[ d + {\textstyle \frac{1}{2}}\cdot(1+2^{-k}\ensuremath{\varepsilon}) 3^{-k-n}\ensuremath{\varepsilon} B_{W_G} \;\subseteq\; \cl{D}^{w^*}, \] but this contradicts our initial assumption that $d(x)=\eta$. \end{proof}
\begin{cor}\label{cor_main_tool} Let $(A_\gamma)_{\gamma \in \Gamma}$ be an admissible family as above, where each $A_\gamma$ is a $\sigma$-$w^*$-LRC and $w^*$-$K_\sigma$ set. Let $\ensuremath{\left\|\cdot\right\|}'$ be an equivalent norm on $X$ and let $B$ be a boundary of $\ensuremath{\left\|\cdot\right\|}'$, such that $\theta(f) < \infty$ whenever $f \in B$. Then $\ensuremath{\left\|\cdot\right\|}'$ can be approximated by both $C^\infty$-smooth norms and polyhedral norms. \end{cor}
\begin{proof} Set $C=B_{(X,\ensuremath{\left\|\cdot\right\|}')^*}$. Given $\ensuremath{\varepsilon}>0$, use Theorem \ref{thm_main_tool} to find a set $D \subseteq X^*$ such that \[ B \;\subseteq\; \cl{D}^{w^*} \;\subseteq\; (1+\ensuremath{\varepsilon})C \qquad\text{and}\qquad \cl{D}^{w^*} \cap A \text{ is a boundary of }\cl{D}^{w^*}, \] where $A:=\lspan(A_\gamma)_{\gamma \in \Gamma}$ is $\sigma$-$w^*$-LRC and $w^*$-$K_\sigma$ by Theorem \ref{thm_linspan}. Define $\ensuremath{{\displaystyle |\kern-.9pt|\kern-.9pt|}\cdot{\displaystyle |\kern-.9pt|\kern-.9pt|}}$ on $X$ by \[
\tn{x} \;=\; \sup\set{|f(x)|}{f \in D}, \qquad\qquad x \in X. \] Then $\n{x}' \leqslant \tn{x} \leqslant (1+\ensuremath{\varepsilon})\n{x}'$ and $B_{(X,\ttri\cdot\ttri)^*} = \cl{\conv}^{w^*}(D \cup (-D))$, and therefore \[ (\cl{D}^{w^*} \cup (-\cl{D}^{w^*})) \cap A, \] is a boundary of $\ensuremath{{\displaystyle |\kern-.9pt|\kern-.9pt|}\cdot{\displaystyle |\kern-.9pt|\kern-.9pt|}}$. We finish the proof by appealing to Theorem \ref{thm_tb}. \end{proof}
The last two results of the section will be of most use in the coming sections. The next corollary is immediate.
\begin{cor}\label{cor_main_tool_2} Let $(A_\gamma)_{\gamma \in \Gamma}$ be as above, and suppose that $\theta(f) < \infty$ for all $f \in X^*$. Then every equivalent norm on $X$ can be approximated by both $C^\infty$-smooth norms and polyhedral norms. \end{cor}
We conclude the section by making one further reduction. Suppose that $P_\gamma X$ is 1-dimensional for all $\gamma \in \Gamma$, or equivalently, that we have a shrinking bounded M-basis $(e_\gamma,e^*_\gamma)_{\gamma \in \Gamma}$. Now we let $A_\gamma = P^*_\gamma X^* = \lspan(e^*_\gamma)$, which is always a $\sigma$-$w^*$-LRC and $w^*$-$K_\sigma$ set. Moreover, (\ref{defn_rho}) simplifies to \begin{equation}\label{defn_rho_reduced}
\rho(F) \;=\; \max\set{\n{\sum_{\gamma \in F}a_\gamma e_\gamma^*}}{a_\gamma \in \mathbb{R} \text{ and }|a_\gamma|\|e^*_\gamma\|\leqslant 1 \text{ whenever }\gamma \in F}. \end{equation}
\begin{cor}\label{cor_main_tool_3} Let $(e_\gamma,e^*_\gamma)_{\gamma \in \Gamma}$ be a shrinking bounded M-basis of $X$, and suppose that $\theta(f) < \infty$ for all $f \in X^*$. Then every equivalent norm on $X$ can be approximated by both $C^\infty$-smooth norms and polyhedral norms. \end{cor}
\section{Spaces having a symmetric basis}\label{sect_symmetric}
The main purpose of this section is to develop tools to allow us to apply Corollary \ref{cor_main_tool_3} to certain spaces having symmetric bases. These tools focus on making it easier to establish whether or not $\theta(f)<\infty$ for all $f \in X^*$. Let $X$ be a Banach space and $\Gamma$ a set. Recall that a family of vectors $(e_\gamma)_{\gamma \in \Gamma}$ is a \emph{symmetric basis} of $X$ if, first, given $x \in X$, there is a unique family of scalars $(a_\gamma)_{\gamma \in \Gamma}$, such that $x = \sum_{\gamma \in \Gamma} a_\gamma e_\gamma$ (where the convergence is necessarily unconditional), and second, given any permutation $\pi$ of $\Gamma$, the sum $\sum_{\gamma \in \Gamma} b_\gamma e_{\pi(\gamma)}$ converges whenever $\sum_{\gamma \in \Gamma}b_\gamma e_\gamma$ converges (again, unconditionally). By the uniform boundedness principle, with such a basis in hand, we have \[ K\;:=\; \sup_{\theta, \pi} \n{T_{\theta, \pi}} \;<\; \infty, \] where, given a permutation $\pi$ of $\Gamma$ and $\theta=(\theta_\gamma)_{\gamma \in \Gamma}$ a choice of signs, the operator $T_{\theta,\pi}$ is defined on $X$ by \[ T_{\theta,\pi}\bigg( \sum_{\gamma \in \Gamma} a_\gamma e_\gamma\bigg) \;=\; \sum_{\gamma \in \Gamma} a_\gamma\theta_\gamma e_{\pi(\gamma)}. \] The number $K$ is known as the \emph{symmetric basis constant} of $(e_\gamma)_{\gamma \in \Gamma}$. If we define a new norm \[ \pn{x}{s} \;=\; \sup_{\theta,\pi} \n{T_{\theta,\pi} x}, \] then $\n{x} \leqslant \pn{x}{s} \leqslant K\n{x}$ for all $x \in X$, and the symmetric basis constant of $(e_\gamma)_{\gamma \in \Gamma}$ with respect to $\pndot{s}$ is equal to $1$ (see, for example \cite[Section 3.a]{lt:77}). Evidently, $\ensuremath{\left\|\cdot\right\|}$ and $\pndot{s}$ are equal if and only if $K=1$.
Hereafter, suppose that the symmetric basis $(e_\gamma)_{\gamma \in \Gamma}$ is shrinking. Let us fix a sequence $(\gamma_n)$ of distinct points in $\Gamma$. Following \cite[Proposition 3.a.6]{lt:77}, given $n \in \mathbb{N}$, define \[ \lambda(n) \;=\; \pn{\sum_{k=1}^n e_{\gamma_k}}{s} \qquad\text{and}\qquad\mu(n) \;=\; \pn{\sum_{k=1}^n e^*_{\gamma_k}}{s}, \]
and set $\lambda(0)=\mu(0)=0$. Since the basis is $1$-symmetric with respect to $\pndot{s}$, the definitions of $\lambda(n)$ and $\mu(n)$ are independent of the choice of initial sequence $(\gamma_n)$. It is also important to note that, by \cite[Proposition 3.a.6]{lt:77}, we have $\lambda(n)\mu(n)=n$ for all $n$. By scaling the basis vectors by the same amount, we can assume that $\pn{e_\gamma}{s}=\|e^*_\gamma\|_s=1$ for all $\gamma \in \Gamma$.
The functions $\lambda$ and $\mu$ will help to simplify the task of establishing whether or not $\theta(f)$ is finite.
\begin{prop}\label{prop_rho_and_mu} Let $X$ have a shrinking symmetric basis as above. Then \[
\theta(f) \;<\; \infty \qquad\text{if and only if}\qquad \sum_{k=1}^\infty q_k(f)\mu(|G_k(f)|) \;<\; \infty. \] \end{prop}
\begin{proof} Let $K$ be the symmetric basis constant. Let $F\subseteq \Gamma$ be finite. The result will follow immediately if we can prove that $K^{-1}\mu(|F|) \leqslant \rho(F) \leqslant K\mu(|F|)$. First, the simplification of (\ref{defn_rho}) to (\ref{defn_rho_reduced}) applies here. Next, if we set $a_\gamma=K^{-1}$, $\gamma \in F$, then $|a_\gamma|\|e^*_\gamma\| \leqslant K^{-1}K\|e^*_\gamma\|_s=1$, and thus \[
K^{-1}\mu(|F|) \;=\; K^{-1}\pn{\sum_{\gamma \in F}e_\gamma^*}{s} \;\leqslant\; \n{\sum_{\gamma \in F}a_\gamma e_\gamma^*} \;\leqslant\; \rho(F). \]
On the other hand, if $|a_\gamma|\|e^*_\gamma\| \leqslant 1$, then $|a_\gamma| \leqslant |a_\gamma|\|e^*_\gamma\|_s \leqslant 1$. Therefore, \[
\n{\sum_{\gamma \in F}a_\gamma e_\gamma^*} \;\leqslant\; K\pn{\sum_{\gamma \in F}a_\gamma e_\gamma^*}{s} \;\leqslant\; K \pn{\sum_{\gamma \in F} e_\gamma^*}{s} \;=\; K\mu(|F|). \qedhere \] \end{proof}
Recall the comments about subspaces of $c_0(\Gamma)$ following Theorem \ref{c0_approx}. The following result uses the function $\lambda$ to provide a straightforward test that will ensure that the examples we present are new.
\begin{prop}\label{prop_not_subspace_c_0} Let $X$ have a symmetric basis $(e_\gamma)_{\gamma\in\Gamma}$ as above. The following statements are equivalent. \begin{enumerate} \item The sequence $(\lambda(n))_{n=1}^\infty$ is bounded. \item The spaces $X$ and $c_0(\Gamma)$ are isomorphic. \item There exists a set $\Delta$ such that $X$ is isomorphic to a subspace of $c_0(\Delta)$. \end{enumerate} \end{prop}
\begin{proof}The implication $(1) \Rightarrow (2)$ follows immediately from the fact that, given a normalized basis $(e_\gamma)_{\gamma \in \Gamma}$ of a Banach space having unconditional basis constant $L$, we have \[
L^{-1}\max_{\gamma \in F} |a_\gamma| \;\leqslant\; \n{\sum_{\gamma \in F} a_\gamma e_\gamma} \;\leqslant\; L\max_{\gamma \in F} |a_\gamma|\n{\sum_{\gamma \in F} e_\gamma}. \] for every finite set $F \subseteq \Gamma$, and real numbers $a_\gamma$, $\gamma \in F$.
The implication $(2) \Rightarrow (3)$ is trivial. Finally, consider $(3) \Rightarrow (1)$. Assume that the bounded linear map $\map{T}{X}{c_0(\Delta)}$ is bounded below. Then $(Te_{\gamma_k})_{k=1}^\infty$ converges weakly to $0$ and satisfies $\inf (\pn{Te_{\gamma_k}}{\infty})_k > 0$, so it admits a subsequence, again labelled $(Te_{\gamma_k})$, that is equivalent to a block basic sequence of $c_0$ \cite[Corollary 4.27]{fhhspz:11}. Consequently, $(Te_{\gamma_k})$, and hence $(e_{\gamma_k})$, is equivalent to the usual basis of $c_0$ \cite[Proposition 4.45]{fhhspz:11}. The boundedness of $(\lambda(n))_{n=1}^\infty$ follows. \end{proof}
We remark that if that $\Gamma$ is uncountable, then the implication $(3) \Rightarrow (2)$ above follows from \cite[Main Theorem]{ht:93}. There, it is shown that if $(u_\alpha)_{\alpha \in A}$ is an uncountable symmetric basic set in an F-space $Y$ having an F-norm and symmetric basis $(v_\beta)_{\beta \in B}$, then there exists a decreasing sequence of non-negative scalars $a_i$, $i \in \mathbb{N}$, and an injection $(\alpha,i) \mapsto \beta_{\alpha,i}$ from $A \times \mathbb{N}$ into $B$, such that $(u_\alpha)_{\alpha \in A}$ is equivalent to $(u'_\alpha)_{\alpha \in A}$, where $u'_\alpha:=\sum_{i=1}^\infty a_i v_{\beta_{\alpha,i}}$ converges in norm for all $\alpha \in A$. Therefore, if $Y=c_0(B)$ and $(v_\beta)_{\beta \in B}$ is its standard basis, then for every finite set $F \subseteq A$ and scalars $c_\alpha$, $\alpha \in F$, we have \[
\pn{\sum_{\alpha \in F} c_\alpha u'_\alpha}{\infty} = \pn{\sum_{\alpha \in F}c_\alpha \sum_{i=1}^\infty a_i v_{\beta_{\alpha,i}}}{\infty} = \pn{\sum_{(\alpha,i) \in F \times \mathbb{N}} c_\alpha a_i v_{\beta_{\alpha,i}}}{\infty} \leqslant a_1 \max\set{|c_\alpha|}{\alpha \in F}. \]
The next theorem is the main result of the section.
\begin{thm}\label{theta_equivalences} Let $X$ have a shrinking symmetric basis $(e_\gamma)_{\gamma \in \Gamma}$. The following statements are equivalent. \begin{enumerate} \item\label{finite_theta} $\theta(f)<\infty$ for all $f \in X^*$; \item\label{eqn2} the quantity \[ \sup\set{\n{\sum_{k=1}^n (\mu(k)-\mu(k-1))e_{\gamma_k}}}{n \in \mathbb{N}}, \] is finite; \item\label{bidual} the series \[ \sum_{k=1}^\infty (\mu(k)-\mu(k-1))e_{\gamma_k}, \] converges in $(X^{**},w^{**})$. \end{enumerate} \end{thm}
\begin{proof} The equivalence of (\ref{eqn2}) and (\ref{bidual}) follows because the basis is shrinking \cite[Proposition 1.b.2]{lt:77}. Now we prove the equivalence of (\ref{finite_theta}) and (\ref{eqn2}). Let $K$ be the symmetric basis constant of $(e_\gamma)_{\gamma \in\Gamma}$. As above, we assume that the basis is normalized with respect to $\pndot{s}$. Let $f \in X^*$. By Proposition \ref{prop_rho_and_mu}, \begin{equation}\label{eqn-1}
\theta(f) \;<\; \infty \qquad\text{if and only if}\qquad \sum_{k=1}^\infty q_k(f)\mu(|G_k(f)|) \;<\; \infty. \end{equation} By Lemma \ref{lem_ran(f)_structure}, there exists a sequence $(\gamma'_i)_{i=1}^\infty$ of distinct points in $\Gamma$, and integers $1=i_1 < i_2 < i_3 < \dots$, such that \[ G_k(f) \;=\; \set{\gamma'_i}{1 \leqslant i < i_{k+1}}, \]
whenever $k \in \mathbb{N}$ and $p_k(f)>0$. Define $a_i = \|P^*_{\gamma'_i} f\| = |f(e_{\gamma'_i})|\|e_{\gamma'_i}^* \|$, $i \in \mathbb{N}$. It follows that $p_k(f)=a_i$ whenever $i_k \leqslant i < i_{k+1}$. Now \begin{align}\label{eqn0} \sum_{i=1}^\infty (a_i-a_{i+1})\mu(i) \;&=\; \sum_{k=1}^\infty \sum_{i=i_k}^{i_{k+1}-1} (a_i-a_{i+1})\mu(i) \nonumber\\
&=\; \sum_{k=1}^\infty (a_{i_k}-a_{i_k+1})\mu(i_k) \;=\; \sum_{k=1}^\infty q_k(f)\mu(|G_k(f)|). \end{align} Observe that \begin{equation}\label{eqn3} \sum_{i=1}^n (a_i-a_{i+1})\mu(i) \;=\; \sum_{i=1}^n a_i(\mu(i)-\mu(i-1)) - a_{n+1}\mu(n). \end{equation} Since $\n{e_\gamma^*} \leqslant K\pn{e_\gamma^*}{s}=K$ for all $\gamma \in \Gamma$, and the basis is 1-symmetric with respect to $\pndot{s}$, we can see that \[
a_{n+1}\mu(n) \;=\; \pn{\sum_{i=1}^n a_{n+1}e^*_{\gamma'_i}}{s} \;\leqslant\; \pn{\sum_{i=1}^n a_ie^*_{\gamma'_i}}{s} \;\leqslant\; K\pn{\sum_{i=1}^n \frac{a_i}{\|e^*_{\gamma'_i}\|}e^*_{\gamma'_i}}{s} \;\leqslant\; K\pn{f}{s}, \] for all $n$. Therefore, by (\ref{eqn-1}) and (\ref{eqn0}), and using the fact that all the terms in the two partial sums in equation (\ref{eqn3}) are non-negative, $\theta(f)<\infty$ if and only if \begin{equation}\label{eqn4} \sum_{k=1}^\infty a_k(\mu(k)-\mu(k-1)) \;<\; \infty. \end{equation} By appealing to the symmetry of the basis, (\ref{eqn4}) holds for all $f \in X^*$ if and only if the family of vectors \[ \set{\sum_{k=1}^n (\mu(k)-\mu(k-1))e_{\gamma_k}}{n \in \mathbb{N}}, \] is weakly bounded, and hence norm bounded, by the uniform boundedness principle. \end{proof}
Armed with Theorem \ref{theta_equivalences}, we turn to our first class of new examples.
\begin{example}\label{ex_Lorentz} Consider a decreasing sequence $w=(w_n)_{n=1}^\infty$ of positive numbers such that $\sum_{n=1}^\infty w_n = \infty$. The {\em Lorentz space} $d(w,1,\Gamma)$ is the space of all functions $\map{f}{\Gamma}{\mathbb{R}}$, such that \begin{equation}\label{defn_Lorentz_norm}
\n{f} \;:=\; \sup\set{\sum_{\gamma \in \Gamma} w_n|f(\gamma_n)|}{(\gamma_n)_{n=1}^\infty \subseteq \Gamma \text{ is a sequence of distinct points}}, \end{equation}
is finite. A treatment of the separable version of these spaces (where $\Gamma=\mathbb{N}$) can be found in \cite[Section 4.e]{lt:77}. It is clear that if $(\gamma_n)_{n=1}^\infty \subseteq \Gamma$ is chosen in such a way that $(|f(\gamma_n)|)_{n=1}^\infty$ is decreasing, then \[
\n{f} \;=\; \sum_{n=1}^\infty w_n |f(\gamma_n)|. \] The {\em predual of Lorentz space}, $d_*(w,1,\Gamma)$, is the set of all functions $\map{x}{\Gamma}{\mathbb{R}}$, such that $\cl{x} \in c_0$, where \[
\cl{x}(k) \;=\; \max\set{\frac{\sum_{i=1}^k |x(\gamma_i)|}{\sum_{i=1}^k w_i}}{\gamma_1,\ldots,\gamma_k \in \Gamma \text{ are distinct}}, \] and $\n{x}=\pn{\cl{x}}{\infty}$. The separable version of these spaces was first considered in \cite{sargent:60}. The families of unit vectors $(e_\gamma)_{\gamma \in \Gamma}$ and $(e_\gamma^*)_{\gamma \in \Gamma}$ form canonical 1-symmetric bases of $d_*(w,1,\Gamma)$ and $d(w,1,\Gamma)$, respectively.
It is straightforward to see that \begin{equation}\label{eqn_Lorentz_mu} \mu(n) \;=\; \pn{\sum_{k=1}^n e^*_{\gamma_k}}{s} \;=\; \n{\sum_{k=1}^n e^*_{\gamma_k}} \;=\; \sum_{k=1}^n w_k, \end{equation} and that Theorem \ref{theta_equivalences} (2) is fulfilled trivially by any space $d_*(w,1,\Gamma)$, as \[ \n{\sum_{k=1}^n (\mu(k)-\mu(k-1))e_{\gamma_k}} \;=\; \n{\sum_{k=1}^n w_k e_{\gamma_k}} \;=\; 1, \] for all $n \in \mathbb{N}$. Hence Corollary \ref{cor_main_tool_3} applies. Finally, \[ \lambda(n) \;=\; \frac{n}{\mu(n)} \;=\; \frac{n}{\sum_{k=1}^n w_k}, \] and this forms a bounded sequence if and only if $w_n \not\to 0$. Therefore, provided $w_n \to 0$, Proposition \ref{prop_not_subspace_c_0} tells us that $d_*(w,1,\Gamma)$ is not isomorphic to a subspace of $c_0(\Delta)$, for any set $\Delta$. \end{example}
The next result provides another test.
\begin{cor}\label{lambda_only} Let $X$ have a shrinking symmetric basis $(e_\gamma)_{\gamma \in \Gamma}$, and suppose that \[ \sup\set{\n{\sum_{k=1}^n \frac{e_{\gamma_k}}{\lambda(k)}}}{n \in \mathbb{N}} \;<\; \infty, \] or equivalently, the series \[ \sum_{k=1}^\infty \frac{e_{\gamma_k}}{\lambda(k)}, \] converges in $(X^{**},w^{**})$. Then $\theta(f)<\infty$ for all $f \in X^*$. \end{cor}
\begin{proof} Because $\lambda(n)\mu(n)=n$ for all $n$, \[ \mu(k) - \mu(k-1) \;=\; \frac{k}{\lambda(k)} - \frac{k-1}{\lambda(k-1)} \;\leqslant\; \frac{k}{\lambda(k)} - \frac{k-1}{\lambda(k)} \;=\; \frac{1}{\lambda(k)}. \] It follows that Theorem \ref{theta_equivalences} (2) is fulfilled. \end{proof}
We provide an application of Corollary \ref{lambda_only}, by considering a symmetric version of the Nakano space. Let $\Gamma$ be a set and let $(p_k)_{k=1}^\infty$ be a non-decreasing sequence, with $p_1 \geqslant 1$. By $h^S_{(p_k)}(\Gamma)$ we denote the space of all real functions $x$ defined on $\Gamma$, such that \[ \phi\bigg(\frac{x}{\rho}\bigg) \;<\; \infty, \] for all $\rho > 0$, where \[
\phi(x) \;:=\; \sup\set{\sum_{k=1}^\infty |x(\gamma_k)|^{p_k}}{(\gamma_k)_{k=1}^\infty \text{ is a sequence of distinct points in }\Gamma}. \] Given $x \in h^S_{(p_n)}(\Gamma)$, we set \[ \n{x} \;=\; \inf\set{\rho>0}{\phi\bigg(\frac{x}{\rho}\bigg) \leqslant 1}. \] It is easy to see that the standard unit vectors $(e_\gamma)_{\gamma \in \Gamma}$ form a $1$-symmetric basis in $h^S_{(p_n)}(\Gamma)$. In \cite{afnst:18}, it is shown that if $p_n \to \infty$, then $h^S_{(p_n)}(\Gamma)$ is isomorphically polyhedral.
\begin{prop}\label{prop_Nakano} Let $\sum_{k=1}^\infty k^{-1}\rho^{-p_k}$ converge for some $\rho>1$. Then \[ \sup\set{\n{\sum_{k=1}^n \frac{e_{\gamma_k}}{\lambda(k)}}}{n \in \mathbb{N}} \;<\; \infty. \] \end{prop}
\begin{proof} Since $\lambda(1)=1$ and the sequences $(p_k)_{k=1}^\infty$ and $(\lambda(k))_{k=1}^\infty$ are non-decreasing, we know that \[ n\lambda(n)^{-p_n} \;\leqslant\; \sum_{k=1}^n \lambda(n)^{-p_k} \;=\; \phi\left( \frac{\sum_{k=1}^n e_{\gamma_k}}{\lambda(n)} \right) \;=\; 1, \] hence \begin{equation}\label{eqn_nakano_1} \lambda(n)^{-p_n} \leqslant n^{-1}, \end{equation} for all $n$. Using the hypothesis, let $m\in\mathbb{N}$ such that $\sum_{k=m+1}^\infty k^{-1} \rho^{-p_k} \leqslant 1$. It follows that \[ \sum_{k=m+1}^\infty \left(\frac{1}{\lambda(k)\rho}\right)^{p_k} \;\leqslant\; 1, \] and thus \[ \n{\sum_{k=m+1}^n \frac{e_{\gamma_k}}{\lambda(k)}} \;\leqslant\; \rho, \] whenever $n \geqslant m+1$. We conclude that \[ \n{\sum_{k=1}^n \frac{e_{\gamma_k}}{\lambda(k)}} \;\leqslant\; \n{\sum_{k=1}^m \frac{e_{\gamma_k}}{\lambda(k)}} + \rho, \] for all $n$. \end{proof}
Corollary \ref{lambda_only} applies to any space $h^S_{(p_n)}(\Gamma)$ satisfying the hypothesis of Proposition \ref{prop_Nakano}. By Proposition \ref{prop_not_subspace_c_0}, $h^S_{(p_n)}(\Gamma)$ does not embed into any space of the form $c_0(\Delta)$, provided $\log(\lambda(n)) \to \infty$. Using (\ref{eqn_nakano_1}), we deduce that the same holds if \begin{equation}\label{eqn_nakano_2} \lim_{k \to\infty} \frac{\log(k)}{p_k} \;=\; \infty. \end{equation}
\begin{example} Set $p_k = 2\log(\log(k)+1)+1$. Then \[ \sum_{k=2}^\infty k^{-1}\ensuremath{\mathrm{e}}^{-p_k} \;<\; \sum_{k=2}^\infty \frac{1}{k\log^2(k)} \;<\; \infty, \] and thus Corollary \ref{lambda_only} applies to $h^S_{(p_n)}(\Gamma)$. Moreover, for large $k$, we have \[ \frac{\log(k)}{p_k} \;=\; \frac{\log(k)}{2\log(\log(k)+1)+1} \;>\; \frac{\log(k)}{3\log(\log(k))} \;\to\; \infty, \] as $k \to \infty$. \end{example}
\begin{prob} Does there exist a non-decreasing sequence $(p_k)_{k=1}^\infty$, with $p_1 \geqslant 1$, $p_k \to \infty$, such that \[ \sup\set{\n{\sum_{k=1}^n \frac{e_{\gamma_k}}{\lambda(k)}}}{n \in \mathbb{N}} \;=\; \infty, \] or \[ \sup\set{\n{\sum_{k=1}^n (\mu(k)-\mu(k-1))e_{\gamma_k}}}{n \in \mathbb{N}} \;=\; \infty, \] with respect to $h^S_{(p_n)}(\Gamma)$? \end{prob}
For some classes of spaces, for example, Orlicz spaces whose standard basis is shrinking, the implication in Corollary \ref{lambda_only} is reversible -- see Theorem \ref{Orliczcondition} below. We close the section by showing that this is not the case in general.
\begin{example} Let $w_n = n^{-1}$, $n \in \mathbb{N}$. Then $d_*(w,1):=d_*(w,1,\mathbb{N})$ satisfies the conditions of Theorem \ref{theta_equivalences}, but not the hypothesis of Corollary \ref{lambda_only}. \end{example}
\begin{proof} The first conclusion is shown in Example \ref{ex_Lorentz}. To see the second, by (\ref{eqn_Lorentz_mu}) we have \[ \log(n) \;\leqslant\; \mu(n) \;=\; \sum_{k=1}^n k^{-1} \;\leqslant\; 1 + \log(n), \] for all $n \in \mathbb{N}$, and hence, by the definition of the norm on $d_*(w,1)$, \begin{align*} \n{\sum_{k=1}^n \frac{e_k}{\lambda(k)}} \;\geqslant\; \frac{\sum_{k=1}^n \lambda(k)^{-1}}{\sum_{k=1}^n k^{-1}} &\;=\; \frac{\sum_{k=1}^n k^{-1}\mu(k)}{\mu(n)}\\ &\;\geqslant\; \frac{1}{1+\log(n)}\sum_{k=1}^n \frac{\log(k)}{k}. \end{align*} The function $\log(t)/t$ is decreasing for $t \geqslant \ensuremath{\mathrm{e}}$, thus given $n\geqslant 3$ \[ \sum_{k=3}^n \frac{\log(k)}{k} \;\geqslant\; \lint{3}{n+1}{\frac{\log(t)}{t}}{t} \;=\; {\textstyle \frac{1}{2}\big(\log^2(n+1)-\log^2(3)\big)}. \] Consequently, \[ \n{\sum_{k=1}^n \frac{e_k}{\lambda(k)}} \;\geqslant\; \frac{\frac{1}{2}\big(\log^2(n) - \log^2(3)\big)}{1+\log(n)} \;\to\; \infty. \tag*{\qedhere} \] \end{proof}
\section{Orlicz spaces and Leung's Condition}
In this section, we consider Orlicz space in the context of norm approximation. Let $M$ be an Orlicz function and let $\Gamma$ be a set. The space $\ell_M(\Gamma)$ is the set of all functions $\map{x}{\Gamma}{\mathbb{R}}$ such that \[
\n{x} :\;=\; \inf\set{\rho>0}{\sum_{\gamma \in \Gamma} M\bigg(\frac{|x(\gamma)|}{\rho}\bigg) \leqslant 1}, \] is finite. The space $h_M(\Gamma)$ is that closed subspace of $\ell_M(\Gamma)$ for which \[
\sum_{\gamma \in \Gamma} M\bigg(\frac{|x(\gamma)|}{\rho}\bigg) \;<\; \infty, \] for \emph{all} $\rho>0$. We denote by $h_M$ the space $h_M(\mathbb{N})$. It is easy to check that the standard unit vectors $(e_\gamma)_{\gamma \in \Gamma}$ form a 1-symmetric basis of $h_M(\Gamma)$. It follows from \cite[Proposition 1.b.2]{lt:77} and the definitions that if this basis is shrinking (and in this section we will always assume so), then $\ell_M(\Gamma)$ is isometric to $h_M(\Gamma)^{**}$. We assume hereafter that $M$ is \emph{non-degenerate}, that is, $M(t)>0$ for all $t>0$. In this case, basic calculation yields \begin{equation}\label{eqn_Orlicz_lambda} \lambda(n) \;=\; \n{\sum_{k=1}^n e_{\gamma_k}} \;=\; \frac{1}{M^{-1}\big(\frac{1}{n}\big)}, \end{equation} and thus $h_M(\Gamma)$ is not isomorphic to a subspace of $c_0(\Delta)$, for any $\Delta$, by Proposition \ref{prop_not_subspace_c_0} (if $M$ is degenerate then $h_M(\Gamma)$ is isomorphic to $c_0(\Gamma)$).
The theory of polyhedrality in Orlicz sequence spaces was initiated by Leung in \cite{leung:94,leung:99}. His work focuses in large measure on a condition on $M$ that we shall call \emph{Leung's Condition}.
\begin{defn}[cf.~{\cite[Theorem 4]{leung:94}}]\label{defn_Leung} We say that a non-degenerate Orlicz function $M$ satisfies {\em Leung's Condition} if there exists $K>1$ such that \[ \lim_{t \to 0} \frac{M(K^{-1}t)}{M(t)} \;=\; 0. \] \end{defn}
The utility of this condition is demonstrated by the following results.
\begin{thm}[{\cite[Theorem 18]{leung:99}}]\label{theorem_Leung} The following statements are equivalent. \begin{enumerate} \item $M$ satisfies Leung's condition; \item $h_M$ embeds isomorphically in $C(\omega^\omega+1)$; \item $h_M$ embeds isomorphically in $C(\alpha+1)$ for some countable ordinal $\alpha$. \end{enumerate} \end{thm}
In particular, if $M$ satisfies Leung's condition then $h_M$ admits an equivalent polyhedral norm. This was first proved in \cite[Theorem 4]{leung:94}, but it follows easily from Theorem \ref{theorem_Leung} since, given a countable compact Hausdorff space $K$, the space $C(K)$ admits a countable boundary and any such space admits an equivalent polyhedral norm \cite[Theorem 3]{fonf:80}. More generally, Leung's condition implies the existence of an equivalent polyhedral norm on $h_M(\Gamma)$ \cite[Corollary 25]{fpst:08}. Concerning the approximation of all equivalent norms on $h_M(\Gamma)$ by polyhedral and $C^\infty$-smooth norms, we have the following result.
\begin{thm}\label{Orliczcondition} Let $\Gamma$ be a set and let $M$ be a non-degenerate Orlicz function. The following statements are equivalent. \begin{enumerate} \item There exists $K>1$ such that \begin{equation}\label{eqn5} \sum_{n=1}^\infty M\bigg(\frac{M^{-1}\big(\frac{1}{n}\big)}{K} \bigg) \;<\; \infty. \end{equation} \item The quantity \[ \sup\set{\n{\sum_{k=1}^n \frac{e_{\gamma_k}}{\lambda(k)}}}{n \in \mathbb{N}}, \] is finite. \item The standard basis of $X:=h_M(\Gamma)$ is shrinking and $\theta(f)<\infty$ for all $f \in X^*$. \end{enumerate} \end{thm}
Before proving the theorem, we present three results, mostly due to D.~Leung (who contacted us after seeing a previous version of the paper), that help us to compare condition (\ref{eqn5}) with Leung's condition. Only the final consequence of Proposition \ref{prop_implies_Leung} had been proved by us (in a more complicated way) before Leung made contact. These results have much enhanced this part of the paper and we are grateful to Leung for giving us permission to include them here.
\begin{lem}[\cite{leung:18}] Condition (\ref{eqn5}) in Theorem \ref{Orliczcondition} is equivalent to the existence of $K>1$ such that \begin{equation}\label{Leung_sum} \sum_{j=1}^\infty \frac{M(K^{-1}2^{-j})}{M(2^{-j})} \;<\; \infty. \end{equation} \end{lem}
\begin{proof} Given $j \in \mathbb{N}$, let $n_j \in \mathbb{N}$ be minimal, subject to the condition $n_j^{-1} \leqslant M(2^{-j})$. By this minimality, and the convexity of $M$, it follows that \begin{equation}\label{Leung_sum_1} 2(n_j-1) \;<\; \frac{2}{M(2^{-j})} \;\leqslant\; \frac{1}{M(2^{-j-1})} \;\leqslant\; n_{j+1} \;<\; \frac{1}{M(2^{-j-1})} + 1. \end{equation} Given $m,n \in \mathbb{N}$, with $n_j \leqslant n < n_{j+1}$, we have $2^{-j-1} < M^{-1}(\frac{1}{n}) \leqslant 2^{-j}$, giving \[ {\textstyle M(2^{-j-m-1}) \;<\; M(2^{-m}M^{-1}(\frac{1}{n})) \;\leqslant\; M(2^{-j-m}),} \] and consequently \[ \sum_{j=1}^\infty (n_{j+1}-n_j)M(2^{-j-m-1}) \;<\; \sum_{n=n_1}^\infty {\textstyle M(2^{-m}M^{-1}(\frac{1}{n}))} \;\leqslant\; \sum_{j=1}^\infty (n_{j+1}-n_j)M(2^{-j-m}). \] Hence the convergence of $\sum_{j=1}^\infty n_{j+1} M(2^{-j-m})$ implies that of $\sum_{n=1}^\infty M(2^{-m}M^{-1}(\frac{1}{n}))$. Together with (\ref{Leung_sum_1}), this in turn implies the convergence of $\sum_{j=1}^\infty n_{j+1} M(2^{-j-m-1})$. Again, given (\ref{Leung_sum_1}), the statements \[ \sum_{j=1}^\infty n_{j+1} M(2^{-j-m}) \;<\; \infty \qquad\text{and}\qquad \sum_{j=1}^\infty \frac{M(2^{-j-m})}{M(2^{-j})} \;<\; \infty, \] are equivalent. This completes the proof. \end{proof}
\begin{prop}[mainly \cite{leung:18}]\label{prop_implies_Leung}Leung's condition is satisfied if and only if \begin{equation}\label{Leung_sum_2} \lim_{j \to \infty} \frac{M(2^{-j-m})}{M(2^{-j})} \;=\; 0, \end{equation} for sufficiently large $m \in \mathbb{N}$. Consequently, if $M$ satisfies condition (\ref{eqn5}) then it satisfies Leung's condition. \end{prop}
\begin{proof}Clearly, if $M$ satisfies Definition \ref{defn_Leung} then it satisfies (\ref{Leung_sum_2}) for a sufficiently large $m$. To see that the converse holds, let $2^{-j-1} \leqslant t \leqslant 2^{-j}$. Then $M(2^{-j-1}) \leqslant M(t)$ and $M(2^{-m-1}t) \leqslant M(2^{-m-1-j})$. Therefore \[ \frac{M(2^{-m-1}t)}{M(t)} \;\leqslant\; \frac{M(2^{-j-1-m})}{M(2^{-j-1})}, \] meaning that we obtain Leung's condition, where $K=2^{m+1}$. The result now clearly follows from Lemma \ref{Leung_sum}. \end{proof}
On the other hand, Leung's condition does not imply condition (\ref{eqn5}).
\begin{example}[\cite{leung:18}] Let $M$ be an Orlicz function satisfying \[ M'(t) \;=\; a_j \;:=\; \prod_{k=1}^j \frac{1}{\log(k+2)}, \] whenever $2^{-j-1} < t < 2^{-j}$. Then $M$ satisfies Leung's condition but not condition (\ref{eqn5}). \end{example}
\begin{proof} We have $M(2^{-j}) \;=\; \sum_{\ell=j}^\infty 2^{-\ell-1} a_\ell$, so $2^{-j-1}a_j \leqslant M(2^{-j}) \leqslant 2^{-j}a_j$. It follows that \[ \frac{M(2^{-j-1})}{M(2^{-j})} \;\leqslant\; \frac{a_{j+1}}{a_j} \;\to\; 0, \] as $j\to\infty$. However, given $m \in \mathbb{N}$, \[ \sum_{j=1}^\infty \frac{M(2^{-j-m})}{M(2^{-j})} \;\geqslant\; \sum_{j=1}^\infty \frac{2^{-j-m-1} a_{j+m}}{2^{-j}a_j} \;=\; 2^{-m-1}\sum_{j=1}^\infty \frac{a_{j+m}}{a_j} \;\geqslant\; 2^{-m-1}\sum_{j=1}^\infty \frac{1}{\log^m(j+3)}, \] which diverges. The result now follows from the previous two. \end{proof}
\begin{proof}[Proof of Theorem \ref{Orliczcondition}] First, we demonstrate the equivalence of (1) and (2). Let $(\gamma_n) \subseteq \Gamma$ be the sequence of distinct points in $\Gamma$ from Section \ref{sect_symmetric}. Suppose that (1) holds. Recall equation (\ref{eqn_Orlicz_lambda}). Set $L= \max\{1,\sum_{n=1}^\infty M\big(M^{-1}(\frac{1}{n})/K \big)\}$. By convexity of $M$ and the fact that $M(0)=0$, we have $\sum_{n=1}^\infty M\big(M^{-1}(\frac{1}{n})/KL \big) \leqslant 1$ and thus \[ \n{\sum_{k=1}^n \frac{e_{\gamma_k}}{\lambda(k)}} \;\leqslant\; KL, \] for all $n$, giving (2). The converse implication follows similarly.
Now we prove that (2) implies (3). Let (2) hold. Since (2) implies (1), using Proposition \ref{prop_implies_Leung}, $M$ satisfies Leung's condition. This implies that $X$ cannot contain an isomorphic copy of $\ell_p$, $p \geqslant 1$ \cite[Theorem 4.a.9]{lt:77}. Since $X$ cannot contain an isomorphic copy of $\ell_1$, the standard basis of $X$ must be shrinking \cite[Theorem 1.c.9]{lt:77}. Now we are in a position to apply Corollary \ref{lambda_only}, giving (3).
Finally, we show that (3) implies (2). Observe that both conditions (2) and (3) hold independently of the choice of (equivalent) norm on $X$ that is used to define $\lambda$ and $\theta$:~if (2) is satisfied with respect to one equivalent norm, then it is satisfied with respect to all others, and likewise for (3). Since the basis of $X$ is shrinking, $X$ cannot contain an isomorphic copy of $\lp{1}$. Therefore, according to \cite[Theorem 4.a.9 and p.~144]{lt:77}, we can deduce that there exists $a>1$ and a differentiable Orlicz function $N$ that is equivalent to $M$ at $0$ and satisfies \[ \lim_{t \to 0} \inf \frac{tN'(t)}{N(t)} \;>\; a. \] We extend the function $N$ for large $t$ in such a way that \[ \frac{tN'(t)}{N(t)} \;>\; a, \] for all $t>0$. Given $w >0$, define $F(w)=wt(w)$, where $t(w)=N^{-1}(\frac{1}{w})$. By the Inverse Function Theorem, \begin{align*} F'(w) \;=\; t(w) + wt'(w) &\;=\; t(w) + w\cdot\frac{1}{N'(t(w))}\cdot\bigg({-\frac{1}{w^2}}\bigg) \\
&\;=\; t(w) - \frac{1}{wN'(t(w))}\\
&\;>\; t(w) - \frac{t}{awN(t(w))} \;=\; \bigg(\frac{a-1}{a}\bigg)t(w) \;=\; \bigg(\frac{a-1}{a}\bigg){\textstyle N^{-1}\big(\frac{1}{w}\big)}. \end{align*} Set $c = (a-1)/a \in (0,1)$. By the Mean Value Theorem, for each $n$ there exists $\tau_n \in (0,1)$ such that \begin{equation}\label{eqn6} F(n) - F(n-1) \;=\; F'(n-\tau_n) \;>\; cN^{-1}\bigg(\frac{1}{n-\tau_n}\bigg) \;>\; c{\textstyle N^{-1}\big(\frac{1}{n}\big)}. \end{equation}
The function $N$ yields an equivalent 1-symmetric norm $\pndot{N}$ on $X$, with respect to which the basis is normalized. Let $\lambda_N$ and $\mu_N$ be the functionals corresponding to $\pndot{N}$. Condition (3) is satisfied with respect to $\pndot{N}$, and hence \[ \sup\set{\pn{\sum_{k=1}^n (\mu_N(k)-\mu_N(k-1))e_{\gamma_k}}{N}}{n \in \mathbb{N}} \;<\; \infty, \] by Theorem \ref{theta_equivalences}. Since $\lambda_N(n) = N^{-1}(\frac{1}{n})$ and $\mu_N(n)=n/\lambda_N(n) = F(n)$, equation (\ref{eqn6}) means that \[ \mu_N(n)-\mu_N(n-1) \;>\; \frac{c}{\lambda_N(n)}, \] for all $n$. Therefore, \[ \sup\set{\pn{\sum_{k=1}^n \frac{e_{\gamma_k}}{\lambda_N(k)}}{N}}{n \in \mathbb{N}} \;<\; \infty, \] from which condition (2) follows. \end{proof}
\begin{example} The Orlicz function \[ M(t) \;=\; \begin{cases}\ensuremath{\mathrm{e}}^{-\frac{1}{t}} & 0 < t \leqslant \frac{1}{2} \\ 0 & t=0,\end{cases} \] (and extended suitably for $t>\frac{1}{2}$) satisfies the condition in Theorem \ref{Orliczcondition}. Given $K>1$, we have \[ M\bigg(\frac{M^{-1}\big(\frac{1}{n}\big)}{K} \bigg) \;=\; \ensuremath{\mathrm{e}}^{-K\log(n)} \;=\; n^{-K}. \tag*{\qedsymbol} \] \end{example}
\section{A class of $C(K)$ spaces having approximable norms}
It is well known that if $\Gamma$ is infinite then $c_0(\Gamma)$ is isomorphic to $C(\Gamma \cup \{\infty\})$, where $\Gamma \cup \{\infty\}$ is the 1-point compactification of $\Gamma$ endowed with the discrete topology. In this section, we use Corollary \ref{cor_main_tool_2} to present a class of compact scattered (Hausdorff) spaces $K$ having the property that any equivalent norm on $C(K)$ can be approximated by both $C^\infty$-smooth norms and polyhedral norms. This class contains, for every ordinal $\alpha$, a space $K$ such that the Cantor-Bendixson derivative $K^{(\alpha)}$ of order $\alpha$ is non-empty. This result was motivated by the following problem, which remains open (see \cite{hp:14} for related results).
\begin{prob}\label{prob_density_omega_1} Let $\alpha \geqslant \omega_1$. Can every equivalent norm on $C(\alpha+1)$ be approximated by $C^1$-smooth norms? \end{prob}
It will be more convenient for us to work with locally compact scattered spaces in the main. Given such a space $M$, we identify the dual space $C_0(M)^*$ with $\ell_1(M)$ in its natural norm $\pndot{1}$. We define a \emph{tree} to be a partially ordered set, such that the set of predecessors of each element of the tree is well-ordered. Our class consists of trees that are locally compact with respect to a certain topology. Given an ordinal $\eta$, let $q(\eta)$ denote the unique ordinal having the property that \[ \omega q(\eta) \;\leqslant\; \eta \;<\; \omega(q(\eta)+1). \] Given ordinals $\alpha$ and $\eta$, define the sets \[ M_\alpha \;=\; \set{\map{t}{n}{\omega\alpha}}{\text{$1\leqslant n < \omega$ and $t(i+1)<\omega q(t(i))$ whenever $i+1<n$}}, \] and \[ K_\eta \;=\; \set{(\eta)^\frown t}{t \in M_{q(\eta)} \cup \{\varnothing\}} \;\subseteq\; M_{q(\eta)+1}, \]
where $n<\omega$ is treated in the definition of $M_\alpha$ as the set of ordinals strictly preceeding $n$, and where $^\frown$ denotes concatenation of sequences. We make $M_\alpha$ into a tree by partially ordering it with respect to end-extension:~$s \preccurlyeq t$ if and only if $|s| \leqslant |t|$ and $t\restrict{|s|}\, = s$, where $|t| := \dom t$. Given $t \in M_\alpha$, the condition $t(i+1)<\omega q(t(i))$ whenever $i+1<|t|$ implies that $t$ is a strictly decreasing sequence of ordinal numbers. It follows that $M_\alpha$ is \emph{well-founded}, that is, it does not admit any infinite totally ordered subsets.
We proceed to equip $M_\alpha$ with a topology. Given $s \in M_\alpha$ and a finite set $F \subseteq s(|s|-1)$, we define the set \[
U_{s,F} \;=\; \{s\} \cup \set{t \in M_\alpha}{s \prec t \text{ and }t(|s|) \notin F}, \]
(here, as above, $s(|s|-1)$ is treated as the set of ordinals strictly preceeding the number itself). The family $\mathscr{U}_s :=\set{U_{s,F}}{F \subseteq s(|s|-1) \text{ is finite}}$ will be a local base of neighbourhoods of $s$, and $\bigcup_{s \in M_{\alpha}} \mathscr{U}_s$ a base of our topology on $M_\alpha$. This topology agrees with the so-called \emph{coarse wedge topology}, that can be defined on an arbitrary tree. Using the Alexander Subbase Theorem, it can be shown that $M_\alpha$ is locally compact with respect to this topology \cite{n:97}. Moreover, if $\eta<\omega\alpha$, then $K_\eta$ is a compact open subtree of $M_\alpha$.
In order to eliminate a potential source of confusion, we should point out that the coarse wedge topology is in general strictly finer than another (quite commonly used) locally compact topology with which one can endow a tree, namely the \emph{interval topology}. This topology has been used in a number of results in renorming theory, including \cite[Theorem 10]{fpst:08}, which characterises exactly when an equivalent polyhedral norm exists on $C_0(T)$, where $T$ is a tree. Since the coarse wedge topology is different, that result is not comparable with the work contained in this section.
Recall that, given a scattered locally compact space $M$, the \emph{scattered height} of $M$ is the least ordinal $\Omega$ for which $K^{(\Omega)}$ is empty. It is easy to compute the Cantor-Bendixson derivatives of the spaces $M_\alpha$.
\begin{prop}Given an ordinal $\xi$, we have \[
M_\alpha^{(\xi)} \;=\; \set{t \in M_\alpha}{\xi \leqslant q(t(|t|-1))}. \] Consequently, the scattered height of $M_\alpha$ equals $\alpha$. \end{prop}
\begin{proof} We proceed by transfinite induction on $\xi$. The result is obvious if $\xi=0$. Suppose that it holds for $\xi$. Let $t \in M_\alpha^{(\xi)}$. If $\xi = q(t(|t|-1))$ then $t$ must be maximal in $M_\alpha^{(\xi)}$, because it is impossible to strictly extend $t$ to $u \in M_\alpha^{(\xi)}$ in such a way that $u(|t|) < \omega q(u(|t|-1)) = \omega \xi$. Instead, if $\xi+1 \leqslant q(t(|t|-1))$ then $t^\frown(\omega\xi+n)$, $n < \omega$, are all strict extensions of $t$ in $M_\alpha^{(\xi)}$, and thus $t \in M_\alpha^{(\xi+1)}$. The result is therefore true for $\xi+1$. The limit ordinal case follows easily. \end{proof}
By \cite[Theorem 3.8]{lancien:95}, given a locally compact space $M$, if $C_0(M)$ is isomorphic to a subspace of $c_0(\Delta)$ for some set $\Delta$, then the scattered height of $M$ equals $n$ for some $n<\omega$. Therefore, provided $\alpha \geqslant \omega$, $C_0(M_\alpha)$ cannot embed isomorphically into any such $c_0(\Delta)$.
Given a locally compact space $M$, let $A(M)$ be the subspace $\lspan\set{\delta_t}{t \in M}\subseteq C_0(M)^*$ of finite linear combinations of Dirac functionals on $M$. All $w^*$-relatively discrete subsets of dual spaces are $w^*$-LRC \cite[Example 6 (1)]{fpst:14}. Therefore, if $M$ is a \emph{$\sigma$-discrete} set, that is, the countable union of countably many relatively discrete subsets, then $\set{\delta_t}{t \in M} \cup \{0\}$ is a $w^*$-compact and $\sigma$-$w^*$-LRC subset of $C_0(M)^*$. Consequently, $A(M)$ is $\sigma$-$w^*$-compact and $\sigma$-$w^*$-LRC, by Theorem \ref{thm_linspan}. It is easy to see that the set of elements of $M_\alpha$ having length $n$ is relatively discrete, thus each $M_\alpha$ is $\sigma$-discrete and so $A(M_\alpha)$ is $\sigma$-$w^*$-LRC and $w^*$-$K_\sigma$.
\begin{thm}\label{thm_tree} Let $\alpha$ be an ordinal. Then every equivalent norm on $C_0(M_\alpha)$ can be approximated by both $C^\infty$-smooth norms and polyhedral norms. \end{thm}
We set up some more machinery in order to prove Theorem \ref{thm_tree}. We will need to consider some isomorphisms. The isomorphisms stated in the result below are well known. We only include an explicit isomorphism in the proof in order to establish the additional connection between the subspaces of the duals.
\begin{prop}\label{prop_unif_bdd} Let $K$ be a compact space admitting a convergent sequence of distinct points. Then given $u \in K$, there exists a surjective isomorphism $\map{S}{C(K)}{C_0(K\setminus\{u\})}$, such that $S^*A(K\setminus\{u\})=A(K)$. \end{prop}
\begin{proof} Let $(t_n)_{n=1}^\infty$ be a convergent sequence of distinct points and let $u \in K$. Without loss of generality, we assume that $u \neq t_n$ for all $n\in\mathbb{N}$. The space $C_0(K\setminus\{u\})$ identifies isometrically with the hyperplane \[ X \;=\; \set{f \in C(K)}{f(u)=0}, \] which we will work with instead. Let $U_n$, $n \in \mathbb{N}$, be a collection of pairwise disjoint open sets such that $t_n \in U_n$ and $u \notin U_n$ for all $n$, and let $\map{\phi_n}{K}{[0,1]}$ be continuous functions satisfying $\phi_n(t_n)=1$ and $\phi_n(s)=0$ whenever $s \in K\setminus U_n$. We have $\phi_n \in X$ for all $n$. Define $\map{S}{C(K)}{X}$ by \[ Sf \;=\; f - f(u)\mathbf{1}_K - f(t_0)\phi_0 + \sum_{n=1}^\infty (f(t_{n-1})-f(t_n))\phi_n. \] The infinite sum is an element of $X$ because the $U_n$ are pairwise disjoint and $f(t_{n-1})-f(t_n) \to 0$ as $n \to \infty$. We see that, given $\delta_s \in X^*$, $s \in K\setminus\{u\}$, \[ S^*\delta_s \;=\; \delta_s - \delta_u - \phi_0(s)\delta_{t_0} + \sum_{n=1}^\infty \phi_n(s)(\delta_{t_{n-1}}-\delta_{t_n}). \] The apparently infinite sum is really a finite sum, because $\phi_n(s)$ can be non-zero for at most one $n$. Hence $S^*A(K\setminus\{u\}) \subseteq A(K)$.
We have $Sf(t_0)=-f(u)$ and $Sf(t_n) = f(t_{n-1})$ whenever $n \geqslant 1$. Using these facts, it is easy to verify that $S^{-1}$ exists and equals $\map{R}{X}{C(K)}$, where \[ Rg \;=\; g-g(t_0)\mathbf{1}_K + g(t_1)\phi_0 + \sum_{n=1}^\infty (g(t_{n+1})-g(t_n))\phi_n. \] Likewise, it can be shown that $R^*\delta_s$ is finitely supported for all $s \in K$, so $R^*A(K) \subseteq A(K\setminus\{u\})$. \end{proof}
Given an infinite compact scattered space $K$, it is obvious that there exists a sequence of isolated points that converges to an element of the first derivative $K'$. Hence the proposition above applies to any such $K$.
Fix an ordinal $\alpha\geqslant 1$. We wish to apply Corollary \ref{cor_main_tool_2} to the space $C_0(M_\alpha)$. This means that we need to build an appropriate system of projections, an admissible family of subspaces (in the sense of Definition \ref{defn_admissible_set}), and ensure that the corresponding function $\theta$ is always finite.
We begin by setting up the projections. Given a non-empty clopen (that is, closed and open) subset $L$ of a locally compact space $M$, we define a projection $P_L$ on $C_0(M)$ by $P_Lf = f\cdot \mathbf{1}_L$, where $\mathbf{1}_L$ is the characteristic function of $L$. It could be that $M$ is a clopen subset of another locally compact space $M'$, and if so, we will use the same notation $P_L$ for the corresponding projections on $C_0(M)$ and $C_0(M')$.
It is evident that $M_\alpha$ is the discrete union of pairwise disjoint compact open sets \[ M_\alpha \;=\; \bigcup_{\eta < \omega\alpha} K_\eta, \] meaning that $C_0(M_\alpha)$ is naturally isometric to the $c_0$-sum of the spaces $C(K_\eta)$, $\eta<\omega\alpha$. Given $\eta<\omega\alpha$, let $P_\eta = P_{K_\eta}$. Having in mind the decomposition of $M_\alpha$, it is clear that $(P_\eta)_{\eta < \omega\alpha}$ fulfills properties (1)--(4) listed at the start of Section \ref{sect_approx_frame}. We will take advantage of the fact that the images $P_\eta C_0(M_\alpha)$ and $P_\eta^* C_0(M_\alpha)^*$ are naturally isometric to $C(K_\eta)$ and its dual, respectively.
Next, we determine $\theta$ and establish that it is always finite. Let $\mu \in C_0(M_\alpha)^*$. Given finite $F \subseteq \omega\alpha$, the definition of $\rho$ in (\ref{defn_rho}) becomes \[
\rho(F) \;=\; |F|, \] because $\pn{\sum_{\eta \in F}\mu_\eta}{1}=\sum_{\eta \in F}\pn{\mu_\eta}{1}$. Moreover, in this case (\ref{eqn_h_less_than_theta}) becomes \[
\pn{h_m(\mu)}{1} \;=\; \sum_{k=1}^m q_k(\mu)\pn{\omega_k(\mu)}{1} \;=\; \sum_{k=1}^m q_k(\mu)|G_k(\mu)| \;=\; \sum_{k=1}^m q_k(\mu)\rho(G_k(\mu)). \] By Lemma \ref{lem_theta_dominate}, the left hand side converges to $\pn{\mu}{1}$, which means that $\theta(\mu) = \pn{\mu}{1}$ and is therefore always finite, regardless of the system of projections with respect to which it is defined. For this reason, we shall remove $\theta$ from all subsequent arguments.
Finally, we define a family of subspaces that we claim is admissible. Observe that \[ A(K_\eta) \;\subseteq\; \cl{\lspan}^{\ensuremath{\left\|\cdot\right\|}}(\delta_t)_{t \in K_\eta}\;=\; C(K_\eta)^* \;\equiv\; P^*_\eta C_0(M_\alpha)^*, \] whenever $\eta<\omega\alpha$. The family we consider is $(A(K_\eta))_{\eta<\omega\alpha}$.
\begin{proof}[Proof of Theorem \ref{thm_tree}] We use transfinite induction on $\alpha\geqslant 1$ to show that $(A(K_\eta))_{\eta<\omega\alpha}$ is an admissible family on $C_0(M_\alpha)$. This is trivial in the case $\alpha=1$, because $K_\eta$ is the singleton $\{(\eta)\}$ and $A(K_\eta)=P^*_\eta(M_1)^*$ whenever $\eta<\omega$. Limit cases are straightforward. Assume that $\alpha$ is a limit ordinal and that $(A(K_\eta))_{\eta<\omega\xi}$ is admissible on $C_0(M_\xi)$ whenever $1 \leqslant \xi < \alpha$. Given a finite set $F \subseteq \omega\alpha$, there is an ordinal $\xi$ such that $1\leqslant \xi < \alpha$ and $F \subseteq \omega\xi$. Hence we can apply the admissibility of $(A(K_\eta))_{\eta<\omega\xi}$ on $C_0(M_\xi)$, which embeds naturally inside $C_0(M_\alpha)$, to see that the conditions in Definition \ref{defn_admissible_set} are fulfilled.
The successor case requires more work. Assume that $\alpha\geqslant 1$ and that $(A(K_\eta))_{\eta<\omega\alpha}$ is admissible on $C_0(M_\alpha)$. Fix a finite set $F \subseteq \omega(\alpha+1)$. There is nothing to prove if $F\subseteq \omega\alpha$, so we assume that $F\setminus \omega\alpha$ is non-empty. Define the subspaces \begin{align*} V\;&=\; \lspan(P_\lambda C_0(M_{\alpha+1}))_{\lambda \in F} \;\equiv\; C\bigg(\bigcup_{\lambda \in F} K_\lambda\bigg),\\ W\;&=\; \lspan(P^*_\lambda C_0(M_{\alpha+1})^*)_{\lambda \in F}, \text{ and}\\ A\;&=\; \lspan(A(K_\lambda))_{\lambda \in F} \;=\; A\bigg(\bigcup_{\lambda \in F} K_\lambda\bigg). \end{align*} It is clear that $W$ is naturally isometric to $V^*$, so we regard them as equal.
Let $\ensuremath{\varepsilon}>0$, and let $C \subseteq W$ be a $w^*$-compact set that is norming with respect to $V$. In order to get the set $D$ we need to fulfil Definition \ref{defn_admissible_set}, we will define a surjective isomorphism $\map{T}{V}{C_0(M_\alpha)}$ such that \begin{equation}\label{eqn_C(K)_condition} T^*A(M_\alpha) \;=\; A\bigg(\bigcup_{\lambda \in F} K_\lambda\bigg) \;=\; A. \end{equation} Then $(T^*)^{-1}C \subseteq C_0(M_\alpha)^*$ is $w^*$-compact and norming. As $(A(K_\eta))_{\eta<\omega\alpha}$ is admissible, by Theorem \ref{thm_main_tool}, there exists $D \subseteq W$ such that \[
(T^*)^{-1}C \;\subseteq\; \cl{(T^*)^{-1} D}^{w^*} \subseteq (T^*)^{-1}C + \|T\|^{-1}\ensuremath{\varepsilon} B_{C_0(M_\alpha)^*} \] and \[ \cl{(T^*)^{-1} D}^{w^*} \cap A(M_\alpha) \text{ is a boundary of } \cl{(T^*)^{-1} D}^{w^*}. \] Using (\ref{eqn_C(K)_condition}), it follows that \[ C \;\subseteq\; \cl{D}^{w^*} \;\subseteq\; C + \ensuremath{\varepsilon} B_W \qquad\text{and}\qquad \cl{D}^{w^*} \cap A \text{ is a boundary of } \cl{D}^{w^*}, \] which is what we need in order to satisfy Definition \ref{defn_admissible_set}.
All that remains is to construct the isomorphism $T$. Let $G=F \cap \omega\alpha$, $H=F\setminus \omega\alpha$ and choose a bijection \[ \map{\pi}{G \cup (H \times \omega\alpha)}{\omega\alpha}, \] such that $\pi(\lambda)=\lambda$ whenever $\lambda \in G$, and $q(\pi(\lambda,\eta))=q(\eta)$ whenever $(\lambda,\eta) \in H \times \omega\alpha$.
Given $\lambda \in H$, define the discrete union \[ L_\lambda \;=\; \bigcup_{\eta < \omega\alpha} K_{\pi(\lambda,\eta)}, \] which is a clopen subset of $M_\alpha$. Because $q(\pi(\lambda,\eta))=q(\eta)$ whenever $\eta < \omega\alpha$, it follows that the map \[ (\eta)^\frown t \;\mapsto\; (\pi(\lambda,\eta))^\frown t, \qquad\qquad t \in M_{q(\eta)} \cup \{\varnothing\}, \] is a homeomorphism of $K_\eta$ and $K_{\pi(\lambda,\eta)}$. Gluing together these homeomorphisms for $\eta < \omega\alpha$ yields a homeomorphism of $M_\alpha$ and $L_\lambda$. Moreover, since $q(\lambda)=\alpha$, the map \[ (\lambda)^\frown t \;\mapsto\; t, \qquad\qquad t \in M_\alpha, \] is a homeomorphism of $K_\lambda\setminus\{(\lambda)\}$ and $M_\alpha$. Together with Proposition \ref{prop_unif_bdd}, this means that there exists a surjective isomorphism \[ \map{S_\lambda}{P_\lambda C_0(M_{\alpha+1}) \equiv C(K_\lambda)}{P_{L_\lambda}C_0(M_\alpha) \equiv C_0(L_\lambda)}, \] such that $S^*_\lambda A(L_\lambda) = A(K_\lambda)$.
Finally, the properties of $\pi$ imply \[ \bigcup_{\lambda \in G} K_\lambda \cup \bigcup_{\lambda \in H} L_\lambda \;=\; M_\alpha. \] Define $\map{T}{V}{C_0(M_\alpha)}$ by \[ T \;=\; \sum_{\lambda \in G} P_\lambda + \sum_{\lambda \in H} S_\lambda P_\lambda. \] This has inverse \[ T^{-1} \;=\; \sum_{\lambda \in G} P_\lambda + \sum_{\lambda \in H} S_\lambda^{-1} P_{L_\lambda}. \] Given $t \in M_\alpha$, we have $T^*\delta_t \in A(K_\lambda)$ if $t \in K_\lambda$ and $\lambda \in G$, and $T^*\delta_t \in A(K_\lambda)$ if $t \in L_\lambda$ and $\lambda \in H$. Thus $T^*A(M_\alpha) \subseteq A$. Arguing similarly using the inverse map yields equality, hence we have satisfied (\ref{eqn_C(K)_condition}), as required. The proof is complete. \end{proof}
Thus we can apply Corollary \ref{cor_main_tool_2} to the spaces $C_0(M_\alpha)$ for all $\alpha \geqslant 1$, as promised. Our initial class, as advertised, consisted of compact spaces. Such a class can be easily arranged by considering $K_{\omega\alpha}$, which is homeomorphic to the 1-point compactification of $M_\alpha$ and thus has Cantor-Bendixson height $\alpha+1$.
We end this section with a problem. Note that if $K$ is a compact space such that $K^{(3)}$ is empty, then $C(K)$ admits an equivalent norm having a locally uniformly rotund dual norm \cite[Theorem VII.4.7]{dgz:93}. According to \cite[Theorem II.4.1]{dgz:93}, every equivalent norm on $C(K)$ can be approximated by such a norm, and all such norms are $C^1$-smooth \cite[Proposition I.1.5]{dgz:93}.
\begin{prob}\label{prob_K^2} Let $K$ be a compact space such that $K^{(3)}$ is empty? Can every equivalent norm on $C(K)$ be approximated by $C^2$-smooth norms or polyhedral norms? \end{prob}
\end{document}
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\title[Multiplicative unimodality for stable laws]
{Multiplicative strong unimodality for positive stable laws}
\author[Thomas Simon]{Thomas Simon}
\address{Laboratoire Paul Painlev\'e, U. F. R. de Math\'ematiques, Universit\'e de Lille 1, F-59655 Villeneuve d'Ascq Cedex. {\em Email} : {\tt [email protected]}}
\keywords{Beta and Gamma variables - Log-concavity - Positive stable law - Strong unimodality}
\subjclass[2000]{60E07, 60E15}
\begin{abstract} It is known that real Non-Gaussian stable distributions
are unimodal, not additive strongly unimodal, and multiplicative
strongly unimodal in the symmetric case. By a theorem of
Cuculescu-Theodorescu \cite{CT}, the only remaining relevant situation for the
multiplicative strong unimodality of stable laws is the one-sided. In this paper, we show that positive $\alpha-$stable laws are multiplicative strongly unimodal iff $\alpha\le 1/2.$ \end{abstract}
\maketitle
\section{The MSU property and stable laws}
A real random variable $X$ is said to be unimodal (or quasi-concave) if there exists $a\in{\mathbb{R}}$ such that the functions ${\mathbb{P}}[X\le x]$ and ${\mathbb{P}}[X>x]$ are convex respectively in $(-\infty,a)$ and $(a,+\infty).$ If $X$ is absolutely continuous, this means that its density increases on $(-\infty,a]$ and decreases on $[a,+\infty).$ The number $a$ is called a mode of $X$, and might not be unique. A well-known example due to Chung shows that unimodality is not stable under convolution, and for this reason the notion of strong unimodality had been introduced in \cite{I}: a real variable $X$ is said to be strongly unimodal if the independent sum $X + Y$ is unimodal for any unimodal variable $Y$ (in particular $X$ itself is unimodal, choosing $Y$ degenerated at zero). In \cite{I} Ibragimov also obtained the celebrated criterion that $X$ is strongly unimodal iff it is absolutely continuous with a log-concave density.
Proving unimodality or strong unimodality properties is simple for variables with given densities, but the problem might turn out complicated when these densities are not explicit. In this paper we will deal with real (strictly) stable variables, where very few closed formulae (given e.g. in \cite{Z} p. 66) are known for the densities. A classical theorem of Yamazato shows that they are all unimodal with a unique mode - see Lemma 1 in \cite{IC} for the previously shown one-sided case and Theorem 53.1 in \cite{S} for the general result, but except in the Gaussian situation it is easy to see that stable laws are not strongly unimodal, because their heavy tails prevent the densities from being everywhere log-concave - see Remark 52.8 in \cite{S}.
Having infinitely divisible distributions, stable variables appear naturally in additive identities, a framework where they hence may not preserve unimodality. Stable variables also occur in multiplicative factorizations as a by-product of the so-called $M$-scheme (or $M$-infinite divisibility), a feature which has been studied extensively by Zolotarev - see Chapter 3 in \cite{Z} and also \cite{P} for the one-sided case. Another concrete example of a multiplicative identity involving a stable law is the following. Suppose that $X, Y$ are two positive variables whose Laplace transforms are connected through the identity ${\mathbb{E}} [e^{-\lambda X}] \, = \, {\mathbb{E}} [e^{-\lambda^\alpha Y}]$ for some $\alpha \in (0,1)$ (up to a reformulation one could also take $\alpha >1$). Then one has $$ X\; \stackrel{d}{=} \; Z_\a\;\times\; Y^{1/\alpha}$$ where $Z_\a$ is an independent positive $\alpha-$stable variable. In such identities, one can be interested in understanding whether the factorization through $Z_\a$ modify or not some basic distributional properties.
From the point of view of unimodality, it is therefore natural to ask whether stable variables are multiplicative strongly unimodal, in other words, whether their independent product with any unimodal variable remains unimodal, or not. A quick answer can be given in the symmetric case because the mode of a symmetric stable variable is obviously zero: Khintchine's theorem entails then that its product with {\em any} independent variable will be unimodal with mode at zero - see Proposition 3.6 in \cite{CT} for details. However, in the non-symmetric case the mode of a stable variable is never zero (this is obvious in the drifted Cauchy case $\alpha = 1$ and we refer to \cite{Z} p. 140 for the case $\alpha\neq 1$), so that such a simple argument cannot be applied. The following criterion established in \cite{CT} Theorem 3.7, is a multiplicative counterpart to Ibragimov's theorem:
\begin{THEO}[Cuculescu-Theodorescu] Let $X$ be a unimodal random variable such
that 0 is not a mode of $X$. Then $X$ is multiplicative strongly unimodal if and only if it is one-sided and absolutely continuous, with a density $f^{}_{X}$ having the property that \begin{equation} \label{CCTD} t\;\mapsto\; f^{}_{X}(e^t) \quad\mbox{is log-concave in ${\mathbb{R}}$} \end{equation} when $X$ is positive (resp. $t\,\mapsto\, f^{}_{X}(-e^t)$ is log-concave in ${\mathbb{R}}$ when $X$ is negative). \end{THEO} With a slight abuse of notation, in the following we will say that a positive random variable is {\em MSU\,} if and only if (\ref{CCTD}) holds. Cases of multiplicative strongly unimodal, positive variables with mode at zero and such that (\ref{CCTD}) does not hold, are hence excluded in this definition. But these cases are particular, and by the above remark no more relevant to the content of this paper which deals with stable laws. Besides, a change of variable and Ibragimov's theorem entail the following useful characterization for positive variables: \begin{equation} \label{Ibr} \mbox{$X$ is MSU}\;\;\Longleftrightarrow\;\; \mbox{$\log X$ is strongly unimodal.} \end{equation} In particular the MSU property is stable by inversion and also, from Pr\'ekopa's theorem, by independent multiplication. Another important feature coming from (\ref{Ibr}) is that the MSU property remains unchanged under rescaling and power transformations, which also comes from the obvious analytical fact that (\ref{CCTD}) holds iff $t \mapsto K_1 e^{a_1 t} f^{}_X (K_2 e^{a_2 t})$ is log-concave for some $a_1\in{\mathbb{R}}$ and $a_2, K_1, K_2 \in{\mathbb{R}}^*.$ Notice however that the MSU property is barely connected to the strong unimodality of $X$ itself (several examples of this difference are given in \cite{CT}).
In this paper we are interested in the MSU property for positive $\alpha-$stable laws. For every $\alpha \in\, ]0,1[,$ consider $f_\a$ the positive $\alpha-$stable density and $Z_\a$ the corresponding random variable, normalized such that \begin{equation} \label{Laplace} \int_0^\infty e^{-\lambda t} f_\a(t) dt\; =\; {\mathbb{E}}\left[ e^{-\lambda Z_\a}\right] \; =\; e^{-\lambda^\alpha}, \qquad \lambda \ge 0. \end{equation} Before studying the MSU property for $Z_\a$, in view of (\ref{Ibr}) one must first ask if $\log Z_\a$ is simply unimodal. A positive answer for all $\alpha \in\, ]0,1[$ had been given by Kanter - see Theorem 4.1 in \cite{K2}, who also deduced from \cite{IC} the decomposition $$\log Z_\a\; =\; \alpha^{-1}\log b_\a(U)\; +\; (\alpha-1)/\alpha\log L$$ where $L$ is a standard exponential variable, $U$ an independent uniform variable over $[0,\pi],$ and $b_\alpha(u) = (\sin(\alpha u)/\sin (u))^\alpha(\sin((1-\alpha) u)/\sin(u))^{1-\alpha}.$ The random variable $\log L$ is easily seen to be strongly unimodal, but $\log b_\a(U)$ is not - at least for $\alpha = 1/2,$ see the Remark before Theorem 4.1 in \cite{K}. This leaves the question of the MSU property for positive stable laws unanswered, and our result aims at filling this gap:
\begin{THEO} The variable $Z_\a$ is MSU if and only if $\alpha \le 1/2.$ \end{THEO}
To conclude this introduction, let us give two further reformulations of (\ref{CCTD}) in the positive stable case. The first one lies at the core of our proof, whereas the second one is probably nothing but a mere curiosity. Since $f_\a$ is smooth, differentiating twice the logarithm entails that (\ref{CCTD}) is equivalent to the inequality \begin{equation} \label{LCE} (x^2 f_\alpha''(x) + xf_\alpha'(x))f_\alpha(x)\; \le \; x^2(f_\alpha'(x))^2, \quad x\ge 0. \end{equation} If $m_\alpha$ stands now for the mode of $Z_\a$ and if $x_\alpha = \inf \{ x > m_\alpha, \; f_\a''(x) = 0\},$ then we know from (53.13) in \cite{S} - an identity which is proved there for a certain class of positive self-decomposable distributions, but it can also be obtained for positive stable laws after taking the weak limit - and the discussion thereafter that (\ref{LCE}) is true for any $x\in [0, x_\alpha].$ Hence, the MSU property amounts to the fact that it remains true for all $x > x_\alpha$. From (53.13) in \cite{S} we also know that (\ref{LCE}) is equivalent to the positivity everywhere of the function $$x\;\mapsto\; \int_0^x (f_\a'(x-y)f_\a(x) - f_\a(x-y)f_\a'(x))y^{-\alpha}dy,$$ but this criterion is not very tractable because of the long memory involved in the integral. Thanks to the Humbert-Pollard representation for $f_\a$ which will be recalled at the beginning of Section 3, we finally mention that (\ref{LCE}) is equivalent to $$\left(\sum_{n\ge 1} (1+\alpha n)^2 \frac{(-1)^n x^{-(1 +\alpha n)}}{\Gamma (-n\alpha)n!} \right)\left(\sum_{n\ge 1} \frac{(-1)^n x^{-(1 +\alpha n)}}{\Gamma (-n\alpha)n!} \right)\; \le\; \left(\sum_{n\ge 1} (1+\alpha n) \frac{(-1)^n x^{-(1 +\alpha n)}}{\Gamma (-n\alpha)n!} \right)^2,$$ a strange inequality which would plainly hold in the opposite direction if the terms of the series had constant signs.
\section{Some particular cases}
In this section we depict some situations where the MSU property can be proved or disproved directly, thanks to more or less explicit representations for the density $f_\a$ or the variable $Z_\a.$ First of all, in the case $\alpha = 1/2$ it is readily seen from the known formula $$f_{1/2}(x)\; =\; \frac{1}{2\sqrt{\pi x^3}}e^{-1/4x}$$ that $Z_{1/2}$ is MSU. When $\alpha = 1/3$, Formula (2.8.31) in \cite{Z} yields $$f_{1/3}(x)\; =\; \frac{1}{3\pi x^{3/2}} K_{1/3}\left( \frac{2}{3\sqrt{3}} x^{-1/2}\right)$$ where $K_{1/3}$ is the Macdonald function of order 1/3, so that (\ref{CCTD}) amounts to show that $t\mapsto K_{1/3} (e^t)$ is log-concave. From (\ref{LCE}) and since $K_{1/3}$ is a solution to the modified Bessel equation $$x^2 K_{1/3}'' \;+\; x K_{1/3}'\; =\; (x^2 + 1/9)K_{1/3}$$ over ${\mathbb{R}}$, this is equivalent to $(x^2 + 1/9)K_{1/3}^2(x)\,\le \, x^2( K_{1/3}'(x))^2$ for every $x\ge 0.$ Because $K_{1/3}'(x) < 0$ for every $x \ge 0,$ the latter is now an immediate consequence of a Turan-type inequality for modified Bessel functions recently established in \cite{Ba} - see (2.2) therein, so that $Z_{1/3}$ is MSU.
With a probabilistic and more concise argument, one can also show that Property (\ref{CCTD}) holds for every $\alpha = 1/p, \, p \ge 2.$ A classical representation originally due to E.~J.~Williams - see Section 2 in \cite{W} - shows indeed that after rescaling $Z_{1/p}^{-1}$ is an independent product of $p$ Gamma variables: $$Z_{1/p}^{-1}\; \stackrel{d}{=}\; p^p\;Y_{1/p}\;\times\; Y_{2/p}\;\times\;\cdots\;\times\; Y_{(p-1)/p}$$ where each $Y_{k/p}$ is MSU since its density is $\Gamma(k/p)^{-1} y^{k/p -1} e^{-y}{\bf 1}_{\{y > 0\}}.$ Hence, since the MSU property is stable by inversion and independent multiplication, it follows that $Z_{1/p}$ is MSU for every $p\ge 2.$
As a matter of fact, a much stronger property is known to hold when $\alpha = 1/p$ for some $p \ge 2.$ In theses cases, it had namely been noticed in \cite{K} thanks to the classical computation of the fractional moments \begin{equation} \label{MellinS} {\mathbb{E}}[Z_\a^s]\; =\; \frac{\Gamma(1 - s/\alpha)}{\Gamma(1-s)} \end{equation} for every $s < \alpha,$ and the duplication formula for the Gamma function, that the kernel $(x,y)\mapsto f_\alpha(e^{x-y})$ is totally positive - see pp. 121-122 and 390 in \cite{K} as well as pp. 11-12 therein for the definition of total positivity. In particular, it is totally positive of order 2 (${\rm TP}_2$) which means precisely that $x\mapsto f_\alpha(e^x)$ is a log-concave function - see e.g. Theorem 4.1.9 in \cite{K} Chap. 4. However, when $\alpha$ is not the reciprocal of an integer, the function $$s\; \mapsto\;\frac{\Gamma(1 -s)}{\Gamma(1-s/\alpha)}$$ is not an entire function of the type ${\mathcal E}_2^*$ - see e.g. \cite{K} p. 336 for a definition - and by Theorem 7.3.2 in \cite{K}, this entails that $(x,y) \mapsto f_\alpha(e^{x-y})$ is no more a totally positive kernel. Karlin raised then the question whether or not it should be totally positive of some finite order - see \cite{K} p. 390, a problem which seems as yet unadressed.
Let us finally show that $Z_{2/3}$ is not MSU, in other words that $(x,y) \mapsto f_{2/3}(e^{x-y})$ is not a ${\rm TP}_2$ kernel. Formula (2.8.33) in \cite{Z} (with a slight normalizing correction therein) yields first the expression $$f_{2/3}(x)\; =\; \sqrt{\frac{3}{\pi x}}\, e^{-2/27x^2} W_{1/2, 1/6} \left( \frac{4}{27} x^{-2}\right)$$ where $W_{1/2, 1/6}$ is a Whittaker function. Hence, from formul\ae \,(6.9.2) and (6.5.2) in \cite{E} we see that (\ref{CCTD}) is equivalent to the log-concavity of $t\mapsto g(e^t)$ with $g(x) = e^{-x} U_4(x)$ and the notation $U_\lambda(x) = \Psi(1/6, \lambda/3, x)$ for all $\lambda >1/2,$ where $$\Psi(a, c, x)\; =\; \frac{1}{\Gamma(1/6)}\int_0^\infty e^{-xs}s^{a-1}(1+s)^{c-a-1} ds$$ is a confluent hypergeometric function ($c > a >0$). We readily see that $g'(x) = - e^{-x}U_7(x)$ and $g''(x) = e^{-x}U_{10}(x),$ so that by (\ref{LCE}) the MSU property for $X_{2/3}$ amounts to $$(x^2 U_{10}(x) - x U_{7}(x)) U_4(x) \; \le\; x^2 U_7^2(x), \quad x\ge 0.$$ Using twice the contiguity relation (6.6.5) in \cite{E} and some simple transformations, we then find the equivalence condition \begin{equation} \label{Whitt} (xU_4(x) - U_1(x)/6)(U_7(x) - U_4(x))\;\ge\; -5U_4^2(x)/6, \quad x\ge 0. \end{equation} Notice that (\ref{Whitt}) is true as soon as $x\ge 1/6$ thanks to the obvious inequalities $U_7(x) \ge U_4(x)\ge U_1(x).$ However, some easy computations yield the asymptotics $$U_7(x)\sim \frac{\Gamma(4/3)}{\Gamma(1/6)}x^{-4/3}, \;\; U_4(x)\sim \frac{\Gamma(1/3)}{\Gamma(1/6)}x^{-1/3}\;\;\mbox{and}\;\; U_1(x) \to \frac{\Gamma(2/3)}{\Gamma(5/6)}$$
when $x\to 0^+,$ so that (\ref{Whitt}) does not hold anymore. This last discussion about the case $\alpha = 2/3$ may look tedious, all the more that the proof that the MSU property does not hold for any $\alpha > 1/2$ is quite simple as we will soon see. But this must be read as a preparatory example for the sequel, since we will need inequalities such as (\ref{Whitt}) for some confluent hypergeometric functions in order to show the MSU property when $\alpha\le 1/2,$ a fact which is more involved.
\section{Proof of the Theorem}
We begin with the only if part, and we will give three different arguments. The first one relies on the aforementioned Humbert-Pollard representation for $f_\a$ - see e.g. formula (14.31) in \cite{S}: $$f_\alpha (x)\; =\; \frac{1}{\pi} \sum_{n\ge 1} \frac{(-1)^{n-1}}{n!}\sin (\pi\alpha n) \Gamma(1+\alpha n) x^{-(1 +\alpha n)}\; =\; \sum_{n\ge 1} \frac{(-1)^n}{\Gamma (-n\alpha)n!} x^{-(1 +\alpha n)}.$$ Because $\alpha < 1$ this expansion may be differentiated term by term on $(0, + \infty),$ yielding $$x f_\alpha' (x)\; =\;\sum_{n\ge 1} \frac{(-1)^{n-1}}{\Gamma (-n\alpha)n!} (1+\alpha n)x^{-(1 +\alpha n)}$$ and $$x^2 f_\alpha'' (x) \; +\; x f_\alpha' (x)\; =\; \sum_{n\ge 1} \frac{(-1)^{n-1}}{\Gamma (-n\alpha)n!} (1+\alpha n)^2 x^{-(1 +\alpha n)}$$ for every $x > 0$ (these three series representations explain the reformulation of (\ref{LCE}) in terms of a reverse Cauchy-Schwarz inequality mentioned at the end of Section 1). Using the expansions up to $n = 2$ and the concatenation formula $\Gamma(z+1) = z\Gamma(z)$ we obtain, after some simplifications, $$x^2(f_\alpha')^2(x)\, - \, (x^2 f_\alpha''(x) + xf_\alpha'(x))f_\alpha(x)\; =\; \frac{\alpha^2 x^{-(2 + 3\alpha)}}{2\Gamma(-\alpha)\Gamma(-2\alpha)}\; +\; {\rm o}(x^{-(2 + 3\alpha)})$$ in the neighbourhood of infinity, which entails that (\ref{LCE}) does not hold when $\alpha > 1/2,$ since the leading term is then negative.
The second one hinges upon an expansion for the density $g_\alpha$ of the random variable $Y_\alpha = \log Z_\a,$ which had been obtained in \cite{BB} - see (3.5) and (6.4) therein - independently of the Humbert-Pollard formula: $$g_\alpha(t)\; =\; e^{-\alpha t - e^{-\alpha t}}\sum_{j \ge 0} b_j \alpha^{j+1} (-1)^jR_j (-e^{-\alpha t})$$ for every $t\in{\mathbb{R}},$ where the coefficients $b_j$ and $R_j(x)$ are defined through the entire series $$\frac{1}{\Gamma(1+z)}\; =\; \sum_{j\ge 0} b_j z^j\quad \mbox{and}\quad e^{z + xe^z}\; =\; \sum_{j\ge 0} \left( \frac{R_j(x) e^x}{j!}\right) z^j.$$ Besides, setting $$P_\alpha(x) \; =\; \sum_{j \ge 0} b_j \alpha^{j+1} (-1)^jR_j (-x)$$ for any $x> 0,$ we know from (6.3) in \cite{BB} that the series converge absolutely, and with exactly the same argument one can show that it can be differentiated term-by-term. On the other hand, simple computations entail that $g_\alpha$ is log-concave over ${\mathbb{R}}$ if and only if $$P_\alpha(x)^2 + x^2 (P'_\alpha(x))^2 \; \ge \; P_\alpha(x) (xP_\alpha'(x) + x^2P_\alpha''(x))$$ for every $x >0.$ Letting $x\to 0^+$ yields $$P_\alpha(x)^2 + x^2 (P'_\alpha(x))^2\; \sim\; xP_\alpha(0)^2\; =\; x\left(\frac{\alpha}{\Gamma(1-\alpha)}\right)^2$$ and $$P_\alpha(x) (xP_\alpha'(x) + x^2P_\alpha''(x))\; \sim\; xP_\alpha(0)P_\alpha'(0)\; =\; x\left(\frac{\alpha}{\Gamma(1-\alpha)}\right)^2\left( 1 - \frac{\Gamma(1-\alpha)}{\Gamma(1-2\alpha)}\right)$$ where the evaluations of $P_\alpha(0)$ and $P_\alpha'(0)$ rest upon the definition of $b_j$ and the fact that $R_j(0) = 1$ and $R_j'(0) = 2^j - 1.$ Similarly as above, we see that the second asymptotic is larger than the first one when $\alpha > 1/2.$
For the third and simplest argument, we will invoke an identity in law connecting two independent copies $Y_\alpha$ and ${\tilde Y}_\alpha$ of the random variable $\log Z_\a,$ which can be readily obtained in changing the variable in Exercise 4.21 (3) of \cite{CY}: \begin{equation} \label{ID} Y_\alpha\, -\, {\tilde Y}_\alpha\; \stackrel{d}{=}\; U_\alpha \end{equation} where $U_\alpha$ is a real random variable with density $$u_\alpha(x)\; =\; \frac{\sin \pi\alpha}{\pi(e^{\alpha x} + 2 \cos \pi\alpha + e^{-\alpha x})}\cdot$$ We compute then the second derivative of $x\mapsto \log (e^{\alpha x} + 2 \cos \pi\alpha + e^{-\alpha x})$ which is $$\frac{4\alpha^2 + \alpha^2\cos \pi\alpha \cosh \alpha x}{(e^{\alpha x} + 2 \cos \pi\alpha + e^{-\alpha x})^2}\cdot$$ Hence, we see that $U_\alpha$ has a log-concave density over ${\mathbb{R}}$ iff $\alpha \le 1/2.$ By Pr\'ekopa's theorem this entails that $\log Z_\a$ does not have a log-concave density over ${\mathbb{R}}$ when $\alpha > 1/2,$ which means that $Z_\a$ is not MSU.
\begin{REMS} {\em (a) This negative result shows that the kernel $(x,y)\mapsto
f_\alpha(e^{x-y})$ is not ${\rm TP}_2$ when $\alpha > 1/2,$ which contradicts the
affirmation made in Lemma 1 (iv) of \cite{G} that this kernel is strictly totally
positive for every $0<\alpha< 1$ (actually the contradiction could already have been seen
in reading \cite{K} p. 390 carefully). Notice that this latter affirmation seems crucial to obtain the so-called
bell-shape property for all $\alpha$-stable variables with index $\alpha < 1$ - see
p. 237 in \cite{G}. However, since this question is quite different from the topic of the present paper, we plan to tackle the problem (if really
any) in some further research.
\noindent (b) Though somewhat more technical, the methods resting upon Humbert-Pollard's and Brockwell-Brown's expansions give some insight on the location where the inequality (\ref{LCE}) breaks down, an information which could not have been obtained by the third argument. I had also believed for a long time that these two expansions would give the if part, but this still eludes me because of the alternate signs.
\noindent (c) It would be quite interesting to see if the third argument could not give the if part either. From the analytical viewpoint this would be the consequence of a positive answer to the following question. If $X$ is a real random variable with density such that the independent difference $X -X$ has a log-concave density, does $X$ have a log-concave density as well? This assertion, a kind of reverse to Pr\'ekopa's theorem for which we found neither references nor counterexamples in the literature, goes somehow the opposite way to the central limit theorem (which leads to a log-concave density after enough convolutions on any probability distribution with finite variance). Franck Barthe wrote me that he would be surprised if it held true in full generality. In our positive $\alpha-$stable framework, the fact that $\logZ_\a$ is unimodal (and log-concave at both infinities when $\alpha\le 1/2$) might add some crucial properties, but overall I could not find any clue in the direction of this statement.}
\end{REMS}
We now consider the if part, using yet another argument since the three above methods turned out fruitless. We first prove two lemmas which are of independent interest. The second one is a generalisation of Williams' result and could be formulated in several ways (we chose the one tailored to our purposes).
\begin{LEMO} Let $X$ be a ${\rm Beta}\, (\alpha, \beta)$ variable and $Y$ an
independent $\,{\rm Gamma}\, (c)$ variable such that $\beta \le 1$ and $\alpha + \beta \ge c.$ Then the product $X\times Y$ is MSU. \end{LEMO}
\noindent {\em Proof}: When $\alpha +\beta = c,$ the result follows easily without further assumption on $\beta$ because \begin{equation} \label{IDG} X\; \times\; Y \;\stackrel{d}{=}\; {\rm Gamma} \, (\alpha), \end{equation} a fact which can be found e.g. in \cite{CY} Exercise 4.2. When $\alpha +\beta \ge c,$ we first compute the density function of $X\times Y$: two changes of variable entail \begin{eqnarray*} f_{X\times Y}(x) & = & \frac{\Gamma (\alpha +\beta)}{\Gamma
(\alpha)\Gamma(\beta)\Gamma(c)}\int_0^1 e^{-x/u} (x/u)^{c-1} u^{\alpha -1} (1-u)^{\beta
-1} \frac{du}{u}\\ & = & \frac{\Gamma (\alpha +\beta) x^{c-1}}{\Gamma
(\alpha)\Gamma(\beta)\Gamma(c)}\left( e^{-x}\int_0^\infty e^{-xu} u^{\beta-1}(u +1)^{c -(\alpha
+\beta)} du\right) \end{eqnarray*} (notice that this recovers (\ref{IDG}) when $\alpha +\beta = c$), so that we will be done as soon as the function $t\mapsto g_{\alpha,\beta,c}(e^t)$ is log-concave, with the notation $$ g_{\alpha,\beta,c}(x)\; =\;e^{-x}\int_0^\infty e^{-xu} u^{\beta-1}(u +1)^{c -(\alpha
+\beta)} du.$$ Using now exactly the same discussion made at the end of Section 2 for the case $\alpha = 2/3$ (with adaptated computations) entails this log-concavity property is equivalent to $$(x g_{\alpha,\beta,c}(x) + (\alpha +\beta -c)g_{\alpha,\beta,c-1}(x)) (g_{\alpha,\beta,c+1}(x) - g_{\alpha,\beta,c}(x))\; \ge \; (\beta -1)(g_{\alpha,\beta,c-1}(x))^2$$ for every $x\ge 0.$ But in the above, the right-hand side is negative because $\beta < 1,$ whereas the left-hand side is positive from the obvious inequality $g_{\alpha,\beta,c+1}(x)\ge g_{\alpha,\beta,c}(x),$ and since by assumption $\alpha +\beta \ge c.$
\fin
\begin{LEMU} For all integers $p, n \ge 2$ such that $n > 2p,$ one has the following representation as an independent product: $$Z_{p/n}^{-p}\; \stackrel{d}{=}\; \frac{n^n}{p^p}\;{\rm Beta}\,(2/n, 1/p - 2/n)\,{\rm Gamma}(1/n)\;\times\;{\rm Beta}\,(4/n, 3/p - 4/n)\,{\rm Gamma}(3/n)\;\times\;\cdots\;$$
\begin{flushright}
$\times\; {\rm Beta}\,(2(p-1)/n, (p-1)/p - 2(p-1)/n)\,{\rm Gamma}((2p-3)/n)$
$\times\; {\rm Gamma}\,((2p-1)/n) \;\times\;\cdots\;\times\;{\rm Gamma}\,((n-1)/n).$ \end{flushright} \end{LEMU}
\noindent {\em Proof}: We first evaluate the fractional moments of $Z_{p/n}^{-p}$ using (\ref{MellinS}), the duplication formula for the Gamma function - see e.g. Formula (1.2.11) in \cite{E}, and some rearrangement involving the crucial assumption $n>2p:$ for every $s > - 1/n$ one obtains \begin{eqnarray*} {\mathbb{E}}\left[ (Z_{p/n}^{-p})^s\right] & = & \frac{\Gamma(ns +1)}{\Gamma(s+1)}\; \times\;\frac{\Gamma(s +1)}{\Gamma(ps+1)} \\ & = & \left( \frac{n^n}{p^p}\right)^{\! s} \frac{\Gamma(s +1/n)\;\ldots\; \Gamma(s +
(n-1)/n)\;\Gamma(1/p)\;\ldots\; \Gamma((p-1)/p)}{\Gamma(s +1/p\,)\;\ldots\; \Gamma(s +
(p-1)/p)\;\Gamma(1/n)\;\ldots\; \Gamma((n-1)/n)}\\ & = & \left( \frac{n^n}{p^p}\right)^{\! s} \left(\frac{\Gamma(s
+2/n)\Gamma(1/p)}{\Gamma(s +1/p)\Gamma(2/n)}\right)\left(\frac{\Gamma(s
+1/n)}{\Gamma(1/n)}\right) \end{eqnarray*} \begin{eqnarray*} & & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times\;\left(\frac{\Gamma(s
+4/n)\Gamma(2/p)}{\Gamma(s +2/p)\Gamma(4/n)}\right)\left(\frac{\Gamma(s
+3/n)}{\Gamma(3/n)}\right)\;\times\;\cdots\\ & & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times\;\left(\frac{\Gamma(s
+ 2(p-1)/n)\Gamma((p-1)/p)}{\Gamma(s
+ (p-1)/p)\Gamma(2(p-1)/n)}\right)\left( \frac{\Gamma(s +
(2p-3)/n)}{\Gamma((2p-3)/n)}\right)\\ & & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times\;\left(\frac{\Gamma(s + (2p-1)/n)}{\Gamma((2p-1)/n)}\right)\;\times\;\cdots\;\times\;\left( \frac{\Gamma(s + (n-1)/n)}{\Gamma((n-1)/n)}\right). \end{eqnarray*}
\noindent On the other hand, it is well-known and easy to see that the fractional moments of the ${\rm Beta}\, (\alpha, \beta)$ and $\,{\rm Gamma}\, (c)$ variables are given by $${\mathbb{E}}\left[ ({\rm Beta}\, (\alpha, \beta))^s\right]\; =\; \frac{\Gamma(s+\alpha)\Gamma(\alpha
+\beta)}{\Gamma(s+\alpha +\beta)\Gamma(\alpha)}\quad\mbox{and}\quad {\mathbb{E}}\left[ ({\rm Gamma}\, (c))^s\right]\; =\; \frac{\Gamma(s+ c)}{\Gamma(c)}\cdot $$ The claim follows now by identification of the Mellin transform.
\fin
\begin{REM} {\em As mentioned before, we see from this proof that analogous product representations for positive $\alpha$-stable laws with any $\alpha$ rational can be obtained accordingly. This might be useful to some other problems. Compare also with Theorem 2.8.4 in \cite{Z} where transforms of the densities $f_{p/q}$ are given as solutions to differential equations of higher order, an analytical representation which seems less tractable than Williams-type representations.}
\end{REM}
\noindent {\bf End of the proof}: We need to show (\ref{LCE}) for any $\alpha\le 1/2$ and $x \ge 0.$ Setting $g_\a (x) = (x^2 f_\alpha''(x) + xf_\alpha'(x))f_\alpha(x) - x^2(f_\alpha'(x))^2,$ we see from the Humbert-Pollard decomposition and its differentiations that the application $\alpha\mapstog_\a(x)$ is continuous on $(0,1)$ for every fixed $x \ge 0.$ By a density argument, it is therefore sufficient to prove (\ref{LCE}) for any $\alpha = p/n$ with $p, n$ integers greater than two such that $n> 2p,$ and every $x \ge 0.$ This amounts to the MSU property for $Z_{p/n}^{-p},$ which now comes easily from Lemmas 1 \& 2, the MSU property for Gamma variables and the stability of the MSU property by independent multiplication.
\noindent {\bf Acknowledgement.} This work was initiated during a stay at the University of Tokyo. I am very grateful to Nakahiro Yoshida for his hospitality, and to the grant ANR-09-BLAN-0084-01 for partial support.
\end{document}
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\begin{document}
\title{Broken Adaptive Ridge Regression for Right-Censored Survival Data \thanks{The research of Gang Li was partly supported by National Institute of Health Grants P30 CA-16042, P50 CA211015, and UL1TR000124-02. The research of Zhihua Sun was partly supported by Natural Science Foundation of China 11871444. The research of Yi Liu was partly supported by Natural Science Foundation of China 11801567. }}
\centerline{\large\bf Broken Adaptive Ridge Regression for Right-Censored Survival Data}
\centerline{Zhihua Sun$^{1}$, Yi Liu$^{1}$, Kani Chen$^{2}$, Gang Li$^{3}$$^{(\textrm{\Letter})}$ }
\centerline{\it $^{1}$Ocean University of China, $^{2}$Hong Kong University of Science and Technology and} \centerline{\it $^{3}$University of California at Los Angeles}
\institute{ Zhihua Sun \at
Department of Mathematics, Ocean University of China, Qingdao, China. \\ \email{[email protected]}
\and Yi Liu \at
Department of Mathematics, Ocean University of China, Qingdao, China.
\\ \email{[email protected]}
\and Kani Chen \at
Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong. \\ \email{[email protected]}
\and \Letter Gang Li \at
Professor of Biostatistics and Computational Medicine, University of California, Los Angeles, CA 90095-1772, USA.
\\ \email{[email protected]} }
\begin{abstract} Broken adaptive ridge (BAR) is a computationally scalable surrogate to $L_0$-penalized regression, which involves iteratively performing reweighted $L_2$ penalized regressions and enjoys some appealing properties of both $L_0$ and $L_2$ penalized regressions while avoiding some of their limitations. In this paper, we extend the BAR method to the semi-parametric accelerated failure time (AFT) model for right-censored survival data. Specifically, we propose a censored BAR (CBAR) estimator by applying the BAR algorithm to the Leurgan's synthetic data and show that the resulting CBAR estimator is consistent for variable selection, possesses an oracle property for parameter estimation {and enjoys a grouping property for highly correlation covariates}. Both low and high dimensional covariates are considered. The effectiveness of our method is demonstrated and compared with some popular penalization methods using simulations. Real data illustrations are provided on a diffuse large-B-cell lymphoma data {and a glioblastoma multiforme data}. \keywords{Accelerated failure time model \and {Grouping effect} \and $L_0$ penalization \and Right censoring \and Variable selection} \end{abstract}
\section{Introduction} \label{section1}
$L_0$-penalized regression, which directly penalizes the cardinality of a model, has been commonly used for variable selection {in the low dimensional setting} via well-known information criteria such as Mallow's $C_p$ \citep{mallows1973some}, Akaike's information criterion (AIC) \citep{akaike1974new}, the Bayesian information criterion (BIC) \citep{schwarz1978estimating,chen2008extended}, and risk inflation criteria (RIC) \citep{foster1994risk}. It has also been shown to possess some optimal properties for variable selection and parameter estimation \citep{shen2012likelihood, lin2010risk}. However, $L_0$-penalization is also known to have some limitations such as being computationally NP-hard, not scalable to high dimensional data, and unstable for variable selection \citep{breiman1996heuristics}. To overcome these shortcomings, the broken adaptive ridge (BAR) method \citep{ dai2018fusedBAR, DAI2018334} has been recently introduced as a surrogate to $L_0$ penalization for simultaneous variable selection and parameter estimation under the linear model \citep{dai2018fusedBAR, DAI2018334}. {It was noted by \citet{dai2018fusedBAR, DAI2018334} that the BAR estimator, defined as the limit of an iteratively reweighted $L_2$ (ridge) penalization algorithm, retains some appealing properties of $L_0$ penalization while avoiding its pitfalls. For instance, BAR generally yields a more sparse, accurate, and interpretable model than some popular $L_1$-type penalization methods such as LASSO and its various variations, while maintaining comparable prediction performance. Moreover, unlike the exact $L_0$ penalization, BAR is computationally scalable to high dimensional covariates and is stable for variable selection. Lastly, in addition to being consistent for variable selection and oracle for parameter estimation, the BAR estimator enjoys a grouping property for highly correlated covariates, a desirable feature not shared by most other oracle variable selection procedures.}
{Because of its appealing properties, the BAR penalization method has been recently extended to the \citet{Cox1972} model with censored survival data \citep{Eric,Jianguo2019} via penalized likelihood. However, it is well known that the \citet{Cox1972} proportional hazards assumption do not always hold in practice. Thus it is desirable to extend the BAR penalization method to other common survival regression models. This paper studies an extension of the BAR penalization method to the semi-parametric accelerated failure time (AFT) model, a popular alternative to the Cox model for right censored survival data. To this end, we note that the semi-parametric AFT model is a linear model for the log-transformed survival time with a completely unspecified error distribution, for which the likelihood approach does not yield a consistent parameter estimator even for the classical uncensored linear regression model. Hence, the BAR penalized likelihood methods of \citet{Eric} and \citet{Jianguo2019} for the \citet{Cox1972} do not apply to the semiparmetric AFT model. A different approach would be required.}
In this paper, we propose an extension of the BAR penalization method to the semi-parametric AFT model by coupling the \citet{Leurgans1987} synthetic data approach with the BAR penalty, study its large sample properties, and demonstrate it effectiveness in comparison with some popular penalization methods using simulations.
Specifically, we first use the \citet{Leurgans1987} synthetic variable method to construct a synthetic outcome variable and then apply the BAR method for uncensored linear regression \citep{ DAI2018334} to the synthetic outcome variable. We then give sufficient conditions under which the proposed censored BAR (CBAR) estimator is consistent for variable selection, behaves asymptotically as well as the oracle estimator based on the true reduced model, {and possesses a grouping property for highly correlated covariates. We also combine BAR with a sure joint screening method to obtain a two-step variable selection and parameter estimation method for ultra-high dimensional covariates. Not surprisingly, our simulations demonstrate that the proposed CBAR method generally yields a more sparse and more accurate model as compared to some other popular penalization methods such as LASSO, SCAD, MCP, and adaptive LASSO within the \citet{Leurgans1987} synthetic data framework, which is consistent with the findings of \citet{DAI2018334} for uncensored data. Lastly, we have implemented the proposed CBAR method in an R package, named CenBAR, and made it publicly available at https://CRAN.R-project.org/package=CenBAR.}
{Before going further, we note that there exist a number of other variable selection methods in the literature for the semiparametric the AFT model with right censored data. These methods are derived by combining various penalization methods such as LASSO with different extensions of the least squares principle for right censored data.
For example, the Lasso, bridge, elastic net or MCP penalties have been combined with the \citet{stute1993consistent} weighted least squares method \citep{Huang2005,Huang2010Variable,Datta2007Predicting};
and the Dantzig, elastic net, Lasso, adaptive Lasso and SCAD penalties have been combined with the \citet{Buckley1979Linear} method \citep{Yi2009Dantzig,Wang2008Doubly,Johnson2008Penalized,Johnson2009On}. This paper makes a unique theoretical contribution since neither the BAR penalization nor the \citet{Leurgans1987} synthetic data method has been previously rigorously studied in the context of variable selection for the semiparametric the AFT model. We also illustrate and compare empirically the BAR penalization versus some popular penalization methods when the \citet{Leurgans1987} synthetic data least squares method is used. We do not compare different penalization methods when they are coupled with different censored least squares methods because different censored least squares methods are derived under different conditions and none is expected to dominate another across all scenarios. \par The rest of the paper is organized as follows. In Section 2, we define our CBAR estimator and state its theoretical properties. We also discuss how to handle ultra-high dimensional covariates. In Section \ref{Numerical}, we evaluate the finite sample performance of CBAR in comparison with other penalization methods via extensive simulations. In Section 4, we illustrate the CBAR method on
a diffuse large-B-cell lymphoma data {and a glioblastoma multiforme data} with high dimensional covariates. Proofs of the theoretical results are provided in the appendix.
\section{Censored broken adaptive ridge (CBAR) regression } \label{section2} \subsection{Notations and preliminaries} \subsubsection{Model and data} Consider the linear regression model \begin{equation} \label{model:1} Y_i= \*x_i^\top \bm \beta+\varepsilon_i, \quad i=1,2,...,n, \end{equation} where for the $i$th subject, $Y_i$ denotes the response variable, $\*x_i$ is the $p_n$-vector random covariates, $\bm\beta = (\beta_1,...,\beta_{p_n})^\top$ is a vector of regression coefficients, and $\varepsilon_i$ is {i.i.d.} error term with an unknown error distribution, $E(\varepsilon_i)=0$ and $Var(\varepsilon_i)=\sigma^2<\infty$. Model (\ref{model:1}) is commonly referred to as the accelerated failure time (AFT) model when $Y$ is the log-transformed survival time \citep{Kalbfleisch2002}.
Without loss of generality, assume that {$ \bm \beta_{0}=( \bm \beta_{01}^\top , \bm \beta_{02}^{\top})^\top $} is the true value of $\bm \beta$, where
$ \bm \beta_{01}$ is a ${q}\times 1$ nonzero vector and $ \bm \beta_{02}$ is a $(p_n-{q})\times 1$ zero vector. We further assume the columns of the design matrix {$\*X= (\*x_1,...,\*x_n)^\top$} have mean zero and unit $L_2$-norm. Throughout the paper, $\|\cdot\|$ represents the Euclidean norm for a vector and spectral norm for a matrix.
Assume that one observes a right censored data consisting of $n$ independent and identically distributed triples $(T_i,\delta_i,\*x_i)$, $i=1,\ldots, n$, where for the $i$th subject, $T_i=\min(Y_i,C_i)$ is the observation time, $\delta_i=I(Y_i \leq C_i)$ is a censoring indicator, $C_i$ is the {i.i.d.} censoring time with the distribution function $H$. $C_i$ is assumed to be independent of $Y_i$ {and $\*x_i$.}
\subsubsection{Broken adaptive ridge (BAR) for uncensored data} For reader convenience, we first briefly review the broken adaptive ridge (BAR) estimator of \citet{DAI2018334} for simultaneous variable selection and parameter estimation with the uncensored data $\*Y$ and $\*X$, where {$\*Y=(Y_1,...,Y_n)^\top$}.
Following the notations of \citet{DAI2018334}, the BAR estimator of $\bm \beta$ based on $\*Y$ and $\*X$ is a surrogate $L_0$-penalized estimator defined as the limit of the following iteratively reweighted ridge regression algorithm: \begin{eqnarray} \nonumber
{\bm \beta}^{(k)}&=&\arg\min_{\bm \beta} \{ \|\*Y-\*X\bm\beta\|^2+\lambda_n \sum_{j=1}^{p_n}\frac{\beta_j^2} {\{{\bm \beta}^{(k-1)}_j\}^2}\} \\ &=&\{\*X^\top\*X+\lambda_n \*D({\bm \beta}^{(k-1)})\}^{-1}\*X^\top\*Y, \quad k=1,2,... \end{eqnarray}
where ${\bm \beta}^{(0)}=\arg\min_{\bm \beta} \{ \|\*Y-\*X\bm\beta\|^2+\xi_n \sum_{j=1}^{p_n} \beta_j^2\} =(\*X^\top\*X+\xi_n\*I)^{-1}\*X^\top\*Y $ is an initial ridge estimator, $\xi_n>0$ and $ \lambda_n\ge 0$ are tuning penalization parameters, and
for any $p_n$-dimensional vector $\bm \theta = (\theta_1, ..., \theta_{p_n})^T$, $\*D(\bm \theta)=\mbox{diag}(\frac{1}{\theta_1^2}, ..., \frac{1}{\theta_{p_n}^2})$. Note that each reweighted $L_2$ {penalty} can be regarded as an adaptive surrogate $L_0$ penalty and the approximation of $L_0$ penalization improves with each iteration. \citet{DAI2018334} showed that the BAR estimator $\hat{\bm \beta} = \lim_{k\to\infty} {\bm \beta}^{(k)}$ is selection consistent and possesses an oracle property: if the true model is sparse with some zero coefficients, then with probability tending to 1, BAR estimates the true zero coefficients as zeros and estimates the non-zero coefficients as well as the scenario when the true sub-model is known in advance.
\subsection{Broken adaptive ridge estimator for censored data (CBAR)} For right censored data, the above BAR algorithm is obviously not applicable since one only observes $(T_i, \delta_i)$ instead of $Y_i$. To overcome the problem, we propose to adopt the \citet{Leurgans1987} synthetic data approach for censored linear regression to variable selection by first transforming $(T_i, \delta_i)$ into a synthetic variable $Y_i^*$ and then applying the BAR method to the synthetic data variable $Y_i^*$. Specifically, the \citet{Leurgans1987} synthetic data $Y_i^*$ is defined as \begin{equation} \label{model:Leurgans} Y_i^*=\int_{-\infty }^{T^n} {\left(\frac{I(T_i\geq s)}{1-\hat{H}(s)}- I(s\textless{0})\right)}ds, \end{equation} where $T^n =\max\{T_1, ..., T_n\}$ and $\hat{H}$ is the Kaplan-Meier estimator of $H$. To apply the BAR method to synthetic data $Y_i^*$, let $\*Y^*= (Y^*_1,...,Y^*_n)^\top$ and define an initial ridge estimator \begin{eqnarray} \hat{\bm \beta}^{(0)} =(\*X^\top\*X+\xi_n\*I)^{-1}\*X^\top\*Y^*, \label{ridge} \end{eqnarray} and then, for $k\ge 1$, \begin{equation}
\hat{\bm \beta}^{(k)}=g(\hat{\bm \beta}^{(k-1)}), \end{equation} where \begin{eqnarray} \label{eq:one}
g(\tilde{\bm\beta}) =\arg\min_{\bm \beta} \{ \|\*Y^*-\*X\bm\beta\|^2+\lambda_n \sum_{j=1}^{p_n}\frac{\beta_j^2}{\tilde{\beta}_j^2}\} =\{\*X^\top\*X+\lambda_n \*D(\tilde{\bm\beta})\}^{-1}\*X^\top\*Y^*. \label{eq:onea} \end{eqnarray} Finally, the CBAR estimator is defined as \begin{equation} \label{BJ BAR} \hat{\bm \beta}^*=\lim_{k\rightarrow\infty} \hat{\bm \beta}^{(k)}. \end{equation} In the next section, we give conditions under which the CBAR estimator $\hat{\bm\beta}^*$ is selection consistent and has an oracle property for estimation of the nonzero component $ \bm \beta_{01}$ of $ \bm \beta$.
\subsection{Large sample properties of CBAR} Similar to \citet{Zhou1992Asymptotic}, define $F_i{(t)}=P\{Y_i \geq t\}$, $ G_{i}(t)=P\{T_i \geq t\}=F_i{(t)}(1-H(t))$, $ K(t)= -\int_{0}^{t}{\frac{1}{\lim{(1/n)}\sum{F_i}}}\frac{dG}{G^2}$ and denote $$\Lambda_i^{+}{(t)}=-\int_{0}^{t}{\frac{dG_i{(s)}}{G_i{(s^-)}}},\ \Lambda_i^D{(t)}=-\int_{0}^{t}{\frac{dF_i{(s)}}{F_i{(s^-)}}},\ \Lambda^C{(t)}=\int_{0}^{t}{\frac{dH{(s)}}{1-H{(s^-)}}}.$$ Then, $$M_{i}^{+}(t)=I_{[T_i\leq t]}-\int_{0}^{t}{I_{[T_i\geq s]}d\Lambda_i^{+}{(s)}},$$ $$M_{i}^{D}(t)=I_{[T_i\leq t; \delta_i=1]}-\int_{0}^{t}{I_{[T_i\geq s]}d\Lambda_i^{D}{(s)}},$$ $$M_{i}^{C}(t)=I_{[T_i\leq t; \delta_i=0]}-\int_{0}^{t}{I_{[T_i\geq s]}d\Lambda_i^{C}{(s)}}$$ are square-integrable martingales and satisfies $M_{i}^{+}=M_{i}^{D}+M_{i}^{C}$ \citep{Zhou1992Asymptotic}. Let $\*\Omega(\tau)=( \sigma_{kl}(\tau))$ be defined by \begin{eqnarray} \label{sigmma_leurgans}
\nonumber \sigma_{kl}(\tau)&=&\lim n \sum_{i=1}^n {\omega_{ki}\omega_{li}}\int_{0}^{\tau}{\left [\int_{t}^{\tau}{F_i}{ds} \right ]^2{\frac{d\Lambda_i^D{(t)}}{G_i}}}\\
&+& \lim n\sum_{i=1}^n{\int_{0}^{\tau}{\prod_{c_{i}=\omega_{ki},\omega_{li}}{\left [ \frac{\sum{c_{j}}\int_{t}^{\tau}{F_j{ds}}}{(1-H)\sum{F_j}}-\frac{c_{k}{\int_{t}^{\tau}{F_i{ds}}}}{G_i} \right ]}G_i}{d\Lambda^C}}, \end{eqnarray} where $\omega_{ji}=((\*X^\top \*X)^{-1}\*X^\top )_{ji}$. Let {$\bm\omega_{i}$ denote the $i$th column of the matrix $(\*X^\top \*X)^{-1}\*X^\top$,} $\*X_1$ denote the first $q_n$ columns of $\*X$, $ \*\Sigma_{n}=n^{-1} \*X^\top \*X$ and $\*\Sigma_{n1}=n^{-1} \*X_1^\top\*X_1$. Write $ \hat{\bm\beta}^*=( \hat{\bm\beta}_{1}^{*^\top }, \hat{\bm\beta}_{2}^{*^\top })^\top$, where $\hat{\bm\beta}_{1}^{*}$ is a ${q}\times 1$ vector estimator of $ \bm \beta_{01}$ and $\hat{\bm\beta}_{2}^{*}$ is a $(p_n-{q})\times 1$ vector estimator of $ \bm \beta_{02}$.
\par The following conditions are needed for our theoretical derivations. \begin{itemize} \itemsep0em \itemsep0em
\item[(C1)]
$\sup_{t} E(\varepsilon_i-t |\varepsilon_i> t )<\infty$, and {\color{black}for any $p_n$-vector $ \*b_n$ satisfying {$\| \*b_n\|\leq 1$}, $\*b_n^\top\*\Omega(\tau)\*b_n$ }is finite for $\tau\in[K,\infty]$ and {$\*b_n^\top\*\Omega(\tau)\*b_n \to \*b_n^\top\*\Omega(\infty)\*b_n$ } as $\tau\to \infty$.
\item[(C2)]
$\sup_n \int_{0}^{\infty}\sum_{i=1}^n {(\*b_n^\top\bm\omega_{i} )}^2 \sum_{i=1}^n {F_{i}^2{dK(t)}}< \infty$ for {any $p_n$-vector $ \*b_n$ satisfying {$\| \*b_n\|\leq 1$}}.
{$X_i$ are bounded, and for some constants $C^*>0$ and $S<1$, $C^*F_i(t)^S\leq 1-H(t)$.}
\item[(C3)]
$\int_{0}^{\infty}\{{\frac{n\sum ({\*b_n^\top\bm\omega_{i} })^{2}F_{i}}{1-H(s)}\}^{\frac{1}{2}}ds}\leq M < \infty$ and $\int_{0}^{\infty}{K^{1/2}(t)|\sum {\*b_n^\top\bm\omega_{i} }F_{i}|dt} <\infty$ for {any $p_n$-vector $ \*b_n$ satisfying {$\| \*b_n\|\leq 1$}}.
\item[(C4)]
There exists a constant $\tilde{C}>1$ such that $0<1/ \tilde{C}<\lambda_{\min}( \*\Sigma_n)\leq \lambda_{\max}(\*\Sigma_n)< \tilde{C} <\infty$ for every integer $n$.
\item[(C5)]
Let
$a_{{{0}}}= \min_{1\leq j\leq {q}}|\beta_{0j}|$ and $a_{{{1}}}= \max_{1\leq j\leq {q}}|\beta_{0j}|$.
As $n\to\infty$, ${{{p_n}}}/{\sqrt{n}} \to 0$, { ${\xi_n}/\sqrt{n} \to 0$} and {$\lambda_n/\sqrt{n} \to 0$}
\end{itemize} Conditions (C1)-(C3) are regularity conditions required to establish the asymptotic properties of the unpenalized synthetic data least squares estimator under diverging dimension. Conditions (C4) and (C5) are additional conditions needed to derive the selection consistency and oracle property of the synthetic data BAR estimator of this paper as stated in Theorem 1 below \begin{theorem}[Oracle property] \label{theorem:CBAR}
Assume conditions (C1)-(C5) hold. For any $q$-dimensional vector $ {\*c}$ satisfying $\| {\*c}\|\leq 1$, define ${z^2= \*c^\top \*\Omega_{1} \*c}$, where $\*\Omega_{1}$ is the first ${q \times q}$ sub-matrix of $\*\Omega(\infty)$. Define {$f( \bm\alpha)= \{ \*X_{1} ^\top \*X_1+\lambda_n \*D_1( \bm\alpha)\}^{-1} \*X_1^\top \*Y^*$}, where $ \*D_1( \bm\alpha) = {\rm diag}(\alpha_1^{-2},\ldots,\alpha_{{q}}^{-2})$. Then, with probability tending to 1, \begin{enumerate} \item[(i)] ${{\widehat{\bm \beta }}^{*}}={{(\widehat{\bm \beta }_{1}^{{{*}^{\top }}},\widehat{\bm \beta }_{2}^{{{*}^{\top }}})}^{\top}}$ exists and is unique, with $\hat{\bm\beta}_{2}^{*}=0$ and $\hat{ \bm \beta}_1^*$ being the unique fixed point of $f(\bm\alpha)$; \item[(ii)]
$\sqrt{n}\, {z^{-1} \*c}^\top(\hat{ \bm \beta}_{1}^{*}- \bm \beta_{01}) \rightarrow_D N(0,1).$ \end{enumerate} \end{theorem} Part (i) of the above theorem guarantees that the CBAR estimator is consistent for variable selection. Part (ii) states that the asymptotic distribution of the nonzero component of the CBAR estimator is the same as the one when the true model is known in advance. The proof of Theorem \ref{theorem:CBAR} is deferred to the Appendix. { \subsection{Grouping effect}\label{group_simulation}
When the true model has a group structure, it would be desirable for a variable selection method to either retain or drop all variables that are clustered within the same group.
Below we establish that the CBAR estimator possesses a grouping property in the sense that highly correlated covariates tend to be grouped together with similar coefficients. \begin{theorem}
\label{theorem:2}
{\it Assume that the columns of matrix $\*X$ are standardized and $\*Y^*$ is centered. Let $\hat{ \bm \beta}^*$ be the CBAR estimator and {$\hat{\beta}_i^*\hat{\beta}^*_j > 0$}, then, with probability tending to $1$, \begin{equation} \label{groupEquation1}
|\hat{\beta}_i^{*-1} - \hat{\beta}_j^{*-1}|\leq \frac{1}{\lambda_n}\, \| \*Y^*\|\sqrt{2(1- r_{ij})}, \end{equation} where ${r_{ij}= \*x_i^\top \*x_j}$ is the sample correlation of $\*x_i$ and $ \*x_j$.} \end{theorem}
The above result implies that the estimated coefficients of two highly positively-correlated variables will be similar in magnitude. The proof of Theorem \ref{theorem:2} is given in the Appendix. Similarly, it can be shown that the estimated coefficients of two highly negatively-correlated variables will also be similar in magnitude. }
{ \subsection{Ultrahigh dimensional covariates}\label{highdim}
Theorem 1 is established under a sufficient condition that $p_n < n$. In many applications, $p_n$ can be much larger than the sample size $n$. For high dimensional problems, a common strategy is to proceed a variable selection method with a sure screening dimension reduction step \citep{Fan2008Sure,Zhu2011Model,Cui2015Model}. This strategy also applies to the semiparametric AFT model with right censored data. For example, one can first apply the sure joint screening method BJASS of \citet{YiLiu} to obtain a lower dimensional model and then apply the CBAR method to the reduced model. We refer to the resulting two-step estimator $\hat{ \bm \beta}^{*}$ as the BJASS-CBAR estimator.
Below we give some additional sufficient conditions under which the BJASS-CBAR estimator $\hat{ \bm \beta}^{*}$ has an oracle property. } { \begin{itemize}
\item [(D1)] $\log(p)=O(n^d)$ for some $0\leq d < 1$.
\item [(D2)] $P(t\leq Y_i \leq C_i)\geq \tau_0>0$ for some positive constant $\tau_0$ and any $t\in [0,\varsigma]$, where $\varsigma$ denotes the maximum follow up time. Furthermore, $\sup\{t:P(Y>t)>0\}\geq \sup\{t:P(C>t)>0\} $. $H(t)$ has uniformly bounded first derivative.
\item [(D3)] $\min_{j\in s^*}|\beta_j^*|\geq \omega_1n^{-\tau_1}$ and $q<k\leq \omega_2n^{\tau_2}$ for some positive constants $\omega_1,~\omega_2$ and nonnegative constants $\tau_1,~\tau_2$ satisfying $\tau_1+\tau_2<1/3$, where $k$ is the size of the screened model from BJASS.
\item [(D4)] For sufficiently large $n$, $\lambda_{\min}(n^{-1} \*X_s ^\top \*X_s)\geq c_1$ for some constant $c_1>0$ and all $s\in S_{+}^{2k}$, where $\lambda_{\min}(\cdot)$ denotes the smallest eigenvalue of a matrix, and $S_{+}^{k}=\{s:s^*\subset s; \|s\|_0\leq k\}$ denotes the collection of the over-fitted models of cardinality $k$ or smaller.
\item [(D5)] Let $\sigma_i^2=\int\int[\frac{G_i(s\vee t)}{(1-H(s))(1-H(t))}-F_i(s)I(t<0)-F_i(t)I(s<0)+I(s<0)I(t<0)]dsdt-E^2(Y_i)$. There exist positive constants $c_2$, $c_3$, $c_4$, $\sigma$ such that $|X_{ij}|\leq c_2$, $|X_i^\top\bm \beta^*|\leq c_3$, $|\sigma_i|\leq \sigma$ and for sufficiently large $n$,
\begin{eqnarray*}
\max_{1\leq j\leq p}\max_{1\leq i \leq n}\left\{\frac{X_{ij}^2}{\sum_{i=1}^nX_{ij}^2\sigma_i^2}\right\}\leq c_4n^{-1}.
\end{eqnarray*}
\item [(D6)] There are positive constants $K_1$, $K_2$ and $\tau_3$ such that
\begin{eqnarray*}
P(|\epsilon|\geq M)\leq K_1\exp(-K_2M^{\tau_3}),
\end{eqnarray*}
for any $M=O(n^{\tau})>0$, where $\tau\geq 0$, $\tau_1+\tau_2+\tau<(1-d)/2$, $\tau_2+d-\tau\tau_3<0$, and $2\tau_2+2\tau+d<1/3$. \end{itemize}
\begin{theorem}[Oracle property of the BJASS-CBAR estimator]\label{BJASS_CBAR} Assume that conditions (D1)-(D6) hold and that the assumptions of Theorem 1 hold for the BJASS reduced model. Then, with probability tending to 1, \begin{enumerate} \item[(i)] $\hat{\bm\beta}_{2}^{*}=0$; \item[(ii)]
$\hat{ \bm \beta}_{1}^{*}$ performs as well as the oracle estimator for the true model $\mathcal{M}_{*}=\{1\le j\le q\}$ in the sense of part (ii) of Theorem 1. \end{enumerate} \end{theorem}
The above result is a direct consequence of Theorems 4 of \citet{YiLiu} and the oracle property of CBAR stated in Theorem 1. In Section \ref{sect:Simulation2}, we present a simulation study to illustrate the advantages of BJASS-BAR in comparison with some other penalization methods under a high dimensional setting. }
\section{Simulations}\label{Numerical} We present some simulations to illustrate the effectiveness of the proposed CBAR estimator for variable selection, prediction, parameter estimation in comparison with some popular penalization methods including Lasso \citep{tib1996}, adaptive Lasso \citep{zou2006}, SCAD \citep{fan2001} and MCP \citep{zhang2010}),
in the context of the \citet{Leurgans1987} synthetic data framework. We use the \textsf{R} package \texttt{glmnet} \citep{friedman2010} for Lasso and adaptive Lasso and \textsf{R} package \texttt{ncvreg} \citep{breheny2011} for SCAD and MCP, performed on the \citet{Leurgans1987} synthetic data outcome. Five-fold cross-validation (CV) is used to select tuning parameters for all methods. For CBAR, we all use 10 equally log-spaced grid points on $[a, b]$ for the paths of $\lambda_n$ and $\xi_n$ where $a=1e^{-4}$ and $b=\mbox{max}\left\{\frac{(\*x^{{ \mathrm{\scriptscriptstyle T} }}_j\*y)^2}{4\*x^{{ \mathrm{\scriptscriptstyle T} }}_j\*x_j}\right\}^p_{j=1}$.
\subsection{Simulation 1: $p_n < n$ }\label{sect:Simulations}
We consider the following two model settings similar to \citep{Tibshirani1997, Fan2002, Cai2009Regularized}: \leftmargini=2.5cm \begin{itemize}
\item[Model 1:] $Y_i= \*x_i^\top \bm\beta_0+\varepsilon_i$, where the covariate vector $ \*x_i$ is generated from a multivariate normal distribution with mean $0$ and variance-covariance matrix $\*\Sigma=(\rho^{|i-j|})$, and the error $\varepsilon_i$ has the standard normal distribution and is independent of the covariates.\\ The true parameter value is $\bm\beta_0 = (3,-2, 0, 0, 6, 0,\ldots, 0)^\top$. \item[Model 2:]
The same as Model 1 except that\\ $\bm\beta_0 = (3,-2, 6, 0.3, -0.2, 0.6, 0, \ldots, 0)^\top $. \end{itemize} Note that Model 1 contains strong signals, whereas Model 2 includes both strong and weak signals. The censoring variable $C_i$ is generated from the normal distribution $ N(c, 2)$, where $c$ is chosen to yield a desired level of censoring rate.
The variable selection performance is assessed using five measures: the mean number of misclassified non-zeros and zeros (MisC), mean of false non-zeros (FP), mean of false zeros (FN), probability that the selected model is identical to the true model (TM), and a similarity measure (SM)
between the selected set $\hat S $ and the true active set $ |S|_{0}$:
$SM= \frac{ |\hat{S}\cap S|_{0}}{\sqrt{|\hat{S}|_{0} |S|_{0}}},$
where $|.|_{0}$ denotes model size. The prediction performance is measured by the mean squared prediction error (MSPE) from the five-fold CV. The parameter estimation performance is measured by the mean of the absolute bias of the parameter estimator (MAB). We have run extensive simulations for a variety of settings by varying $n$, $p$, $\rho$ and the censoring rate, with {1,000} Monte Carlo replications for each setting. Part of the findings are presented in Table \ref{table:1}.
\begin{center} [Insert Table \ref{table:1} approximately here] \end{center}
\begin{table}
\centering \caption{\small {Comparison of CBAR with Lasso, SCAD, MCP, and Adaptive Lasso (ALasso) when coupled with the \citet{Leurgans1987} synthetic data procedure based on {{1,000}} Monte-Carlo replications. Data settings: $n=100$, $p \in \{10, 50, 80, 90 \}$. (MisC = mean number of misclassified non-zeros and zeros; FP = mean of false positives (non-zeros); FN = mean of false negatives (zeros); TM = probability that the selected model is exactly the true model; SM = similarity measure; MSPE = mean squared prediction error from five-fold CV or five-jointly CV and MAB = mean of the absolute bias of the parameter estimator.)}}
\scalebox{0.8}{ \begingroup \setlength{\tabcolsep}{5pt} \renewcommand{1}{0.9} \begin{tabular}{clllllllll} \hline\hline
Model& p & Method & MisC & FP & FN & TM & SM & MSPE & MAB \\ \hline
1& 10 & CBAR & $\bm{0.60}$ & $\bm{0.60}$ & 0 & $\bm{74\%}$ & $\bm{0.94}$ &$\bm{8.86}$ & 1.50 \\
& & Lasso & 3.05 & 3.05 & 0 & 6.6$\%$ & 0.73 & 9.28 & 2.29\\
& & SCAD & 1.11 & 1.11 & 0 & 46.2$\%$ &0.89 & 9.03 & 1.47 \\
& & MCP & 0.76 & 0.76 & 0 & 63.6$\%$ & 0.92 & 9.01 & 1.45 \\
& & Alasso & 1.12 &1.12 & 0 & 49.2$\%$ & 0.89 & 8.91 & 1.68 \\
\cline{2-10}
& 50 & CBAR & $\bm{0.73}$ & $\bm{0.71}$ & 0.02 & $\bm{74.80\%}$ & $\bm{0.94}$ & $\bm{8.9}$ & 1.69\\
& & Lasso & 7.33 & 7.33 & 0 & 1.7$\%$ &0.58 & 9.77 & 3.36 \\
& & SCAD & 2.96 & 2.96 & 0 & 21.3$\%$ & 0.76 & 9.09 & 1.72\\
& & MCP & 1.24 & 1.23 & 0.01 & 47.7$\%$ & 0.88 & 9.03 & 1.56 \\
& & Alasso & 6.09 & 6.09 & 0 & 15.2$\%$ & 0.67 & 8.69& 3.04 \\
\cline{2-10}
& 80 & CBAR & $\bm{0.86}$ & $\bm{0.84}$ & 0.02 & $\bm{72.3\%}$ &$\bm{0.93}$ & 8.84 & 1.81 \\
& & Lasso & 9.40 & 9.40 & 0 & 1.30$\%$ & 0.54 & 10.06 & 3.79\\
& & SCAD & 3.90 & 3.90 & 0 &15.2$\%$ &0.72 & 9.33 & 1.89\\
& & MCP & 1.41 & 1.40 & 0.01 & 45.6$\%$ & 0.87 & 9.26 & 1.65 \\
& & Alasso & 11.09 & 11.08 & 0.01 & 11.8$\%$ & 0.59 & $\bm{8.74}$ & 4.51\\
\cline{2-10}
& 90 & CBAR & $\bm{0.94}$ & $\bm{0.92}$ & 0.02 & $\bm{69.7\%}$ &$\bm{0.93}$ & $\bm{8.93}$ & 1.88 \\
& & Lasso & 9.36 & 9.36 & 0 & 1.4$\%$ &0.54 & 10.05 & 3.82 \\
& & SCAD & 4.09 & 4.09 & 0 & 13.5$\%$ & 0.71 & 9.27 & 1.91\\
& & MCP & 1.44 & 1.43 & 0.01 & 43.2$\%$ &0.87 & 9.20 & 1.64 \\
& & Alasso & 4.29 & 4.27 & 0.02 & 10.5$\%$ &0.70 &9.13 & 2.69 \\
\hline
2 & 10 & CBAR & 2.61 & $\bm{0.65}$ & 1.96 & 0.9$\%$ & 0.77 & 9.36 & 2.36 \\
& & Lasso & 3.00 & 2.14 & $\bm{0.86 }$ & 2.2$\%$ &0.78 & 9.50 & 2.74 \\
& & SCAD & 2.64 & 1.10 & 1.54 & 2.3$\%$ & 0.78 & 9.35 & 2.35\\
& & MCP & 2.64 & 0.86 & 1.78 & 1.9$\%$ & 0.77 & 9.34 & 2.36\\
& & Alasso & $\bm{2.50}$ & 0.92 & 1.58 & $\bm{3.1}\%$ & $\bm{0.79}$ & $\bm{9.14}$ & 2.37\\
\cline{2-10}
& 50 & CBAR & $\bm{3.65}$ & $\bm{1.03}$ & 2.62 & 0.1$\%$ & $\bm{0.69}$ & 9.41 & 2.92\\
& & Lasso & 9.75 & 7.99 & $\bm{1.76}$ & 0$\%$ & 0.52 & 10.40 & 4.37 \\
& & SCAD & 5.57 & 3.40 & 2.17 & 0$\%$ & 0.61 & 9.84 & 2.77 \\
& & MCP & 3.92 & 1.46 & 2.46 & 0$\%$ &0.67 & 9.80 & 2.64 \\
& & Alasso & 9.18 & 7.22 & 1.96 & 0.1$\%$ &0.55 & $\bm{9.21}$ & 4.27 \\
\cline{2-10}
& 80 & CBAR & $\bm{3.89}$ & $\bm{1.19}$ & 2.70 & 0$\%$ & $\bm{0.68}$ & 9.11 & 3.02 \\
& & Lasso & 11.61 & 9.69 & $\bm{1.92}$ & 0$\%$ & 0.48 & 10.39 & 4.70 \\
& & SCAD & 6.48 & 4.21 & 2.27 & 0$\%$ & 0.57 & 9.66 & 2.86 \\
& & MCP & 4.06 &1.49 & 2.57 & 0$\%$ & 0.66 & 9.60 & 2.64 \\
& & Alasso & 13.82 & 11.78 & 2.04 & 0$\%$ & 0.48 & $\bm{8.99}$ & 5.55\\
\cline{2-10}
& 90 & CBAR &$\bm{3.85}$ & $\bm{1.16}$ & 2.69 &0$\%$ & $\bm{0.68}$& 9.31 & 3.05 \\
& & Lasso & 12.44 & 10.47 & $\bm{1.97}$ & 0$\%$ & 0.46 & 10.20 & 4.86 \\
& & SCAD & 6.92 & 4.67 & 2.25 & 0$\%$ &0.56 & 9.40 & 2.92 \\
& & MCP & 4.24 & 1.68 & 2.56 & 0$\%$ & 0.65 & 9.36& 2.68 \\
& & Alasso & 7.00 & 4.50 & 2.50 & 0$\%$ & 0.54 & $\bm{9.27}$ & 3.73 \\
\hline \end{tabular} \endgroup } \label{table:1} \end{table}
It is seen from Table \ref{table:1} that CBAR stands out as the top or top two performers with respect to almost all variable selection performance measures (MisC, FP, TM and SM). {In particular, CBAR generally yields a more sparse and accurate model with the largest TM and SM, and much lower MisC and FP}. Also, using fewer active features, CBAR achieves comparable prediction accuracy as other methods that use more features. For estimation, CBAR, SCAD and MCP are comparable with similar bias (MAB), whereas Lasso and {Adaptive lasso} can be substantially worse.
\subsection{Simulation 2: $p_n >> n$ }\label{sect:Simulation2} In this simulation, we consider the same models as in Simulation 1, except in a high dimensional setting with $ n=200$, $p=1000$. We again compared the same five penalization methods, with each method proceeded with the sure joint screening method BJASS of \citet{YiLiu} with $k=2log(n)*n^{(1/4)}$ for the semi-parametric AFT model to yield a two-step sparse estimator. We denote these methods by BJASS-CBAR, BJASS-Lasso, BJASS-SCAD, BJASS-MCP and BJASS-ALasso. The censoring rate is 0.2. The results are summarized in Table \ref{table:3}. \begin{center} [Insert Table \ref{table:3} approximately here] \end{center}
\begin{table}[h!]
\centering
\caption{\small Comparison of BJASS-CBAR with CBAR with BJASS-Lasso, BJASS-SCAD, BJASS-MCP, and BJASS-ALasso when coupled with the \citet{Leurgans1987} synthetic data procedure in a high-dimensional setting: $ n=200$, $p=1000$. (MisC= mean number of misclassified non-zeros and zeros; FP = mean of false positives (non-zeros); FN = mean of false negatives (zeros); TM = probability that the selected model is exactly the true model; SM = similarity measures; MSPE = mean squared prediction error from five-fold CV or five-jointly CV and MAB = mean of the absolute bias of the parameter estimator.)}
\scalebox{0.8}{
\begingroup
\setlength{\tabcolsep}{4pt}
\renewcommand{1}{1}
\begin{tabular}{clllllllr}
\hline
Model & Method & MisC & FP & FN & TM & SM & MAB & MSPE\\
\hline
1 & {BJASS-CBAR} & $\bm{2.24 }$ & $\bm{2.15 }$ & 0.09 &$\bm{63\%}$ & $\bm{0.93}$ & 2.87 & 10.40 \\
& {BJASS-Lasso} & 12.61 & 12.55 & 0.06 & 0$\%$ & 0.63 & 4.79 & 10.87 \\
& {BJASS-SCAD} & 4.23 & 4.14 & 0.09 & 20$\%$ & 0.82 & 2.79 & 10.46 \\
& {BJASS-MCP} & 2.82 & 2.73 & 0.09 & 43$\%$ & 0.88 & $\bm{2.69}$ & 10.45 \\
& {BJASS-ALasso} & 8.08 & 8.00 & 0.08 & 12$\%$ & 0.73 & 4.05 & 10.35 \\
\hline
2 & {BJASS-CBAR} & $\bm{6.15 }$ & $\bm{3.15}$ & 3 & $\bm{41\%}$ & $\bm{0.69}$ & 2.51 & 12.17 \\
& {BJASS-Lasso} & 17.14 & 14.14 & 3 & 0$\%$ & 0.49 & 4.49 & 12.64\\
& {BJASS-SCAD} & 8.68 & 5.68 & 3 & 7$\%$ & 0.62 & 2.09 & 12.39 \\
& {BJASS-MCP} & 6.38 & 3.38 & 3 & 26$\%$ & 0.68 & $\bm{1.96}$ & 12.38 \\
& {BJASS-ALasso} & 12.78 & 9.78 & 3 & 3$\%$ & 0.54 & 3.75 & 11.91 \\
\hline
\end{tabular}
\endgroup
}\\
\label{table:3} \end{table} {It is observed from Table \ref{table:3} that although most penalization methods had comparable performance in terms of estimation bias (MAB) and prediction error (MSPE), BJASS-CBAR outperformed the other methods in the variable selection domain with the lowest MisC, FP and the largest TM and SM, which are consistent with the simulation results for the low-dimension $p_n<n$ settings in Simulation 1.
}
\section{Real data examples} We illustrate the CBAR method on two real datasets with high dimensional covariates.
\subsection{Diffuse large-B-cell lymphoma data} The diffuse large-B-cell lymphoma (DLBCL) data includes $n=240$ patients and $p=7399$ gene features, which was downloaded from \url{http://statweb.stanford.edu/~tibs/superpc/staudt.html}. We first apply the BJASS sure joint screening method of \citet{YiLiu} to reduce data dimension to $k=2log(n) n^{\frac{1}{4}}=43$ and then apply CBAR and four other popular penalization methods. The results are summarized in Table \ref{DLBCL}.
\begin{center} [Insert Table \ref{DLBCL} approximately here] \end{center}
\begin{table}[h!]
\centering
\caption{Estimated coefficients of {BAJSS-CBAR, BAJSS-Lasso, BAJSS-SCAD, BAJSS-MCP and BAJSS-Alasso for the DLBCL data}.}
\scalebox{0.85}{
\setlength{\tabcolsep}{2pt}
\renewcommand{1}{1}
\begin{tabular}{lccccr}
Parameter & BAJSS-CBAR & BAJSS-Lasso &BAJSS-SCAD & BAJSS-MCP & BAJSS-Alasso\\
\hline $1456$ &-0.0591 &$-0.394$&$-0.609$&-0.630 &$-0.513$\\ $1819$ & &$-0.069$ & & &\\ $1863$ & &-0.006 & & &\\ $2603$ & &$-0.025$ & & &\\ $2672$ & &$-0.062$ & & &\\ $3236$ &-0.480 &-0.348 &-0.394 &-0.426 &-0.399\\ $5775$ &-0.261 &$-0.143$ &-0.133 &-0.131 &$-0.111$\\ $6566$ & &-0.088 & -0.061& -0.004&\\
\hline
Tuning parameters & \tabincell{l}{\ $\xi_n=43$\\ $\lambda_n=5.721$}&$\lambda=0.197$&\tabincell{l}{\ $\gamma=3.7$,\\ $\lambda=0.211$}& $\lambda=0.260$ &\tabincell{c}{\ $\gamma=3.598$,\\ $\lambda= 2.058$}\\
\hline
{Number of selected} &3 & 8& 4 & 4& 3\\
\hline
{CV error} &6.399 & 6.731 & 6.496 & 6.515& 6.472\\
\hline
\end{tabular}} \label{DLBCL} \end{table} It is seen that BJASS-CBAR is among the most sparse model and has the smallest CV error, which is consistent with the findings in the simulation studies. { \subsection{Glioblastoma multiforme data} \label{GBMdata}
The glioblastoma multiforme (GBM) methylation data was downloaded from the TCGA program (https://www.cancer.gov/tcga) using TCGA-Assembler 2 (TA2). The initial data consists of 577 patients and 20,156 GBM methylation variables. After removing missing data, the complete case data includes $n=136$ patients and $p=20,037$ methylation variables. Applying the method described in Section 2.5, we first performed sure joint screening using the BJASS method of \citet{YiLiu} reduce data dimension to $k=2 log(n) n^{\frac{1}{4}} =34$ before applying the CBAR penalization method and four other penalization methods (Lasso, SCAD, MCP and Alasso). The final variable selection results are summarized in the Table \ref{GBM_M_1}. \begin{center} [Insert Table \ref{GBM_M_1} approximately here] \end{center} \begin{table}[h!]
\centering
\caption{{Estimated coefficients of BJASS-CBAR, BJASS-Lasso, BJASS-SCAD, BJASS-MCP and BJASS-Alasso for the TCGA GBM methylation data}}
\scalebox{0.85}{
\setlength{\tabcolsep}{2pt}
\renewcommand{1}{1}
\begin{tabular}{lccccr}
Variables & BJASS-CBAR & BJASS-Lasso &BJASS-SCAD & BJASS-MCP & BJASS-Alasso\\
\hline BCL2L10& & 0.051& 0.038& & 0.038\\ CDCP2& -0.272& -0.077& -0.057 & & -0.068\\ HES5& & -0.139 & -0.153& -0.265 & -0.162\\ HLA.E && 0.104 & 0.117 & 0.167 & 0.098\\ HRH3& & 0.021 & & & \\ IRX6& & 0.014 & & & \\ KIF5C& & 0.004& & & \\ NIPSNAP3B& & 0.034 & & & 0.017\\ NPM2 &0.230 & 0.087& 0.065& 0.089 & 0.078\\ OXGR1& & 0.059 & 0.066& & 0.045\\ SLC12A5 &0.282 & 0.144& 0.104& 0.072 & 0.167\\ SMIM11A& 0.417 & 0.349 & 0.469& 0.507& 0.418\\
\hline
Tuning parameters & \tabincell{l}{\ $\xi_n=19$\\ $\lambda_n=1.642$}&$\lambda=0.122$&\tabincell{l}{\ $\gamma=3.7$,\\ $\lambda=0.154$}& $\lambda=0.190$ & $\lambda= 0.625$\\
\hline
{Number of selected} &4 & 12& 9 & 5& 9\\
\hline
{CV error} &3.793 &3.832 &3.804& 3.835& 3.620\\
\hline
\end{tabular}} \label{GBM_M_1} \end{table}
It is seen from Table \ref{GBM_M_1} that our BJASS-CBAR selected the sparsest model with 4 variables while achieving a comparable CV error as compared to the other four methods, which is consistent with our findings in simulation studies. It is interesting to note that the four features selected by BJASS-CBAR have also been selected by three other methods. Among the four selected features, NPM2 and IRX6 have been previously discussed in the literature to possibly play critical roles with human diseases \citep{Eirin2006Long,Box2016Nucleophosmin,Daphna2019Germline,MUMMENHOFF2001193}.
} \section{Discussion}\label{section5} We have rigorously extended the broken adaptive ridge (BAR) penalization method for simultaneous variable selection and parameter estimation to the semiparametric AFT model with right-censored data by coupling BAR penalization with the \citet{Leurgans1987} synthetic data. We have established that the resulting CBAR estimator is asymptotically consistency for variable selection, has an oracle estimation property, {and enjoys a grouping property for highly correlated covariates}. We consider both low and high dimensional covariate settings. Our empirical studies demonstrate that CBAR generally produces a more sparse and accurate model as compared to some popular $L_1$-based penalization methods, which corroborates previous findings in the literature for uncensored data.
We note that coupling the BAR method with the \citet{Leurgans1987} synthetic variable is only one of several possible ways of extending the BAR method to right censored linear model for simultaneous variable selection and parameter estimation. For example, one may couple the BAR method with the \citet{Koul1981Regression} synthetic data method, the \citet{stute1993consistent} weighted least squares method, or the \citet{Buckley1979Linear} iterative imputation method. Our limited numerical studies (not reported here) indicate that using \citet{Koul1981Regression} synthetic data is generally inferior to using \citet{Leurgans1987} synthetic variable, whereas iteratively performing BAR using the \citet{Buckley1979Linear} imputation may sometimes improve the performance of the CBAR method based on the \citet{Leurgans1987} synthetic variable. However, asymptotic properties of each of these distinct approaches require different theoretical developments. Thorough investigations and comparisons of these alternative approaches are needed in future research.
{Lastly, missing data often occurs in real world applications. Although there is a vast amount literature on missing data problems, little has been done to deal with missing data in the context of variable selection for survival data. Further research in this domain is waranteed.
} {
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\end{thebibliography} \appendix \section{Proofs of the theorem} \label{sec: appdx} We first introduce notations and lemmas used to prove Theorem \ref{theorem:CBAR}. \par Using \citet{Leurgans1987} method, we transform $\*Y$ into synthetic data $\*Y^*$. Let $ \bm \beta=( \bm\alpha^\top, \bm\gamma^\top)^\top$, where $ \bm\alpha$ and $ \bm\gamma$ are $q_n\times 1$ and $(p_n-q_n) \times 1$ vector respectively, $\*\Sigma_n=\*X^\top \*X/n$. \begin{equation} \label{eq:t}
g( \bm \beta)=\{ \*X^\top \*X+\lambda_n \*D( \bm \beta)\}^{-1}\* X^\top \*Y^* =( \bm\alpha^{*}( \bm \beta)^\top, \bm\gamma^*( \bm \beta)^\top)^\top. \end{equation} For simplicity, we write $ \bm\alpha^*( \bm \beta)$ and $ \bm\gamma^*( \bm \beta)$ as $ \bm\alpha^*$ and $ \bm\gamma^*$ hereafter. $\*\Sigma_n^{-1}$ can be partitioned as \[ \*\Sigma_n^{-1}=\begin{pmatrix}
\*A_{11} & \*A_{12}\\
\*A^\top_{12} & \*A_{22} \end{pmatrix} \] where the $A_{11} $ is a {$q\times q$ }matrix. Multiplying $( \*X^\top \*X)^{-1}( \*X^\top \*X+\lambda_n \*D( \bm \beta))$ to equation (\ref{eq:t}) \begin{equation} \label{eq:t1} \begin{pmatrix}
{\bm\alpha}^*-{\bm\beta}_{01}\\
{\bm\gamma}^*
\end{pmatrix} +\frac{\lambda_n}{n}\begin{pmatrix} \*A_{11}\*D_1(\bm\alpha)\bm\alpha^*+\*A_{12}\*D_2(\bm\gamma)\bm\gamma^*\\ \*A_{12}^\top\*D_1(\bm\alpha)\bm\alpha^*+\*A_{22}\*D_2(\bm\gamma)\bm\gamma^* \end{pmatrix}=(\*X^\top\*X)^{-1}\*X^\top \bm\varepsilon^* {=\hat{\bm \beta}_{\rm Z}-\bm \beta_0}, \end{equation} where $\bm\varepsilon^*=\*Y^*-\*X{\bm \beta_0}$, {$\hat\bm \beta_{\rm Z}=(\*X^\top\*X)^{-1}\*X^\top\*Y^*$}, $ \*D_1( \bm\alpha)=\mbox{diag }(\alpha_1^{-2},...,\alpha_{{q}}^{-2})$ and $ \*D_2( \bm\gamma)=\mbox{diag }(\gamma_1^{-2},...,\gamma_{p_n-{q}}^{-2})$. \begin{lemma}
\label{lemmma:1}
{ Let $\delta_n$ be a sequence of positive real numbers satisfying $\delta_n \to \infty$ and $p_n\delta_n^2/\lambda_n \to 0$. }
Define \textcolor{black}{$\*H_n = \{ \bm \beta\in \mathbb{R}^{p_n}: \|\bm\beta-\bm\beta_0\| \leq \delta_n\sqrt{p_n/n}\}$ and $\*H_{n1} = \{ \bm\alpha\in \mathbb{R}^{{q}}: \|\bm\alpha-\bm\beta_{01}\| \leq \delta_n\sqrt{p_n/n}\}$}. Assume conditions (C1)-(C5) hold. Then, with probability tending to $1$, we have \begin{itemize}
\item [(a)]
$\sup_{ \bm\beta \in \*H_n} {\| \bm\gamma^*\|}/{\| \bm\gamma\|}< {1}/{C_0},\mbox{ for some constant } C_0>1$;
\item [(b)]
$g $ is a mapping from $ \*H_n$ to itself. \end{itemize} \end{lemma}
\begin{proof} We first prove part (a).
{First, under $\lambda_n/\sqrt{n} \to 0$ and $p_n\delta_n^2/\lambda_n \to 0$, we have $\delta_n\sqrt{p_n/n} \to 0$.}
Let $\hat\bm \beta_{\rm Z}=(\*X^\top\*X)^{-1}\*X^\top\*Y^*$, $\omega_{ji}=((\*X^\top \*X)^{-1}\*X^\top )_{ji}$, $\mu_j^*=\sum_i \omega_{ji}\int_{0}^{T_n}{F_i dt}$ and $\bm \mu =(\mu_1^*, \mu_2^*, ..., \mu_{{pn}}^*)$.
{For any $p_n$-vector $\*b_n$ which $\| \*b_n\|\leq 1$, define $t_n^2= \*b_n^\top \*\Omega(\infty)) \*b_n$. Then, we have $\sqrt{n} \, t_n^{-1} \*b_n^\top (\hat\bm \beta_{\rm Z}-\bm\mu) \rightarrow_D N(0,1).$ This result can be proved using similar techniques to those used in the proof of Theorem 3.1 of \citet{Zhou1992Asymptotic} along the same lines as outlined below: First, we separate $\*b_n^\top (\hat\bm \beta_{\rm Z}-\bm\mu)$ like (3.6) in \citet{Zhou1992Asymptotic} with a main term $S_{\bm \beta}(T^n)$ and a remainder term $SS_{\bm \beta}(T^n)$, i.e., $\*b_n^\top (\hat\bm \beta_{\rm Z}-\bm\mu)=S_{\bm \beta}(T^n)+SS_{\bm \beta}(T^n)$, where $S_{\bm \beta}(T^n)$ is a weighted sum of $\hat H(t)-H(t)$ and $\hat G(t)-G(t)$; and $SS_{\bm \beta}(T^n)$ is a weighted sum of $(\hat H(t)-H(t))(\hat G(t)-G(t))$ and $(\hat H(t)-H(t))(\hat H(t)-H(t))$. Second, under conditions (C2) and (C3), one can show that $\sqrt{n}SS_{\bm \beta}(T^n)$ is negligible. Finally, by applying the martingale central limit theorem and conditions (C1) and (C4), we establish the asymptotic normality of $\sqrt{n}S_{\bm \beta}(T^n)$.
{ By condition (C1) and (C2), we have $ \sqrt{n}t_n^{-1}\*b_n^\top(\bm \beta_0 - \bm\mu) = o_p(1)$, for $\*b_n=\*e_i=(0,...,1,0,...,0)$}. Hence, we have
$\|\hat{\bm \beta}_{\rm Z}-\bm \beta_0\|^2=O_p(p_n/n)$.}
It then follows from (\ref{eq:t1}) that
\begin{equation}
\label{super:3}
\sup_{ \beta \in \*H_n}\big\| \bm\gamma^*+ \lambda_n \*A_{12}^\top \*D_1( \bm\alpha) \bm\alpha^*/n + \lambda_n \*A_{22} \*D_2( \bm\gamma) \bm\gamma^*/n \big\|=O_p(\sqrt{{p_n}/{n}}).
\end{equation}
Note that \textcolor{black}{$\|\bm\alpha - \bm\beta_{01}\|\leq \delta_n(p_n/n)^{1/2}$ and $\| \bm\alpha^*\|\leq \|g( \bm \beta)\|\leq \|\hat{ \bm \beta}_{\rm Z}\|=O_p({\sqrt{p_n}})$}. By assumptions (C4) and (C5), we have
\begin{equation}
\label{super:4}
\begin{split}
\sup_{ \bm \beta \in \*H_n}\left\| \lambda_n \*A_{12}^\top \*D_1( \bm\alpha) \bm\alpha^*/n \right\| & \leq \frac{\lambda_n}{n} \, \| \*A_{12}^\top \|\sup_{ \bm \beta \in \*H_n}\| \*D_1( \bm\alpha)
\bm\alpha^*\|\\ & \leq{\color{black} \sqrt{2}\, \tilde{C}\, \frac{\lambda_n}{n} \, {\frac{a_{1}}{a_{0}^2}}\sup_{ \bm \beta \in \*H_n}\| \bm\alpha^*\|}= o_p(\sqrt{{p_n}/{n}}),
\end{split}
\end{equation}
where the second inequality uses the fact $\|\*A_{12}^\top \|\leq \sqrt{2}\, \tilde{C}$, which follows from the inequality $\|\*A_{12}\*A_{12}^\top \|-\|\*A_{11}^2\|\leq \|\*A_{11}^2+\*A_{12}\*A_{21}\|\leq\|\*\Sigma_n^{-2}\|<\tilde{C}^2.$
Combining (\ref{super:3}) and (\ref{super:4}) gives
\begin{equation}
\label{superr:1}
\sup_{ \bm \beta \in \*H_n}\left\| \bm\gamma^*+ \lambda_n \*A_{22} \*D_2( \bm\gamma) \bm\gamma^*/n \right\|=O_p(\sqrt{{p_n}/{n}}).
\end{equation}
Note that $ \*A_{22}=\sum_{i=1}^{p_n-{q}}\tau_{2i} \*u_{2i} \*u_{2i}^\top $ is positive definite and by the singular value decomposition, , where $\tau_{2i}$ and $ \*u_{2i}$ are eigenvalues and eigenvectors of $ \*A_{22}$. Then, since $1/\tilde{C} <\tau_{2i}< \tilde{C}$, we have
\begin{equation}\nonumber
\begin{split}
\frac{\lambda_n}{n} \, \| \*A_{22} \*D_2( \bm\gamma) \bm\gamma^*\|&=\frac{\lambda_n}{n}\left\|\sum_{i=1}^{p_n-{q}}\tau_{2i} \*u_{2i} \*u_{2i}^\top \*D_2( \bm\gamma) \bm\gamma^*\right\|
= \frac{\lambda_n}{n}\left\{\sum_{i=1}^{p_n-{q}}\tau_{2i}^2\| \*u_{2i}^\top \*D_2( \bm\gamma) \bm\gamma^*\|^2\right\}^{1/2}\\
&\geq \frac{\lambda_n}{n}\frac{1}{\tilde{C}}\left\{\sum_{i=1}^{p_n-{q}}\| \*u_{2i}^\top \*D_2( \bm\gamma) \bm\gamma^*\|^2\right\}^{1/2} = \frac{1}{\tilde{C}} \left\| \lambda_n \*D_2( \bm\gamma) \bm\gamma^* /n \right\|.
\end{split}
\end{equation}
This, together with (\ref{superr:1}) and (C4), implies that {with probability tending to $1$,}
\begin{equation}
\label{superr:2}
\frac{1}{\tilde{C}}\left\|\lambda_n \*D_2( \bm\gamma) \bm\gamma^*/n \right\|-\| \bm\gamma^*\|\leq \delta_n\sqrt{{p_n}/{n}}.
\end{equation}
\par Let $ \*d_{\gamma*/\gamma}=(\gamma^*_1/\gamma_1, \ldots,\gamma^*_{p_n-{q}}/\gamma_{p_n-{q}})^\top $. Because $\| \bm\gamma\|\leq\delta_n\sqrt{p_n/n}$, we have
\begin{equation}
\label{superr:21}
\frac{1}{\tilde{C}}\left\|\frac{\lambda_n}{n}\, \*D_2( \bm\gamma) \bm\gamma^*\right\| =\frac{1}{\tilde{C}}\frac{\lambda_n}{n}\left\|{ \{\*D_2( \bm\gamma)}\}^{1/2} \*d_{\bm\gamma*/\bm\gamma}\right\| \geq \frac{1}{\tilde{C}}\frac{\lambda_n}{n}\frac{\sqrt{n}}{\delta_n\sqrt{p_n}} \, \| \*d_{\bm\gamma*/\bm\gamma}\|
\end{equation}
and
\begin{equation}
\label{superr:22}
\| \bm\gamma^*\| =\|{ \*D_2( \bm\gamma)}^{-1/2} \*d_{\bm\gamma*/\bm\gamma}\|\leq \frac{\delta_n\sqrt{p_n}}{\sqrt{n}}\, \| \*d_{\bm\gamma*/\bm\gamma}\|.
\end{equation}
Combining (\ref{superr:2}), (\ref{superr:21}) and (\ref{superr:22}), we have
that with probability tending to $1$,
\begin{equation}
\label{superr:23}
\| \*d_{\bm\gamma*/\bm\gamma}\|\leq \frac{1}{{\lambda_n}/({p_n}\delta_n^2 \tilde{C})-1}<{1}/{C_0}
\end{equation}
{for some constant $C_0 > 1$ provided that $\lambda_n/({p_n}\delta_n^2) \to \infty$}.
It is worth noting that $\Pr(\| \*d_{\bm\gamma*/\bm\gamma}\| \to 0) \to 1$, as $n \to \infty$.
Furthermore, with probability tending to $1$,
\begin{equation*}
\label{superr:24}
\| \bm\gamma^*\|\leq\| \*d_{\bm\gamma^*/\bm\gamma}\|\max_{1\leq j\leq (p_n-{q})}|\bm\gamma_j|\leq\| \*d_{\bm\gamma^*/\bm\gamma}\|\times\|\bm\gamma\|\leq \| \bm\gamma\|/C_0.
\end{equation*}
This proves part (a).
Next we prove part (b). First, it is easy to see from (\ref{superr:22}) and (\ref{superr:23}) that, as $n \to \infty$,
\begin{equation}
\label{lemma1:i1}
\Pr \Big (\|
\bm\gamma^*\|\leq \delta_n\sqrt{p_n/n} \Big ) \to 1.
\end{equation}
Then, by (\ref{eq:t1}), we have
\begin{equation}
\label{super:6}
\sup_{ \bm\beta \in \*H_n}\left\| \bm\alpha^*- \bm \beta_{01}+ \lambda_n \*A_{11} \*D_1( \bm\alpha) \bm\alpha^*/n +\lambda_n \*A_{12} \*D_2( \bm\gamma)\bm\gamma^* /n \right\|=O_p(\sqrt{{p_n}/{n}}).
\end{equation}
Similar to (\ref{super:4}), it is easily to verify that
\begin{equation}
\label{super:61}
\sup_{ \bm \beta \in \*H_n}\left\| \lambda_n \*A_{11} \*D_1( \bm\alpha)\bm\alpha^*/n \right\| = o_p(\sqrt{{p_n}/{n}}).
\end{equation}
Moreover, with probability tending to $1$,
\begin{equation}
\label{super:62}
\sup_{ \bm \beta \in \*H_n} \left\| \lambda_n \*A_{12} \*D_2( \bm\gamma) \bm\gamma^*/n \right\| \leq\frac{\lambda_n}{n} \sup_{ \bm \beta \in \* H_n} \left\| \*D_2( \bm\gamma) \bm\gamma^*\right\| \times \| \*A_{12}\|\leq 2\sqrt{2}\tilde{C}^2\delta_n\sqrt{{p_n}/{n}},
\end{equation}
where the last step follows from (\ref{superr:2}), (\ref{lemma1:i1}), and the fact that $\|\* A_{12}\|\leq \sqrt{2}\tilde{C}$. It follows from (\ref{super:6}), (\ref{super:61}) and (\ref{super:62}) that
with probability tending to $1$,
\begin{equation}
\label{superr:3}
\sup_{ \bm \beta \in \*H_n} \| \bm\alpha^*- \bm \beta_{01}\|\leq { \big (2\sqrt{2}\tilde{C}^2+1 \big )\delta_n n^{-1/2}\sqrt{p_n}}.
\end{equation}
Because $\delta_n\sqrt{p_n}/\sqrt{n} \to 0$, we have, as $n \to \infty$,
{\color{black}\begin{equation}
\label{lemma1:i2}
\Pr ( \bm\alpha^*\in \*H_{n1}) \to 1.
\end{equation}}
Combining (\ref{lemma1:i1}) and (\ref{lemma1:i2}) completes the proof of part (b). \end{proof} \begin{lemma}
\label{lemmma:2}
Assume that (C1)-(C5) hold.
For any ${q}$-vector $ {\*c}$ satisfying $\| {\*c}\|\leq 1$, define ${z^2}= {\*c}^\top \*\Omega_{1} {\*c}$ as in Theorem \ref{theorem:CBAR}.
Define
\begin{equation}\label{eq:7}
f( \bm\alpha)=\{ \*X_{1} ^\top \*X_1+\lambda_n \*D_1( \bm\alpha)\}^{-1} \*X_1^\top \*Y^*.
\end{equation}
Then, with probability tending to $1$,
(a) $f( \bm\alpha)$ is a contraction mapping from {\color{black}$\*B_{n} \equiv\{ \bm\alpha\in \mathbb{R}^{{q}}: \|\bm\alpha-\bm\beta_{01}\| \leq \delta_n\sqrt{p_n/n}\}$} to itself;
(b)
$\sqrt{n} \, {z^{-1} \*c^\top}(\hat{ \bm\alpha}^{\circ}- \bm \beta_{01}) \rightsquigarrow \mathcal{N}(0,1),$
where $\hat{ \bm\alpha}^{\circ}$ is the unique fixed point of $f(\bm\alpha)$ defined by $$
\hat{ \bm\alpha}^{\circ}= \{ \*X_{1}^\top { \*X_1}+\lambda_n \*D_1( \hat{ \bm\alpha}^{\circ})\}^{-1}{ \*X}_1^\top \*Y^*. $$ \end{lemma}
\begin{proof} We first prove part (a). Note that (\ref{eq:7}) can be rewritten as \begin{equation}\nonumber f( \bm\alpha)- \bm \beta_{01}+\frac{\lambda_n}{n} \bm\Sigma_{n1}^{-1} \*D_1( \bm\alpha)f( \bm\alpha) = {\hat{\bm \beta}_{1\rm Z}- \bm \beta_{01}}. \end{equation} {where $\hat{\bm \beta}_{1\rm Z}=( \*X_1^\top \*X_1)^{-1} \*X_1^\top \* Y^*$}. Then, \begin{equation} \label{lemma2:m1}
\sup_{ \bm\alpha\in \*B_n}\left\|f( \bm\alpha)- \bm \beta_{01}+(\lambda_n/n) \*\Sigma_{n1}^{-1} \*D_1( \bm\alpha)f( \bm\alpha) \right\|= O_p({1/\sqrt{n}}). \end{equation} \begin{equation} \label{lemma2:m2}
\sup_{ \bm\alpha\in \*B_n}\left\|(\lambda_n/n) \*\Sigma_{n1}^{-1} \*D_1( \bm\alpha)f( \bm\alpha) \right\| =o_p({1/\sqrt{n}}). \end{equation} It follows from (\ref{lemma2:m1}) and (\ref{lemma2:m2}) that \begin{equation} \label{lemma2:m3}
\sup_{ \bm\alpha\in \*B_n}\left\|f( \bm\alpha)- \bm \beta_{01} \right\|\leq \delta_n{/\sqrt{n}}, \end{equation} where $\delta_n \to\infty$ and $\delta_n{/\sqrt{n}}\rightarrow 0$. Then we can get \begin{equation} \label{map} \mbox{Pr}(f( \bm\alpha)\in \*B_n)\rightarrow 1, \mbox{ as } n\rightarrow \infty. \end{equation} This means that $f$ is a mapping from the region $\*B_n$ to itself.
Rewrite (\ref{eq:7}) as $\{ \*X_{1}^\top{ \*X_1}+\lambda_n \*D_1( \bm\alpha)\}f( \bm\alpha)= \*X_{1}^\top \*Y^*$, then, we have \begin{equation} ( \*\Sigma_{n1}+(\lambda_n/n){ \*D_1}(\bm \alpha))\dot{f}( \bm\alpha)+(\lambda_n/n)\mbox{ diag } \{-2f_j( \bm\alpha)/{\alpha_j^3}\} ={ 0}, \end{equation} where $\dot{f}( \bm\alpha)={\partial f( \bm\alpha)}/{\partial { \bm\alpha^\top}}$ and $\mbox{ diag } \{\frac{-2f_j( \bm\alpha)}{\alpha_j^3}\}= \mbox{ diag }\{\frac{-2f_1( \bm\alpha)}{\alpha_1^3},...,\frac{-2f_{{q}}( \bm\alpha)}{\alpha_{{q}}^3}\}.$ With the assumption $\lambda_n/\sqrt{n}\rightarrow 0$, \begin{equation} \label{eq:a}
\sup_{ \bm\alpha\in \*B_n }\|\{ \*\Sigma_{n1}+\frac{\lambda_n}{n}{ \*D}_1( \bm\alpha)\}\dot{f}( \bm\alpha)\| = \sup_{ \bm\alpha\in \*B_n} \frac{2\lambda_n}{n}\|\mbox{ diag } \{\frac{f_j( \bm\alpha)}{\alpha_j^3}\}\|=o_p(1). \end{equation} Write $ \*\Sigma_{n1} = \sum_{i=1}^{{q}}\tau_{1i} \*u_{1i} \*u_{1i}^\top$, where $\tau_{1i}$ and $ \*u_{1i}$ are eigenvalues and eigenvectors of $ \*\Sigma_{n1}$. Then, by (C4), $1/\tilde{C}<\tau_{1i}< \tilde{C}$ for all $i$ and \begin{equation} \label{superr:38} \begin{split}
\| \*\Sigma_{n1}\dot{f}( \bm\alpha)\|&=\sup_{\| \*x\|=1, \*x\in R^{{q}}}\| \*\Sigma_{n1}\dot{f}( \bm\alpha) \*x\|=\sup_{\| \*x\|=1, \*x\in R^{{q}}}\left\|\sum_{i=1}^{{q}}\lambda_{1i} \*u_{1i} \*u_{1i}^\top\dot{f}( \bm\alpha) \*x\right\|\\
&= \sup_{\| \*x\|=1, \*x\in R^{{q}}}\left(\sum_{i=1}^{{q}}\lambda_{1i}^2\| \*u_{1i}^\top\dot{f}( \bm\alpha) \*x\|^2\right)^{1/2}
\geq \sup_{\| \*x\|=1, \*x\in R^{{q}}}\frac{1}{\tilde{C}}\left(\sum_{i=1}^{{q}}\| \*u_{1i}^\top\dot{f}(\bm \alpha) \*x\|^2\right)^{1/2} \\
&= \sup_{\| \*x\|=1, \*x\in R^{{q}}}\frac{1}{\tilde{C}} \|\dot{f}( \bm\alpha) \*x\|=\frac{1}{\tilde{C}}\|\dot{f}( \bm\alpha)\|. \end{split} \end{equation} Therefore, it follows from $\bm \alpha\in \*B_n$, (\ref{superr:38}) and (C4) that \begin{align*}
\left\|\left\{ \*\Sigma_{n1}+(\lambda_n/n){\* D}_1( \bm\alpha)\right\}\dot{f}( \bm\alpha)\right\|
&\geq \left\| \*\Sigma_{n1}\dot{f}( \bm\alpha)\right\|-\left\|(\lambda_n/n){ \*D}_1( \bm\alpha)\dot{f}( \bm\alpha)\right\|\\
&\geq (1/\tilde{C})\|\dot{f}( \bm\alpha)\|-(\lambda_n/n)\cdot {a_{0}^{-2}}\|\dot{f}( \bm\alpha)\|, \end{align*} This, together with (\ref{eq:a}) and the fact $\lambda_n/n\rightarrow 0$, implies that \begin{equation} \label{eq:b}
\sup_{ \bm\alpha\in \*B_n}\|\dot{f}( \bm\alpha)\|=o_p(1). \end{equation} Finally, we can get the conclusion in part (a) from (\ref{map}) and (\ref{eq:b}).
Next we prove part (b). Write \begin{equation} \label{eq:decomposition} \begin{split} n^{1/2} \, {z^{-1} \*c^\top} (\hat{ \bm\alpha}^\circ- \bm \beta_{01})&=n^{1/2} \, {z^{-1} \*c^\top} \left[ \left\{ \*\Sigma_{n1}+\frac{\lambda_n}{n} \*D_1(\hat{ \bm\alpha}^\circ)\right\}^{-1} \*\Sigma_{n1}- \*I_{q_n}\right] \bm \beta_{01}\\ &+ n^{-1/2}\, {z^{-1} \*c^\top} \left\{ \*\Sigma_{n1} +\frac{\lambda_n}{n} \*D_1(\hat{ \bm\alpha}^\circ)\right\}^{-1} \*X_{1}^\top {\bm\varepsilon}^* \equiv I_1 +I_2. \end{split} \end{equation} By the first order resolvent expansion formula \[( \*H+ \*\Delta)^{-1}= \*H^{-1}- \*H^{-1}\*\Delta( \*H+\*\Delta)^{-1},\] the first term on the right hand side of equation (\ref{eq:decomposition}) can be rewritten as \begin{equation}\nonumber I_1 = - {z^{-1} \*c^\top} \*\Sigma_{n1}^{-1}\frac{\lambda_n}{\sqrt{n}} \*D_1(\hat{ \bm\alpha}^\circ)\left\{ \*\Sigma_{n1} +\frac{\lambda_n}{n} \*D_1(\hat{ \bm\alpha}^\circ)\right\}^{-1}\* \Sigma_{n1}\bm \beta_{01}. \end{equation} Hence, by the assumption (C4) and (C5), we have \begin{equation}\label{eq:27a}
\|I_1\|\leq {\color{black}\frac{\lambda_n}{\sqrt{n}}{z^{-1}a_{0}^{-2}}\|\*\Sigma_{n1}^{-1}\bm \beta_{01}\|=O_p\bigg({\lambda_n/\sqrt{n}}\bigg) \to 0.}
\end{equation} Furthermore, applying the first order resolvent expansion formula, it can be shown that \begin{equation}\label{eq:A28} \begin{split} I_2&={\frac{z^{-1} }{\sqrt{n}}\*c^{{ \mathrm{\scriptscriptstyle T} }} \*\Sigma_{n1}^{-1}\*X_1^{{ \mathrm{\scriptscriptstyle T} }}{\bm\varepsilon}^*+o_p(1)}\\ &={\frac{z^{-1} }{\sqrt{n}}\*c^{{ \mathrm{\scriptscriptstyle T} }} \*\Sigma_{n1}^{-1}\*X_1^{{ \mathrm{\scriptscriptstyle T} }}(\*Y^*-\*X_1 \bm\mu+\*X_1 \bm\mu-\*X_1\bm \beta_{01})+o_p(1)}\\ &={\sqrt{n}z^{-1}\*c^{{ \mathrm{\scriptscriptstyle T} }}(\hat{\bm \beta}_{1\rm Z}-\bm\mu_1+\bm\mu_1- \bm \beta_{01}) +o_p(1) }\\
\end{split} \end{equation} {where $\bm \mu_1 =(\mu_1^*, \mu_2^*, ..., \mu_{q}^*)$. $I_2$} converges in distribution to $N(0,1)$ by the Lindeberg-Feller central limit theorem. Finally, combining (\ref{eq:decomposition}), (\ref{eq:27a}), and (\ref{eq:A28}) proves part (b). \end{proof}
\noindent {\bf Proof of Theorem \ref{theorem:CBAR}.} Given the initial ridge estimator $\hat{ \bm \beta}^{(0)}$ in (\ref{ridge}), we have \begin{equation} \begin{split} \hat{\bm\beta}^{(0)}-\bm\beta_0&=[ ( \*\Sigma_{n}+\frac{\xi_n}{n} \*I_{p_n})^{-1} \*\Sigma_{n}- \*I_{p_n}] \bm \beta_{0}+( \*\Sigma_{n} +\frac{\xi_n}{n} \*I_{p_n})^{-1} \*X^\top {\bm\varepsilon}^*/n.\\ &\equiv \*T_1+\*T_2. \end{split} \end{equation} By the first order resolvent expansion formula and $\xi_n/\sqrt{n}\rightarrow 0$, \begin{equation}
\|\*T_1\|=\left\|- \*\Sigma_{n}^{-1}\frac{\xi_n}{{n}} ( \*\Sigma_{n}
+\frac{\xi_n}{n}\*I_{p_n})^{-1}\* \Sigma_{n}\bm \beta_{0}\right\|\leq \tilde{C}^3\frac{\xi_n {a_{1}}\sqrt{p_n}}{n} =o_p(\sqrt{\frac{p_n}{n}}). \end{equation} It is easy to see that $
\|\*T_2\|=O_p(\sqrt{{p_n}/{n}}). $
Thus $\|\hat{ \bm \beta}^{(0)}- \bm \beta_0\|=O_p((p_n/n)^{1/2})$. This, combined with part (a) of Lemma 1, implies that \begin{equation}\label{eq:A29} \mbox{Pr}(\lim_{k\rightarrow\infty}{\hat{ \bm\gamma}^{(k)}}= 0)\rightarrow 1. \end{equation} Hence, to prove part (i) of Theorem 1, it is sufficient to show that \begin{equation}\label{eq:A30}
\mbox{Pr}(\lim_{k\rightarrow \infty}\| {\hat{ \bm\alpha}^{(k)}}-\hat{ \bm\alpha}^\circ\|=0)\rightarrow 1, \end{equation} where $\hat{ \bm\alpha}^\circ$ is the fixed point of $f(\bm \alpha)$ defined in part (b) of Lemma 2.
Define $ \bm\gamma^*= 0$ if $\bm\gamma = 0$, for any $\bm \alpha\in \*B_n$, \begin{equation}\label{eq:A31} \lim_{ \bm\gamma\rightarrow 0} \bm\gamma^*( \bm\alpha,\bm \gamma)= 0. \end{equation} Combining (\ref{eq:A31}) with the fact \begin{equation}\nonumber \begin{pmatrix}
\*X_1^\top{ \*X_1}+\lambda_n \*D_1( \bm\alpha) & \*X_1^\top{\* X_2}\\
\*X_2^\top{ \*X_1}& \*X_2^\top{ \*X_2}+\lambda_n \*D_2( \bm\gamma) \end{pmatrix} \begin{pmatrix}
\bm\alpha^*\\
\bm\gamma^* \end{pmatrix} =\begin{pmatrix}
\*X_1^\top \*Y^*\\
\*X_2^\top \*Y^* \end{pmatrix}, \end{equation} implies that for any $\bm \alpha\in \*B_n$, \begin{equation} \label{lim:111} \lim_{ \bm\gamma\rightarrow 0} \bm\alpha^*(\bm \alpha, \bm\gamma)=\{ \*X_1^{{ \mathrm{\scriptscriptstyle T} }}{\* X_1}+\lambda_n \*D_1( \bm\alpha)\}^{-1}\* X_1 \*Y^*{=f( \bm\alpha)}. \end{equation} Therefore, $g(\cdot)$ is continuous and thus uniformly continuous on the compact set $ \bm \beta\in \*H_n$. This, together with (\ref{eq:A29}) and (\ref{lim:111}), implies that
as $k\rightarrow\infty$, \begin{eqnarray} \label{eq:c}
\eta_k\equiv \sup_{\bm \alpha\in \*B_n}\left\|f( \bm\alpha)- \bm\alpha^*( \bm\alpha,{\hat{ \bm\gamma}^{(k)}})\right\|\longrightarrow 0, \end{eqnarray} with probability tending to 1.
Note that \begin{eqnarray} \nonumber
\| {\hat{ \bm\alpha}^{(k+1)}}-\hat{\bm \alpha}^\circ\| &=& \left\| \bm\alpha^*({\hat{ \bm\beta}^{(k)}})-\hat{ \bm\alpha}^\circ\right\|
\le \left\| \bm\alpha^*({\hat{\bm \beta}^{(k)}})-f({\hat{\bm \alpha}^{(k)}})\right\|+\|f({\hat{ \bm\alpha}^{(k)}})-\hat{ \bm\alpha}^\circ\|\\
&\le & \eta_k + \frac{1}{\tilde{C}}\|{\hat{ \bm\alpha}^{(k)}}-\hat{ \bm\alpha}^\circ\|, \label{eq:A34} \end{eqnarray} where the last step follows from
$\|f({\hat{ \bm\alpha}^{(k)}})-\hat{\bm \alpha}^\circ\|=\|f({\hat{ \bm\alpha}^{(k)}})-f(\hat{ \bm\alpha}^\circ)\|\leq (1/\tilde{C})\|{\hat{ \bm\alpha}^{(k)}}-\hat{ \bm\alpha}^\circ\|$. Let $a_k=\| {\hat{ \bm\alpha}^{(k)}}-\hat{\bm \alpha}^\circ\|$, for all $k\geq 0$. From (\ref{eq:c}), we can induce that with probability tending to 1, for any $ \epsilon>0$, there exists an positive integer $N$ such that for all $k> N$, $|\eta_k|<\epsilon$ and \begin{align*} a_{k+1} &\leq \frac{a_{k-1}}{\tilde{C}^2} + \frac{\eta_{k-1}}{\tilde{C}}+\eta_k\\ & \leq \frac{a_1}{\tilde{C}^k}+\frac{\eta_1}{\tilde{C}^{k-1}}+ \cdots+\frac{\eta_N}{\tilde{C}^{k-N}}+ (\frac{\eta_{N+1}}{\tilde{C}^{k-N-1}}+\cdots +\frac{\eta_{k-1}}{\tilde{C}}+\eta_k)\\ &\le (a_1+\eta_1+...+\eta_N) \frac{1}{\tilde{C}^{k-N}} + \frac{1-(1/\tilde{C})^{k-N}}{1-1/\tilde{C}} \epsilon
\rightarrow 0, \mbox{ as } k\rightarrow\infty. \end{align*} This proves (\ref{eq:A30}).
Therefore, it immediately follows from (\ref{eq:A29}) and (\ref{eq:A30}) that the with probability tending to 1, $ \lim_{k\to\infty} \bm \beta^{(k)}= \lim_{k\to\infty} (\hat{ \bm\alpha}^{(k)\top} , \hat{ \bm\gamma}^{(k)\top})^\top=(\hat{ \bm\alpha}^{\circ\top} , 0)^{{ \mathrm{\scriptscriptstyle T} }}$, which completes the proof of part (i). This, in addition to part (b) of Lemma 2, proves part (ii) of Theorem \ref{theorem:CBAR}. \qed
{{\bf Proof of Theorem \ref{theorem:2}.}
Recall that $\hat{\bm \beta}^* =\lim_{k\to\infty}\hat{ \bm\beta}^{(k+1)}$ and
$\hat{ \bm\beta}^{(k+1)}=\arg\min_{ \bm\beta} \{ Q(\bm \beta| \hat{ \bm\beta}^{(k)})\}$, where $$
Q(\bm \beta| \hat{ \bm\beta}^{(k)})= \| \*Y^*-\* X \bm \beta\|^2+\lambda_n
\sum_{\ell=1}^{p_n} {\beta_\ell^2}/{\{\hat{\beta}^{(k)}_\ell\}^2}. $$ If $\beta_\ell^*\ne 0$ for $\ell \in \{ i,j\}$, then $\hat{\bm \beta}^* $ must satisfy the following normal equations for $\ell \in \{ i,j\}$: \begin{equation}\nonumber -2 \*x_\ell^\top \{\*Y^*- \*X\hat{ \bm \beta}^{(k+1)}\}+2\lambda_n {\hat{\beta}_\ell^{(k+1)}}/{\{\hat{\beta}^{(k)}_\ell\}^2} =0. \end{equation} Thus, for $\ell \in \{ i, j \}$, \begin{equation} \label{eqss:2} {\hat{\beta}_\ell^{(k+1)}}/{\{\hat{\beta}^{(k)}_\ell \}^2} = {\*x_\ell^\top \hat{\bm\varepsilon}^{*(k+1)}}/{\lambda_n}, \end{equation} where $\hat{\bm\varepsilon}^{*(k+1)}= \*Y^*- \*X\hat{ \bm \beta}^{(k+1)}$. Moreover, because
$$\|\hat{\bm\varepsilon}^{*(k+1)}\|^2+ \lambda_n\sum_{i=1}^{p_n}\frac{\hat{\beta}_i^{2}}{\tilde{\beta}_i^2}=Q(\hat{ \bm\beta}^{(k+1)}| \hat{ \bm\beta}^{(k)})\leq Q(0|\hat{ \bm\beta}^{(k)}) =\| \*Y^*\|^2,$$ we have \begin{equation} \label{eq:A36}
\|\hat{\bm\varepsilon}^{*(k+1)}\|\leq \| \*\*Y^*\| \end{equation}
Letting $k\to \infty$ in (\ref{eqss:2}) and (\ref{eq:A36}), we have, for $\ell \in \{i, j\}$ and $\|\hat{\bm\varepsilon}^{*}\|\leq \| \*Y^*\|$, $\hat{\beta}_\ell ^{*-1}= \*x_\ell ^\top \hat{\bm\varepsilon}^{*}{\lambda_n}$, where $\hat{\bm\varepsilon}^{*} = \*Y^*- \*X\hat{ \bm \beta}^*$. Therefore,
\[\big|\hat{\beta}_i^{*-1}-\hat{\beta}_j^{*-1}\big|\leq \frac{1}{\lambda_n} \, \| \*Y^*\| \times \|\*x_i - \*x_j\| = \frac{1}{\lambda_n}
\, \| \*Y^*\|\sqrt{2(1-\rho_{ij})}. \]
$\Box$}
\begin{comment}
\end{comment}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\author{\name David P.\ Hofmeyr
{\small \textmd{ Department of Statistics and Actuarial Science}}\\ \name Francois Kamper
{\small \textmd{Stellenbosch University}}\\ \name Michail C. Melonas
{\small \textmd{7600, South Africa}} }
\title{Optimal Projections for Gaussian Discriminants}
\begin{abstract} The problem of obtaining optimal projections for performing discriminant analysis with Gaussian class densities is studied. Unlike in most existing approaches to the problem, the focus of the optimisation is on the multinomial likelihood based on posterior probability estimates, which directly captures discriminability of classes. In addition to the more commonly considered problem, in this context, of classification, the unsupervised clustering counterpart is also considered. Finding optimal projections offers utility for dimension reduction and regularisation, as well as instructive visualisation for better model interpretability. Practical applications of the proposed approach show considerable promise for both classification and clustering. Code to implement the proposed method is available in the form of an {\tt R} package from \url{https://github.com/DavidHofmeyr/OPGD}. \end{abstract}
\paragraph{Keywords:} Classification; clustering; dimension reduction; feature extraction; visualisation
\section{Introduction} \label{sec:intro}
In this paper we study the problem of finding optimal linear projections of a set of data, for the purpose of discriminant analysis. We study both the supervised classification problem, in which a set of known {\em class labels} is used to guide the selection of the projection to one which enhances the discriminability of the points in different classes from one another; and the unsupervised clustering problem, in which the relative spatial relationships between points are used to guide the projection to one on which subsets of points in the data (clusters) each appear more internally cohesive, while simultaneously (as entire groups) more separated from other cohesive groups. We focus specifically on the case in which the discriminability of classes/clusters is measured in terms of the multinomial likelihood based on estimates of the posterior probabilities of class/cluster membership. These posterior probability estimates arise from the standard application of Bayes' theorem, where we model each class/cluster conditional distribution with a multivariate Gaussian density.
The remaining paper is organised as follows. In Section~\ref{sec:background} we provide an explicit introduction to the problem of discriminant analysis for classification, as well as covering existing literature on the topic. In Section~\ref{sec:methodology} we elaborate on our adopted methodology and cover explicitly the practical issues associated with its implementation. In Section~\ref{sec:classificationexperiments} we discuss the results of experiments using our method when applied to publicly available data sets from diverse applications areas, and in comparison with existing methods from the literature. Then, in Section~\ref{sec:clustering}, we introduce a simple modification to our approach which allows us to apply it to the problem of clustering, and investigate the practical relevance of this modification for improving the performance of Gaussian Mixture Models. We then conclude our work in Section~\ref{sec:conclusions} with a discussion of our findings and experiences with the method.
\section{Discriminant Analysis} \label{sec:background}
Discriminant analysis in a probabilistic framework is concerned with estimating the probabilities $P(Y = k |\mathbf{X} = \mathbf{x}); k = 1, ..., K,$ via a simple reformulation based on Bayes' theorem, i.e.,
\begin{align}\label{eq:bayes}
P(Y = k | \mathbf{X} = \mathbf{x}) &= \frac{\pi_kf_{X|Y=k}(\mathbf{x})}{\sum_{l=1}^K \pi_lf_{X|Y=l}(\mathbf{x})}, \end{align}
where we have used $f_{X|Y=k}$ to denote the density of the random variable $X | Y = k$, and $\pi_k$ is used to represent the {\em prior class probability} $P(Y = k)$. Different classification models in this framework differ in how they estimate the functions $f_{X|Y=k}$. it is almost universal that the prior probabilities are estimated as $\hat \pi_k = \frac{n_k}{n}; n_k:= \sum_{i=1}^n I(y_i=k)$, where $I$ is the indicator function and $\{y_1, ..., y_n\}$ are the labels associated with the data, $\{\mathbf{x}_1, ..., \mathbf{x}_n\}$. A very popular approach is to estimate the {\em posterior probabilities} in Eq.~(\ref{eq:bayes}) by replacing $f_{X|Y=k}$ with the Gaussian density with mean $ \boldsymbol{\mu}_k$ and covariance matrix $ \Sigma_k$, using suitably chosen estimates of these parameters, denoted by $\hat \boldsymbol{\mu}_k$ and $\hat \Sigma_k$. That is,
\begin{align}
\widehat{P(Y = k| \mathbf{X} = \mathbf{x})} &= \frac{\hat \pi_k \phi_{\hat\boldsymbol{\mu}_k, \hat \Sigma_k}(\mathbf{x})}{\sum_{l=1}^K \hat \pi_l \phi_{\hat\boldsymbol{\mu}_l, \hat \Sigma_l}(\mathbf{x})} \label{eq:bayesGMM}\\
\phi_{\boldsymbol{\mu}, \Sigma}\left(\mathbf{x}\right) &:= \frac{1}{(2\pi)^{d/2}| \Sigma^{1/2}|}\exp\left(-\frac{1}{2}(\mathbf{x} - \boldsymbol{\mu})^\top \Sigma^{-1}(\mathbf{x} - \boldsymbol{\mu})\right). \end{align}
Of these methods, Linear Discriminant Analysis~\citep[LDA]{citeLDA}, in which a common covariance matrix is used for all values of $k$; and Quadratic Discriminant Analysis~(QDA), in which a freely estimated covariance matrix is given separately to each class, are the most well known, and are often seen to represent opposite ends of a spectrum of Gaussian discriminant models of varying complexity. Intermediaries on this spectrum include, for example, regularised discriminant models, such as those described by~\cite{regularisedDA}; and models where classes share a common principal component structure for their covariance matrices~\citep{CPCDA}. Slightly tangential to this spectrum is the diagonal discriminant analysis model, in which each class is assumed to have a diagonal covariance matrix.
The most flexible class of discriminant models places no restriction on the estimates for $f_{X|Y=k}$, and includes kernel density discriminant analysis~\citep{hand1982}, in which a full description of each density, $f_{X|Y=k}, k = 1, ..., K$, is obtained using kernel density estimation (or some other non-parametric method). A popular restriction to this class includes variants of the kernel Na{\"i}ve Bayes classifier~\citep[NB]{john1995}, in which the estimates of $f_{X|Y=k}$ are assumed to factorise over their margins (i.e., the elements of $X|Y=k$ are treated as independent). Notice that the diagonal discriminant analysis model corresponds to the combination of QDA and the NB assumption. Other discriminant models which offer more flexibility than the class of Gaussian discriminants described in the previous paragraph include Mixture Discriminant Analysis~\citep[MDA]{MDA}, in which each class density is estimated using a simple mixture model; and Flexible Discriminant Analysis~\citep[FDA]{FDA}, where a Gaussian discriminant model is fit to a non-linear transformation of the data.
Of similar motivation to our approach, {\em discriminant feature extraction}~\citep{zhuHastie} is the process of identifying a collection of univariate projections of the data along which classes appear optimally discriminable, under a chosen model. By far the most well known approach of this type is the reduced rank Linear Discriminant Analysis approach. It is straightforward to show (see, for example, \cite{ESL}) that there exist at most $K-1$ univariate projections on which exact discrimination under the LDA model is possible. This comes from the fact that the $K$ class means lie in a subspace of dimension $K-1$, and since the sharing of a common covariance matrix means that a single scaling of the data projected into this $K-1$ dimensional subspace allows for exact discrimination based solely on the Euclidean distances between a point and the projected means, as well as the estimated prior probabilities. Due to this fact, and the ubiquity with which this approach is applied, the reduced rank version of LDA has become standard. It can also be shown that the discriminant features correspond with the leading eigenvectors of the matrix $\hat \Sigma_W^{-1}\hat \Sigma_B$, where $\hat \Sigma_W$ is the shared covariance matrix for the classes, and $\hat \Sigma_B = \frac{1}{n}\sum_{k=1}^K n_k (\hat \boldsymbol{\mu}_k - \bar \boldsymbol{\mu})(\hat \boldsymbol{\mu}_k - \bar \boldsymbol{\mu})^\top$ is the contribution to the total data covariance arising from the relative locations of the class means. Here $\bar \boldsymbol{\mu} = \frac{1}{n}\sum_{i=1}^n \mathbf{x}_i$ is the overall mean of the data. This is a convenient fact, as when fewer than $K-1$, say ${p^\prime}$, features are desired, one simply selects the first ${p^\prime}$ eigenvectors; these being those which contribute the most to the total discrimination of the classes. In the framework of quadratic discriminant analysis, Sliced Average Variance Estimation~\cite[SAVE]{save} has been successfully applied to obtain discriminant features. In this context the discriminant features are taken as the leading eigenvectors of the matrix $M = \sum_{k=1}^K \hat\pi_k (\mathbf{I} - \hat \Sigma_k)^2$, where here the entire data set is first {\em sphered}, i.e., transformed to jointly (combining all classes) have identity covariance matrix. Notice that, with the fact that the data have identity covariance, we have $\hat \Sigma_B + \hat \Sigma_W = \mathbf{I}$, and so $M = \sum_{k=1}^K \hat \pi_k \left(\hat{\Sigma}_B + \hat{\Sigma}_W - \hat{\Sigma}_k\right)^2$. ~\cite{cook2000identifying} have shown that the span of $M$ is equal to the span of the class mean vectors, $\hat \mu_1, ..., \hat \mu_K$, and the differences $\hat{\Sigma}_k - \hat{\Sigma}_{k-1}$ for $k = 2, ..., K$, which provides sufficient information for classification with QDA. The leading eigenvectors of $M$ are thus sensible candidates for discriminant features using the QDA model. Methods like LDA and SAVE are appealing for their fast computation, since their objectives are formulated as quadratic forms based on the class covariance matrices and the between class covariance, $\hat \Sigma_B$, and so fast eigen-solvers can be used to obtain optimal solutions.
More general frameworks, such as those adopted by~\cite{zhuHastie} and~\cite{calo2007gaussian}, use semi- or non-parametric estimates for the class conditional densities, and attempt to obtain discriminant features using a forward procedure based on projection pursuit. Specifically, a projection vector $\mathbf{v}$ is found by maximising the disciminatory capacity in the estimated class conditional densities, $\hat f_{\mathbf{v}^\top X|Y = 1}, ..., \hat f_{\mathbf{v}^\top X |Y = K}$. The data are then modified in such a way that they are no longer easily discriminable along the direction $\mathbf{v}$, and then another projection vector is obtained using the modified data. This process can be iterated in order to obtain the desired number of discriminant features. The reason for modifying the data before obtaining the next projection vector is to ensure that it is not simply re-discovering the discrimination of the classes already explained by the features obtained previously. The form of the density estimates $\hat f_{\mathbf{v}^\top X|Y = 1}, ..., \hat f_{\mathbf{v}^\top X |Y = K}$, as well as the measure used to determine the discriminatory capacity represented therein, is what separates different methods of this type. For example,~\cite{zhuHastie} use fully non-parametric density estimation, and maximise an objective based on the Likelihood Ratio statistic for the test $H_0: f_{X|Y=k} = f_{X|Y=j} \ \forall j, k$, against $H_A: f_{X|Y=k} \not = f_{X|Y=j}$ for some $j, k$. Specifically, they focus on maximising over $\mathbf{v} \in \mathbb{R}^p; ||\mathbf{v}||=1$, the objective
\begin{align}\label{eq:zhu}
LR(\mathbf{v}) = \prod_{i=1}^n\frac{\hat f_{\mathbf{v}^\top X|Y=y_i}(\mathbf{v}^\top\mathbf{x}_i)}{\hat f_{\mathbf{v}^\top X}(\mathbf{v}^\top\mathbf{x}_i)}. \end{align}
Notice that reduced rank LDA, described above, can be formulated as a special case of this formulation, when each of the densities $f_{\mathbf{v}^\top X}$ is fit using a Gaussian density with mean $\mathbf{v}^\top\hat \boldsymbol{\mu}_k$ and variance $\mathbf{v}^\top \hat \Sigma_W \mathbf{v}$, and the null model is fit with a Gaussian density with mean $\mathbf{v}^\top\bar \boldsymbol{\mu}$, and variance $\mathbf{v}^\top \hat \Sigma_W \mathbf{v}$. For this reason the LDA discriminant features can also be seen as those which maximise the ANOVA statistic for the projected data~\citep{citeLDA}.
It is sensible to expect that by applying a discriminant rule based on Eq.~(\ref{eq:bayes}), applied to the data projected onto the features extracted by optimising~(\ref{eq:zhu}), one is likely to obtain reasonably accurate classification. However, if we examine the total discriminatory power which these features capture, as measured by the objective in~(\ref{eq:zhu}), we see that this approach sacrifices access to information available through interactions between these derived features during optimisation. Specifically, if $\mathbf{v}_1, ..., \mathbf{v}_{p^\prime}$ are the identified projections, and $\mathbf{V} \in \mathbb{R}^{p\times {p^\prime}}$ is the matrix with these features as columns, then in general,
\begin{align}\label{eq:biasZhu}
\prod_{i=1}^n \hat \pi_{y_i}\prod_{j=1}^{p^\prime} \frac{\hat f_{\mathbf{v}_j^\top X|Y = y_i}(\mathbf{v}_j^\top\mathbf{x}_i)}{\hat f_{\mathbf{v}_j^\top X}(\mathbf{v}_j^\top \mathbf{x}_i)} \not = \prod_{i=1}^n \frac{\hat \pi_{y_i}\hat f_{\mathbf{V}^\top X | Y=y_i}(\mathbf{V}^\top \mathbf{x}_i)}{\hat f_{\mathbf{V}^\top X}(\mathbf{V}^\top \mathbf{x}_i)}. \end{align}
Notice that in relation to the context of Eq.~(\ref{eq:bayes}), the denominator terms above would be ideally captured by the densities $\hat f_{\mathbf{v}_j^\top X} = \sum_{k=1}^K \hat \pi_k \hat f_{\mathbf{v}_j^\top X | Y = k}$ and $\hat f_{\mathbf{V}^\top X} = \sum_{k=1}^K \hat \pi_k \hat f_{\mathbf{V}^\top X | Y = k}$. With a similar justification to that applied in the context of na{\"i}ve Bayes, it is quite reasonable to expect that features can be found along which the two numerator terms in~(\ref{eq:biasZhu}) are similar. However, it is unlikely that the null model, i.e., the denominator term in the likelihood ratio, represents a density which factorises over its margins, which would be necessary for~(\ref{eq:biasZhu}) to be a reasonable approximation. Indeed, applying a cursory thought to the problem, one finds that for a mixture of discriminable components to allow such factorisation, it would likely require the arrangement of the components to be fairly contrived; for example, lying on a precise grid or lattice.
\iffalse
A limitation of the approach described by~\cite{zhuHastie} is that the overall objective being optimised does not have a direct connection with the original classification rule described in Eq.~(\ref{eq:bayes}). This is because unless the so-called {\em null model}, i.e., the denominator term in the likelihood ratio, represents a density which factorises over its margins, the the total disciminatory power in the discriminant features described by the projections $\mathbf{v}_1, ..., \mathbf{v}_{p^\prime}$ is not likely to accurately capture the discimination under the assumed model. That is,
\begin{align}\label{eq:biasZhu}
\prod_{i=1}^n \hat \pi_{y_i}\prod_{j=1}^{p^\prime} \frac{\hat f_{X^\top \mathbf{v}_j|Y = y_i}(\mathbf{x}_i^\top \mathbf{v}_j)}{\hat f_{X^\top \mathbf{v}_j}(\mathbf{x}_i^\top \mathbf{v}_j)} \not = \prod_{i=1}^n \frac{\hat \pi_{y_i}\hat f_{X^\top \mathbf{V} | Y=y_i}(\mathbf{x}_i^\top \mathbf{V})}{\hat f_{X^\top \mathbf{V}}(\mathbf{x}_i^\top \mathbf{V})}, \end{align}
where $\mathbf{V} \in \mathbb{R}^{p \times {p^\prime}}$ is the matrix with columns given by the vectors $\mathbf{v}_1, ..., \mathbf{v}_{p^\prime}$, and the notation ``$f_Y$'' is used in a general manner to represent the density of the random variable $Y$ (which could be uni- or multi-variate). Notice that in relation to the context of Eq.~(\ref{eq:bayes}), the denominator terms above would be ideally captured by the densities $\hat f_{X^\top \mathbf{v}_j} = \sum_{k=1}^C \hat \pi_k \hat f_{X^\top \mathbf{v}_j | Y = k}$ and $\hat f_{X^\top \mathbf{V}} = \sum_{k=1}^C \hat \pi_k \hat f_{X^\top \mathbf{V} | Y = k}$. Although it is reasonable to expect a degree of discrimination in each of the derived features when optimising the left hand side of~(\ref{eq:biasZhu}), the approach sacrifices access to information available through interactions between these derived features during optimisation.
\fi
\iffalse
In the flexible non-parametric approach taken by~\cite{zhuHastie}, the estimated margins are likely to be very similar to those of the mixtures $\sum_{k=1}^C \hat \pi_k \hat f_{X^\top \mathbf{v}_j | Y = k}, j = 1, ..., {p^\prime}$, but their product will not closely capture the density $\sum_{k=1}^C \hat \pi_k \hat f_{X^\top \mathbf{V} | Y = k}$ without extremely strict restrictions. Indeed, it is even difficult to construct an example of a mixture density whose components are well disciminable and which factorises over its margins, unless one applies extremely precise geometric rules to the component locations. It is also worth noting that the factorisation that would be implied by assuming~(\ref{eq:biasZhu}) represents an approximate equality is in general far more restrictive than that na{\"i}ve Bayes factorisation, which only requires that each component density in the denominators factorises over its margins, not that the entire mixture density factorises.
Although the formulation in Eq.~(\ref{eq:bayes}) can be fit in the likelihood ratio framework; where the null model (denominator term) describes the likelihood under the assumption of a single population (here described as a mixture), and the numerator captures the likelihood under a class specific sub-population, what it measures is considerably different. While in the flexible non-parametric case employed by~\cite{zhuHastie}, each of the marginal likelihood-ratios will be similar
\fi
\section{Optimal Projections for Gaussian Discriminants} \label{sec:methodology}
We study the problem of obtaining an optimal projection matrix, $\mathbf{V} \in \mathbb{R}^{p\times p'}$, for the purpose of discriminant analysis, in which each class conditional density is fit with a Gaussian. We use the word ``projection'' in a slightly more liberal sense than in some other contexts. In particular, we place no explicit restrictions, such as orthonormality, on the matrix $\mathbf{V}$, and so allow the magnitudes of the columns of $\mathbf{V}$ to also optimally accommodate scaling factors for the features of the projected data, $\mathbf{X}\mathbf{V} \in \mathbb{R}^{n\times {p^\prime}}$; where $\mathbf{X} \in \mathbb{R}^{n\times p}$ is the data matrix, with observations stored row-wise.
In the remainder we will occasionally refer to the projection of the data on $\mathbf{V}$, as well as to the projection of the data into the subspace defined by $\mathbf{V}$, which we mean to be interpreted practically equivalent.
The fundamental difference between ours and many existing approaches is that we focus on optimising the likelihood objective based on the original classification rule, i.e.,
\begin{align}\label{eq:classlik}
\mathcal{L}(\mathbf{V}) := \prod_{i=1}^n \frac{\hat \pi_{y_i}\hat f_{\mathbf{V}^\top X | Y=y_i}(\mathbf{V}^\top \mathbf{x}_i)}{\sum_{l=1}^K\hat \pi_{l}\hat f_{\mathbf{V}^\top X | Y=l}(\mathbf{V}^\top \mathbf{x}_i)}. \end{align}
That is, the multinomial likelihood in which the probabilities associated with each observation are given by the posterior estimates arising from the standard application of Bayes' theorem in Eq.~(\ref{eq:bayes}). In the remainder we will refer to this as the {\em classification likelihood}. Of the existing methods of which we are aware, only those of~\cite{peltonen2005discriminative} and \cite{peltonen} use the classification likelihood in order to optimise projections. Both of these estimate each class conditional density with a Gaussian mixture, as in MDA. It is surprising to us that this problem has apparently not been considered explicitly for the simpler Gaussian case. Although it is clear that this is a special case of these existing methods, their utilisation of mixtures allows them to avoid the practical difficulty of accommodating different covariance matrices for the components. Specifically, these methods circumvent this difficulty by either using a fixed isotropic covariance for each mixture component~\citep{peltonen2005discriminative} or by using an alternating optimisation procedure~\citep{peltonen} in which $\mathbf{V}$ is updated based on a gradient step in which the parameters of the mixture components within the projected densities are assumed constant, and a step in which these mixture parameters are updated for the new projection. This alternating procedure allows the authors to avoid having to obtain an exact expression for the gradient of the overall objective, since by ignoring the effect of varying $\mathbf{V}$ on the mixture parameters the problem is vastly simplified. However, the objectives used in the two alternating steps are not the same. In particular the mixture parameters are updated using the standard maximum likelihood objective for mixtures. As a result there is no guarantee that this alternating approach will lead to an increase in the objective of interest, i.e.,~(\ref{eq:classlik})\footnote{We note that it is not only pathological examples where such failure occurs, and we encountered this phenomenon frequently in experimentation with their approach even when each class density is fit with a single Gaussian component.}.
The approach which we adopt is to directly optimise~(\ref{eq:classlik}) using gradient based optimisation techniques. We give explicit details of the necessary derivations in the next subsection.
Although this could be generalised\footnote{A combination of a simple modification of our derived gradients and the model formulation given by~\cite{peltonen} would allow considerable additional flexibility.}, for simplicity we focus here only on the case in which each $\hat f_{\mathbf{V}^\top X|Y=k}$ above is represented by a Gaussian density. We also add a simplifying (and regularising) restriction that, within the subspace defined by the projection on $\mathbf{V}$, the covariance matrix of each of the components is diagonal. Because we have freedom over the selection of $\mathbf{V}$, this is very similar to the restriction imposed by the common principal components model.
The difference between our approach of optimising the classification likelihood and the common principal components model is similar to the difference between maximising the conditional likelihood of $\{y_1, ..., y_n\}$ given $\{\mathbf{x}_1, ..., \mathbf{x}_n\}$, as opposed to that of $\{\mathbf{x}_1, ..., \mathbf{x}_n\}$ given $\{y_1, ..., y_n\}$. That is, where the common principal components model focuses on how well the resulting covariance matrices for the classes best describe their associated observations, our objective is motivated by obtaining the transformation which best aligns the posterior probabilities for the observations with their observed class labels.
\subsection{Optimising $\mathcal{L}(\mathbf{V})$}
In order to obtain an optimal projection for classification under the Gaussian discriminant model, we focus on maximising the classification likelihood given in~(\ref{eq:classlik}), in which each $\hat f_{\mathbf{V}^\top X|Y=k}$ represents a Gaussian density with diagonal covariance matrix. As is common, we directly optimise the logarithm of this likelihood, which is thus given by
\begin{align*}
\ell(\mathbf{V}) &= \sum_{i=1}^n \log\left(\frac{\hat \pi_{y_{i}} \phi_{\mathbf{V}^\top \hat\boldsymbol{\mu}_{y_i}, \Delta(\mathbf{V}^\top\hat \Sigma_{y_i}\mathbf{V})}(\mathbf{V}^\top \mathbf{x}_i)}{\sum_{l=1}^K \hat \pi_l \phi_{\mathbf{V}^\top \hat\boldsymbol{\mu}_l, \Delta(\mathbf{V}^\top \hat \Sigma_l \mathbf{V})}(\mathbf{V}^\top \mathbf{x}_i)}\right)\\
&= \sum_{i=1}^n\left( \log\left(\hat \pi_{y_{i}} \phi_{\mathbf{V}^\top \hat\boldsymbol{\mu}_{y_i}, \Delta(\mathbf{V}^\top\hat \Sigma_{y_i}\mathbf{V})}(\mathbf{V}^\top \mathbf{x}_i)\right) - \log\left(\sum_{l=1}^K \hat \pi_l \phi_{\mathbf{V}^\top \hat\boldsymbol{\mu}_l, \Delta(\mathbf{V}^\top \hat \Sigma_l \mathbf{V})}(\mathbf{V}^\top \mathbf{x}_i)\right)\right), \end{align*}
where for a square matrix $A$ we use $\Delta (A)$ to be the diagonal matrix with $\Delta(A)_{ii} = A_{ii}$ and $\Delta(A)_{ij} = 0$ for $i \not = j$. By considering the effect of $\mathbf{V}$ on the parameters of each component, i.e., $\mathbf{V}^\top \boldsymbol{\mu}_k$ and $\Delta(\mathbf{V}^\top \hat \Sigma_k \mathbf{V})$, during determination of the gradient of $\ell(\mathbf{V})$, we are able to directly apply gradient ascent on this objective. For convenience in the following derivations, we use the notation $\phi^{(k)}_\mathbf{V}$ for the density $\phi_{\mathbf{V}^\top\hat\boldsymbol{\mu}_{k}, \Delta(\mathbf{V}^\top\hat \Sigma_{k}\mathbf{V})}$, for $k=1,2,\hdots,K$.
The log-likelihood then becomes \begin{align*}
\ell(\mathbf{V}) &= \sum_{i=1}^{n}\text{log}(\hat{\pi}_{y_{i}}) +\sum_{i=1}^n \log\left( \phi^{(y_{i})}_{\mathbf{V}}(\mathbf{V}^\top\mathbf{x}_{i})\right) - \sum_{i=1}^{n}\log\left(\sum_{l=1}^K \hat \pi_l \phi^{(l)}_{\mathbf{V}}(\mathbf{V}^\top\mathbf{x}_{i})\right) \\
&= \sum_{i=1}^{n}\text{log}(\hat{\pi}_{y_{i}}) + \ell_{1}(\mathbf{V}) - \ell_{2}(\mathbf{V}), \end{align*} where $\ell_{1}(\mathbf{V}) = \sum_{i=1}^n \log\left( \phi^{(y_{i})}_{\mathbf{V}}(\mathbf{V}^\top\mathbf{x}_{i})\right) $ and $\ell_{2}(\mathbf{V}) = \sum_{i=1}^{n}\log\left(\sum_{l=1}^K \hat \pi_l \phi^{(l)}_{\mathbf{V}}(\mathbf{V}^\top\mathbf{x}_{i})\right)$. \\ \indent First we show that $\ell_{1}(\mathbf{V}) = c - \frac{1}{2}\sum_{k=1}^{K}n_{k}\sum_{t=1}^{p'}\text{log}(\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t})$, where $c$ does not depend on $\mathbf{V}$. Consider: \begin{align} \ell_{1}(\mathbf{V}) &= \sum_{k=1}^{K}\sum_{y_{i}=k} \text{log}(\phi^{(k)}_{\mathbf{V}}(\mathbf{V}^\top\mathbf{x}_{i})) \notag \\ &= -\frac{np'\text{log}(2 \pi)}{2} - \frac{1}{2}\sum_{k=1}^{K}n_{k}
\text{log}|\Delta(\mathbf{V}^\top\hat \Sigma_{k}\mathbf{V})| \notag \\ &- \frac{1}{2}\sum_{k=1}^{K}\sum_{y_{i}=k}(\mathbf{V}^\top\mathbf{x}_{i}-\mathbf{V}^\top\hat{\boldsymbol{\mu}}_{k})^\top(\Delta(\mathbf{V}^\top\hat \Sigma_{k}\mathbf{V}))^{-1}(\mathbf{V}^\top\mathbf{x}_{i}- \mathbf{V}^\top\hat{\boldsymbol{\mu}}_{k}). \notag \end{align} Note that \begin{align}
&(\mathbf{V}^\top\mathbf{x}_{i}-\mathbf{V}^\top\hat{\boldsymbol{\mu}}_{k})^\top(\Delta(\mathbf{V}^\top\hat \Sigma_{k}\mathbf{V}))^{-1}(\mathbf{V}^\top\mathbf{x}_{i}- \mathbf{V}^\top\hat{\boldsymbol{\mu}}_{k}) \notag \\
=& (\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top\mathbf{V}(\Delta(\mathbf{V}^\top\hat \Sigma_{k}\mathbf{V}))^{-1}\mathbf{V}^\top(\mathbf{x}_{i}- \hat{\boldsymbol{\mu}}_{k}) \notag \\
=& \sum_{t=1}^{p'}\frac{\mathbf{v}_{t}^\top(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top\mathbf{v}_{t}}{\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}}. \end{align} The result follows from \begin{align*} \sum_{k=1}^{K}&\sum_{y_{i}=k}(\mathbf{V}^\top\mathbf{x}_{i}-\mathbf{V}^\top\hat{\boldsymbol{\mu}}_{k})^\top(\Delta(\mathbf{V}^\top\hat \Sigma_{k}\mathbf{V}))^{-1}(\mathbf{V}^\top\mathbf{x}_{i}- \mathbf{V}^\top\hat{\boldsymbol{\mu}}_{k}) \notag \\ &=\sum_{k=1}^{K}\sum_{y_{i}=k}\sum_{t=1}^{p'}\frac{\mathbf{v}_{t}^\top(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top\mathbf{v}_{t}}{\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}} =\sum_{k=1}^{K}\sum_{t=1}^{p'}\sum_{y_{i}=k}\frac{\mathbf{v}_{t}^\top(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top\mathbf{v}_{t}}{\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}}\\ &= \sum_{k=1}^{K}\sum_{t=1}^{p'}\frac{1}{\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}} \mathbf{v}_{t}^\top\bigg{(}\sum_{y_{i}=k}(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top\bigg{)}\mathbf{v}_{t} = \sum_{k=1}^{K}\sum_{t=1}^{p'}\frac{n_{k} \mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}}{\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}}
\\ &= \sum_{k=1}^{K}\sum_{t=1}^{p'}n_{k} = p'n, \end{align*}
and $|\Delta(\mathbf{V}^\top\hat \Sigma_{k}\mathbf{V})| = \prod_{t=1}^{p'}\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}$. The gradient of $\ell_{1}$ with respect to the $j$-th column of $\mathbf{V}$, $\mathbf{v}_{j}$, is therefore \begin{equation} \frac{\partial }{\partial \mathbf{v}_{j}} \ell_{1}(\mathbf{V}) = -\bigg{(}\sum_{k=1}^{K}\frac{n_{k}}{\mathbf{v}_{j}'\hat{\Sigma}_{k}\mathbf{v}_{j}} \hat{\Sigma}_{k}\bigg{)}\mathbf{v}_{j}. \end{equation} For the differentiation of $\ell_{2}$ with respect to $\mathbf{v}_{j}$ we note that \begin{equation} \frac{\partial }{\partial \mathbf{v}_{j}} \ell_{2}(\mathbf{V}) = \sum_{i=1}^{n}\bigg{(}\frac{1}{\sum_{l=1}^K \hat \pi_l \phi^{(l)}_{\mathbf{V}}(\mathbf{V}^\top\mathbf{x}_{i})}\sum_{k=1}^{K}\hat{\pi}_{k}\frac{\partial}{\partial \mathbf{v}_{j}}\phi^{(k)}_{\mathbf{V}}(\mathbf{V}^\top\mathbf{x}_{i}) \bigg{)}. \label{LS17.5} \end{equation} Now consider that \begin{align} \frac{\partial}{\partial \mathbf{v}_{j}} & \phi^{(k)}_{\mathbf{V}}(\mathbf{V}^\top\mathbf{x}_{i}) \notag \\
=& (2\pi)^{-\frac{p'}{2}} \frac{\partial}{\partial \mathbf{v}_{j}} |\Delta(\mathbf{V}^\top\hat \Sigma_{k}\mathbf{V})|^{-\frac{1}{2}}\text{exp}\bigg{(}-\frac{1}{2}\sum_{t=1}^{p'}\frac{\mathbf{v}_{t}^\top(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top\mathbf{v}_{t}}{\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}}\bigg{)} \notag \\
=& (2\pi)^{-\frac{p'}{2}} \text{exp}\bigg{(}-\frac{1}{2}\sum_{t=1}^{p'}\frac{\mathbf{v}_{t}^\top(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top\mathbf{v}_{t}}{\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}}\bigg{)}\frac{\partial}{\partial \mathbf{v}_{j}} |\Delta(\mathbf{V}^\top\hat \Sigma_{k}\mathbf{V})|^{-\frac{1}{2}} \notag \\
&+ (2\pi)^{-\frac{p'}{2}} |\Delta(\mathbf{V}^\top\hat \Sigma_{k}\mathbf{V})|^{-\frac{1}{2}} \frac{\partial}{\partial \mathbf{v}_{j}} \text{exp}\bigg{(}-\frac{1}{2}\sum_{t=1}^{p'}\frac{\mathbf{v}_{t}^\top(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top\mathbf{v}_{t}}{\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}}\bigg{)}. \label{LS17.1} \end{align}
Next we evaluate the gradient of $|\Delta(\mathbf{V}^\top\hat \Sigma_{k}\mathbf{V})|^{-\frac{1}{2}} = \prod_{t=1}^{p'}\frac{1}{\sqrt{\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}}}$ with respect to $\mathbf{v}_{j}$. Specifically, \begin{align} \frac{\partial}{\partial \mathbf{v}_{j}}\prod_{t=1}^{p'}\frac{1}{\sqrt{\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}}} &= \bigg{(}\prod_{t\neq j}\frac{1}{\sqrt{\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}}} \bigg{)} \frac{\partial}{\partial \mathbf{v}_{j}} \frac{1}{\sqrt{\mathbf{v}_{j}^\top\hat{\Sigma}_{k}\mathbf{v}_{j}}} \notag = -\bigg{(}\prod_{t\neq j}\frac{1}{\sqrt{\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}}} \bigg{)} \frac{\hat{\Sigma}_{k}\mathbf{v}_{j}}{(\mathbf{v}_{j}^\top\hat{\Sigma}_{k}\mathbf{v}_{j})^{\frac{3}{2}} }
\notag \\ &= -\bigg{(}\prod_{t=1}^{p'}\frac{1}{\sqrt{\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}}} \bigg{)} \frac{\hat{\Sigma}_{k}\mathbf{v}_{j} }{\mathbf{v}_{j}^\top\hat{\Sigma}_{k}\mathbf{v}_{j}}
= -\frac{|\Delta(\mathbf{V}^\top\hat \Sigma_{k}\mathbf{V})|^{-\frac{1}{2}}}{\mathbf{v}_{j}^\top\hat{\Sigma}_{k}\mathbf{v}_{j}}\hat{\Sigma}_{k}\mathbf{v}_{j}. \label{LS17.2} \end{align} To complete the gradient in (\ref{LS17.1}) we need to evaluate \begin{align} \frac{\partial}{\partial \mathbf{v}_{j}} &\text{exp}\bigg{(}-\frac{1}{2}\sum_{t=1}^{p'}\frac{\mathbf{v}_{t}^\top(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top\mathbf{v}_{t}}{\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}}\bigg{)} \notag \\ =&-\frac{1}{2}\text{exp}\bigg{(}-\frac{1}{2}\sum_{t=1}^{p'}\frac{\mathbf{v}_{t}^\top(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top\mathbf{v}_{t}}{\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}}\bigg{)} \frac{\partial}{\partial \mathbf{v}_{j}} \frac{\mathbf{v}_{j}^\top(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top\mathbf{v}_{j}}{\mathbf{v}_{j}^\top\hat{\Sigma}_{k}\mathbf{v}_{j}}. \notag \\ =& -\text{exp}\bigg{(}-\frac{1}{2}\sum_{t=1}^{p'}\frac{\mathbf{v}_{t}^\top(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top\mathbf{v}_{t}}{\mathbf{v}_{t}^\top\hat{\Sigma}_{k}\mathbf{v}_{t}}\bigg{)} \notag \\ &\times \bigg{(} (\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top - \frac{\mathbf{v}_{j}^\top(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top \mathbf{v}_{j}}{\mathbf{v}_{j}^\top\hat{\Sigma}_{k}\mathbf{v}_{j}}\hat{\Sigma}_{k} \bigg{)}\frac{\mathbf{v}_{j}}{\mathbf{v}_{j}^\top\hat{\Sigma}_{k}\mathbf{v}_{j}}. \label{LS17.3} \end{align} Substitution of (\ref{LS17.2}) and (\ref{LS17.3}) into (\ref{LS17.1}) yields \begin{align} \frac{\partial}{\partial \mathbf{v}_{j}}\phi^{(k)}_{\mathbf{V}}(\mathbf{V}^\top\mathbf{x}_{i}) =& \frac{\phi^{(k)}_{\mathbf{V}}(\mathbf{V}^\top\mathbf{x}_{i})}{\mathbf{v}_{j}^\top \hat{\Sigma}_{k}\mathbf{v}_{j}} \notag \\ &\times \bigg{(} \bigg{(}\frac{\mathbf{v}_{j}^\top (\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top\mathbf{v}_{j}}{\mathbf{v}_{j}^\top \hat{\Sigma}_{k}\mathbf{v}_{j}} - 1 \bigg{)}\hat{\Sigma}_{k} - (\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top \bigg{)} \mathbf{v}_{j}. \label{LS17.4} \end{align} Now let $p_{ik}(\mathbf{V}) = \frac{\hat{\pi}_{k}\phi^{(k)}_{\mathbf{V}}(\mathbf{V}^\top\mathbf{x}_{i})}{\sum_{l=1}^{K}\hat{\pi}_{l}\phi^{(l)}_{\mathbf{V}}(\mathbf{V}^\top\mathbf{x}_{i})}$ and $\alpha_{ik}(\mathbf{v}_{j}) = \frac{\mathbf{v}_{j}^\top (\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top\mathbf{v}_{j}}{\mathbf{v}_{j}^\top \hat{\Sigma}_{k}\mathbf{v}_{j}}$, so that substitution of (\ref{LS17.4}) into (\ref{LS17.5}) gives \begin{align} \frac{\partial \ell_{2}}{\partial \mathbf{v}_{j}} &= \bigg{(}\sum_{i=1}^{n}\sum_{k=1}^{K} \frac{p_{ik}(\mathbf{V})}{\mathbf{v}_{j}^\top \hat{\Sigma}_{k}\mathbf{v}_{j}} \big{(} (\alpha_{ik}(\mathbf{v}_{j}) - 1 )\hat{\Sigma}_{k} - (\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top \big{)} \bigg{)}\mathbf{v}_{j}. \end{align} Finally, setting $\mathbf{S}_{k}(\mathbf{V}) = \sum_{i=1}^{n}p_{ik}(\mathbf{V})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top$, we find that \begin{align} &\sum_{i=1}^{n}\sum_{k=1}^{K} \frac{p_{ik}(\mathbf{V})}{\mathbf{v}_{j}^\top \hat{\Sigma}_{k}\mathbf{v}_{j}} \big{(} (\alpha_{ik}(\mathbf{v}_{j}) - 1 )\hat{\Sigma}_{k} - (\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top \big{)} \notag \\ =& \sum_{k=1}^{K} \frac{1}{\mathbf{v}_{j}^\top \hat{\Sigma}_{k}\mathbf{v}_{j}}\bigg{(}\sum_{i=1}^{n}p_{ik}(\mathbf{V})\alpha_{ik}(\mathbf{v}_{j}) -\sum_{i=1}^{n}p_{ik}(\mathbf{V}) \bigg{)}\hat{\Sigma}_{k} \notag \\ &- \sum_{k=1}^{K} \frac{1}{\mathbf{v}_{j}^\top \hat{\Sigma}_{k}\mathbf{v}_{j}}\sum_{i=1}^{n}p_{ik}(\mathbf{V})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})(\mathbf{x}_{i}-\hat{\boldsymbol{\mu}}_{k})^\top \notag \\ =& \sum_{k=1}^{K} \frac{1}{\mathbf{v}_{j}^\top \hat{\Sigma}_{k}\mathbf{v}_{j}}\bigg{(} \frac{\mathbf{v}_{j}^\top\mathbf{S}_{k}(\mathbf{V})\mathbf{v}_{j}}{\mathbf{v}_{j}^\top \hat{\Sigma}_{k}\mathbf{v}_{j}} -\sum_{i=1}^{n}p_{ik}(\mathbf{V}) \bigg{)}\hat{\Sigma}_{k} - \sum_{k=1}^{K} \frac{1}{\mathbf{v}_{j}^\top \hat{\Sigma}_{k}\mathbf{v}_{j}}\mathbf{S}_{k}(\mathbf{V}). \end{align}
\subsection{The Problem of Non-concavity, and Initialisation of $\mathbf{V}$} The objective $\ell(\mathbf{V})$ is a non-concave function of $\mathbf{V}$, and hence the performance of our method will depend on the intialisation of $\mathbf{V}$. We have found that obtaining a warm start is frequently preferable to multiple random initialisations. This especially because the complexity of the gradient evaluations discussed above means that the practical running time of the method is considerably slower than competing approaches such as LDA and SAVE, and hence only few initialisations may frequently be reasonable. In practice, we found that taking the leading $p'$ eigenvalues of the matrix \begin{equation} \hat \Sigma_W^{-1}\hat \Sigma_B + \epsilon \hat{\Sigma}, \label{eq::ini1} \end{equation} where $\hat{\Sigma} = \frac{1}{n}\sum_{i=1}^{n}(\mathbf{x}_{i} - \bar{\boldsymbol{\mu}})(\mathbf{x}_{i} - \bar{\boldsymbol{\mu}})^\top$ is the covariance matrix of all of the data and $\epsilon$ is a small positive constant, frequently yields strong performance. Notice that the first term in the above is the matrix used to obtain discriminant features in LDA, and hence for small $\epsilon$ the leading $K-1$ eigenvectors will be very similar to the LDA features. Since this matrix has rank at mosk $K-1$, however, it is necessary to modify the objective so that more than $K-1$ dimensions can be obtained. The second term being proportional to the total data covariance means that these extra columns in the initialisation of $\mathbf{V}$ will be similar to the principal components within the null space of the leading $K-1$. In practice, to ensure a solution exists, we also add a small ridge to $\hat \Sigma_W$.
\indent
\subsection{Ordering the Columns of $\mathbf{V}$} A limitation of our approach is that, unlike in the case of LDA and SAVE, whose final solutions are given by the eigenvectors of a matrix, there is not a natural ordering on the columns of $\mathbf{V}$. This is of primary importance in the context of visualisation, since it is necessary to produce instructive scatter plots based on the discriminant features. The first visualisation should be given by the features which contribute the most to the discrimination of the classes, and only if necessary will additional visualisations be sought.
We use a simple greedy approach to order the columns in the final solution. Specifically, we evaluate the classification likelihood for the univariate projections given by each of the columns of $\mathbf{V}$. The greatest likelihood determines the leading column of $\mathbf{V}$. We then iteratively augment the current set of ordered columns with each of the columns not yet utilised, and add that which yields the highest likelihood when added to the set already in place.
\section{Classification Experiments} \label{sec:classificationexperiments} In this section an empirical comparison of our aproach, which we henceforth refer to as OPGD (Optimal Projection for Gaussian Discriminants), to popular existing discriminant models based on Gaussian class densities is conducted. Depending on the size of the data set, we use either cross-validation or a standard ``training/validation'' split of the training data to select appropriate hyper-parameters. Performance is then compared based on prediction accuracy on an independent test set.
The methods included, and their respective ranges of hyper-parameters, are \begin{enumerate}
\item (Reduced rank) LDA, with selection of the number of discriminant features from 1 to $K-1$. We used the implementation in the {\tt R} package {\tt MASS}~\citep{MASSpackage}.
\item Regularised Discriminant Analysis (RDA), in which the class covariance matrices are estimated by $\hat \Sigma_k = \alpha \tilde \Sigma_k + (1-\alpha) \hat \Sigma_W$. Here we have used $\tilde \Sigma_k$ to be the maximum likelihood estimate for the $k$-th class and $\hat{\Sigma}_W$ is the pooled estimate of the class covariance, as used in LDA. The value of $\alpha$ was selected from the set $\{0, \frac{1}{p-1}, \frac{2}{p-1}, ...,\frac{p-2}{p-1}, 1\}$, where $p$ is the dimension of the input space. Note that when $\alpha = 1$ this corresponds with the standard QDA model, and with $\alpha = 0$ this is equivalent to LDA.
\item SAVE, with number of discriminant features selected from between 1 and $p$.
\item Discriminant analysis with the Common Principal Components model (CPCDA), with the number of discriminant features (rank of the covariance matrices) selected from between 1 and $p$. We used the implementation in the {\tt R} package {\tt multigroup}~\citep{multigrouppackage} to obtain the estimate of the common principal components and corresponding scaling factors.
\item The proposed method (OPGD) with the number of features (columns of $\mathbf{V}$) selected from between 1 and $p$. \end{enumerate} All data sets under consideration are available from the UCI machine learning repository~\citep{UCI}. These are \begin{enumerate}
\item Vowel recognition. \item Statlog (landsat satellite). \item Image segmentation. \item Optical recognition of handwritten digits. \end{enumerate}
We discuss the specifics for each application explicitly below.
\subsection{Vowel Recognition} Here the objective is the recognition of vowel sounds from utterances taken from multiple speakers. There are 11 vowel sounds and 15 speakers. Each speaker spoke each vowel sound 6 times giving 990 utterances in total. The speech signals of each utterance have been preprocessed into 10 values, yielding an input space of 10 dimensions. We follow the same experimental design as described in~\cite{ESL}, in which four males and four females are chosen for training (528 utterances) and four males and three females for testing (462 utterances).
A cross-validation procedure is applied to the training data in order to select the hyper-parameters of the methods under consideration. Eight folds are used and each fold corresponds to a different speaker. The hyper-parameters chosen are those which yield the smallest cross-validation error (misclassification error is used). Each model is then retrained on the complete training data set using its chosen hyper-parameter.
The results are summarised in Table \ref{tab:vowel}. OPGD provides the smallest test error, followed by RDA and then LDA. SAVE and CPCDA performed relatively poorly.
\begin{table}
\centering
\begin{tabular}{c| c c c c c}
& OPGD & LDA & RDA & SAVE & CPCDA\\
\hline
Test error & 0.4480519& 0.491342& 0.4718615& 0.5281385& 0.5649351 \\
Hyper-parameter& 3 & 2 & 0.4444444& 10& 7
\end{tabular}
\caption{Test errors and chosen hyper-parameters of the cross-validation procedure on the vowel recognition data.}
\label{tab:vowel} \end{table}
\subsection{Statlog} The data consist of multi-spectral values of pixels in $3 \times 3$ neighbourhoods of a satellite image. Each pixel is represented by 4 multi-spectral values yielding an input space of dimension 36. The objective is to classify the multi-spectral values of each neighbourhood into one of 7 classes. The data set consist of 6435 observations. It should be noted that only 6 of the 7 classes are present in the data (the ``mixture'' class is not present).
A split of 75\% training and 25\% testing was use. A third of the training data were used for validation, in order to select hyper-parameters, leading to an overall 50/25/25 training/validation/test split.
Once hyper-parameters have been selected, the final models are trained on the entire training set. For our method, we initialise the optimisation of the final model with the solution which gives the lowest validation error to exploit the computations already performed.
Finally, the models are then assessed on the test set using misclassification error.
The entirety of the above procedure was performed 20 times, each corresponding to a different randomly generated training/validation/test split.
The test errors corresponding to the different methods are summarized in the form of boxplots in the top-left graph of Figure \ref{class::figs}. We see that OPGD outperforms others by a considerable margin. RDA and SAVE yield similar performance to one another, both significantly outperfroming LDA and CPCDA.
\subsection{Image Segmentation} Each input observation consists of 19 values derived from a $3 \times 3$ neighbourhood of an image. These neighbourhoods were drawn randomly from a data base of 7 images and each neighbourhood corresponds to one of 7 classes. The objective is to classify the input observations into one of these 7 classes. \\ \noindent \indent The region-pixel-count input is removed from the analysis since it is constant over all the observations. The CPCDA method is particularly sensitive to the short-line-density-5 and short-line-density-2 inputs, and typically fails to provide output. We added small Gaussian perturbations to these data so that all models could be fit. These do not affect the output of the other methods appreciably.
The same proportions of training/validation/test as in the statlog data were applied. Again, 20 different random splits were generated and the test errors recorded. The results are displayed in the top-right graph of Figure \ref{class::figs}.
Once more OPGD obtains encouraging performance in terms of classification accuracy. In this case LDA and RDA are similar in performance to one another. SAVE and CPCDA have similar average performance to one another, but with the performance of CPCDA model is considerably more variable. A possible reason for this is its sensitivity to the two short-line-density input variables discussed previously.
\subsection{Optical Recognition of Handwritten Digits}
In the final classification experiment we consider a data set containing images of handwritten digits, compressed to $8\times 8$ pixels, yielding 64 dimensions. The objective is to classify an image to one of the digits between 0 and 9 (10 classes in total).
We repeat the approach used for training and assessing models in the statlog and image segmentation data. The first and fortieth input variables were excluded since they were constant over all observations. To the remainder of the variables, independent Gaussian perturbations were added. This was done in order to obtain output for the CPCDA model. Both OPGD an RDA obtain substantially better accuracy than the other three models considered, with OPGD itself being the most accurate.
\begin{figure}
\caption{Results of the 20 training/validation/test splits applied to the statlog, image segmentation and
optimal digit recognition data sets. The boxplots represent the test errors obtained over the different splits. All training/validation/test splits were done according to the 50/25/25 ratio.}
\label{class::figs}
\end{figure}
\subsection{Remarks} OPGD was compared to LDA, RDA, SAVE and CPCDA based on four different data sets. In terms of test performance, the results are encouraging, and OPGD outperformed its competitors in all applications. This improvement in test error does not come without a cost, however, since the computation requirements of OPGD are considerably more demanding than these competing methods.
Everything considered, we believe that the potential for considerably improved accuracy in classification compared with other popular Gaussian discriminant models justifies the consideration of OPGD as an alternative to popular alternatives for problems of moderate size (up to thousands of points in tens of dimensions, or fewer), and that methods of accelerating the algorithm should be considered in further research.
\section{Optimal Projections for Clustering with Gaussian Mixtures} \label{sec:clustering}
In this section we perform a preliminary exploration into the potential of the proposed method to enhance the clustering capabilities of Gaussian Mixture Models (GMMs). We use a slight modification of the objective which is better suited to the clustering context; i.e., where the labels are unknown. Specifically, suppose we have an initial estimate for the parameters of a GMM, denoted by $\hat \pi_1, ..., \hat \pi_K$; $\hat \boldsymbol{\mu}_1, ..., \hat \boldsymbol{\mu}_K$; and $\hat{\Sigma}_1, ..., \hat{\Sigma}_K$. We then focus on maximising the objective,
\begin{align*}
\sum_{i=1}^n \log\left(\frac{\max_{l \in \{1, ..., K\}} \hat \pi_l \phi_{\mathbf{V}^\top \hat\boldsymbol{\mu}_l, \Delta(\mathbf{V}^\top\hat \Sigma_l\mathbf{V})}(\mathbf{V}^\top \mathbf{x}_i)}{\sum_{l=1}^K \hat \pi_l \phi_{\mathbf{V}^\top \hat\boldsymbol{\mu}_l, \Delta(\mathbf{V}^\top \hat \Sigma_l \mathbf{V})}(\mathbf{V}^\top \mathbf{x}_i)}\right) - \lambda ||\mathbf{V}^\top \mathbf{V} - \mathbf{I}||_F^2, \end{align*}
where the difference from the objective used in the classification context is that the numerator terms are now taken as the maxima over the components. This effectively allows the points' assignments to the different components of the GMM to change during optimisation; which is appropriate given the unsupervised context. We have also added the term $-\lambda ||\mathbf{V}^\top \mathbf{V} - \mathbf{I}||_F^2$, where $\lambda > 0$ is a chosen parameter, to penalise deviations of $\mathbf{V}$ from being orthonormal. We have found the performance of this method quite insensitive to the setting of $\lambda$, and simply set it equal to the number of data, $n$, since the first term in the objective scales linearly with $n$ as well. Enforcing or encouraging orthogonality of a projection is common in projection pursuit for clustering~\citep{bolton2003,niu2011, hofmeyrSPDC}, and is useful in making the method more robust to poor intialisation. This was not necessary in the classification context since the class labels are known, and hence the estimates of the model parameters are far more stable.
Intuitively this approach can be thought of as asking the question ``Can we find a projection, $\mathbf{V}$, upon which a GMM provides a strong discrimination of the clusters induced by the model?'' Here we mean the clustering induced by the transformed GMM, defined by the parameters $\mathbf{V}^\top \hat\boldsymbol{\mu}_1, ..., \mathbf{V}^\top \hat\boldsymbol{\mu}_K$; $\Delta(\mathbf{V}^\top\hat \Sigma_1\mathbf{V}), ..., \Delta(\mathbf{V}^\top\hat \Sigma_K\mathbf{V})$; and mixing proportions as before. If the parameters of the initial GMM reasonably capture the denser regions of the data, then it is sensible to expect that the additional flexibility in optimising over $\mathbf{V}$ might further enhance this capability. After optimising the projection, as a final step, we re-estimate the parameters of the GMM, but now within the subspace defined by the projection on $\mathbf{V}$.
We experimented with this approach on numerous popular benchmark data sets taken from the UCI machine learning repository~\citep{UCI}. Our main take away from these experiments is that, especially when the number of dimensions is high relative to the number of observations, frequently the clustering solution changes only slightly from the initial solution. Our understanding of this is that given sufficient degrees of freedom, the application of OPGD serves primarily to further enhance the discrimination of the clusters induced by the initial solution, rather than expose cluster structure in the data which is less identifiable in the full dimensional data set. In a sense OPGD is overfitting to the initial solution. It is important to note, however, that even when no substantial changes to the initial solution are made, this approach still offers the benefit of a reduced representation of the model, through dimension reduction by projection on $\mathbf{V}$, and also instructive visualisation of the clustering solution which may not be available otherwise. Furthermore, it is encouraging that the majority of the time the changes made to the initial solution, although slight, tend to result in an improved solution.
Table~\ref{tb:cluster} contains a summary of the clustering results from this approach. In each case we used the {\tt R} package {\tt mclust}~\citep{CRANmclust} to obtain the initial GMM solutions, and subsequently applied OPGD to obtain an optimal projection of dimension equal to one fewer than the number of clusters. As in the case of classification, we added very small Gaussian perturbations to the data in some cases in order for a GMM to be fit. Since the {\tt mclust} implementation uses a random initialisation, we supplied to our method twenty potential initial solutions for each data set. Note that for some of the data sets {\tt mclust}, despite its own random initialisation, obtained the same solution in all twenty replications. In these cases the output of our method was also the same in all twenty. The table reports the performance of the clustering results based on Adjusted Rand Index~\citep{adjustedrand} and Normalised Mutual Information~\citep{nmi}, both multiplied by 100. These are popular evaluation metrics which allow us to numerically assess the similarity between the clustering result and the known class labels. In both cases higher values (close to 100 on our scale) suggest a closer agreement between the clusters and the true classes. For both metrics, and for each data set, we report the average (Avg) performance of the GMM used for initialisation, as well as the average difference induced by the subsequent application of OPGD. In addition we report the single best and worst relative performance of the OPGD output to the initial GMM solution. For example, on the Opt. Digits data set the GMM had an average Adjusted Rand Index score of 49.5 and the average improvement due to OPGD was to increase it to 52.6. The best performance of OPGD was to increase the GMM performance by 10.0, from 54.2 to 64.2, while in the worst instance it decreased the performance very slightly from 42.2 to 41.6. On the other hand, the application of OPGD caused a substantial deterioration in the performance on the Seeds data set, where in every run the Adjusted Rand Index was decreased from 81.2 to 63.8.
\begin{table}[h]
\centering
\scalebox{0.8}{
\begin{tabular}{lccccccc}
& & \mcl{3}{Adjusted Rand Index} & \mcl{3}{Normalised Mutual Information} \\
Data Set & $n,d,k$ & Avg & Best & Worst & Avg & Best & Worst\\
\hline
Wine & 178,13,3 & 94.9+3.4 & 94.9+3.4 & 94.9+3.4 & 92.8+4.6 & 92.8+4.6 & 92.8+4.6 \\
Iris & 150,4,3 & 90.4+1.8 & 90.4+1.8 & 90.4+1.8 & 90.0+1.4 & 90.0+1.4 & 90.0+1.4\\
Vowel & 990,10,11 & 17.5+3.1 & 17.5+3.1 & 17.5+3.1 & 38.6+4.1 & 38.6+4.1 & 38.6+4.1\\
Satellite & 6435,36,6 & 40.3+2.6 & 40.2+7.2 & 39.6-0.9 & 53.9+3.0 & 59.5+6.5 & 54.0+1.3\\
Image Seg. & 2310,19,7 & 43.5-0.3 & 43.2+1.4 & 46.0-4.8 & 58.0-0.1 & 57.4+1.3 & 57.4-3.3\\
Opt. Digits & 5620,64,10 & 49.5+3.1 & 54.2+10.0 & 42.2-0.6 & 65.6+1.8 & 70.0+4.8 & 60.5-0.7\\
Pen Digits & 10992,16,10 & 57.2+3.8 & 54.2+6.8 & 66.8+1.4 & 75.5+0.5 & 75.2+0.2 & 73.0-1.0\\
Texture & 5500,40,11 & 88.5-0.1 & 86.6+2.3 & 89.3-1.1 & 93.5-0.3 & 92.4+1.2 & 94.1-1.2\\
Libras & 360,90,15 & 30.6+1.0 & 32.1+3.4 & 34.0-0.9 & 59.0+1.1 & 60.2+3.3 & 61.3-0.2\\
Forest & 523,27,4 & 16.8+4.6 & 16.8+4.6 & 16.8+4.6 & 22.8+5.1 & 22.8+5.1 & 22.8+5.1\\
Yeast & 698,72,5 & 50.4+0.1 & 48.5+2.1 & 55.8-5.1 & 54.5-0.0 & 52.7+1.9 & 56.1-4.5\\
Glass & 214,9,6 & 10.9+3.6 & -1.8+15.7 & 28.7-9.0 & 28.1+2.7 & 13.2+14.3 & 42.1-5.6\\
Dermatology & 366,34,6 & 66.5+1.7 & 77.5+6.7 & 45.0-3.5 & 77.2+2.6 & 80.8+7.1 & 72.4-4.0\\
Seeds & 210,7,3 & 81.2-17.4 & 81.2-17.4 & 81.2-17.4 & 77.1-15.2 & 77.1-15.2 & 77.1-15.2\\
M.F. Digits & 2000,216,10 & 59.9+3.7 & 70.4+5.4 & 52.0+2.1 & 70.2+3.7 & 71.1+4.7 & 65.2+2.5\\
\end{tabular}
}
\caption{Clustering results from enhancing Gaussian Mixture Models. The values in the table show the performance of the GMMs and the differences in performance resulting from the application of OPGD. The results are based on 20 replications. The ``Avg'' is the average performance of the GMM and the average difference due to applying OPGD, while the ``Best'' and ``Worst'' correspond respectively to the single best and worst performance of the OPGD enhancement relative to the GMM.}
\label{tb:cluster} \end{table}
\subsection{Visualisation of Clustering Solutions}
In this section we explore the visualisation of clustering solutions based on the projected data, $\mathbf{X}\mathbf{V}$. As arguably the most popular general purpose method for visualisation, PCA is a natural comparison. Visualisation of clustering solutions is important for, among other things, validation of the solutions obtained. Figure~\ref{fig:clust_vis} shows the examples of the Texture\footnote{For a clearer visualisation we have only included a random subset of 2000 of the total 5500 points in the data set} and Forest data sets. In both cases we provide two 2-dimensional scatter plots for each of PCA and OPGD. In the case of PCA we show the first two and third-and-fourth principal components in the two plots respectively. Here the colours and point characters indicate the clustering based on the initial GMM solution. For OPGD, we do the same for the Texture data set, but since we only seek a 3-dimensional projection for the Forest data set, the two plots in that case are of $\mathbf{X}[\mathbf{v}_1 \ \mathbf{v}_2]$ and $\mathbf{X}[\mathbf{v}_1 \ \mathbf{v}_3]$. In these cases the colours and point characters indicate the final solution. In the PCA plots of the Texture data set one would be hard-pressed to validate more than three of the clusters in the solution. Note that going to lower order principal components does not yield any better visualisation of the different clusters than in the first two pairs depicted in the figure. For the case of OPGD one can easily make out the distinction of at least six different clusters, which agree by eye with how we would intuit clusters in a GMM. Although in the interest of brevity we have not included more than these first two plots, lower order columns in $\mathbf{V}$ can be used to visualise the distinction of the remaining clusters clearly. In the case of the Forest data set, one would not be able to sensibly validate any of the four clusters based on the PCA visualisation, whereas arguably at least three clusters are clearly visible as distinct groups of points in the OPGD projections.
\begin{figure}
\caption{Visualisation of Texture and Forest data sets through two-dimensional scatter plots. Clusters are far more easily distinguished in the case of the OPGD projections than when using PCA.}
\label{fig:clust_vis}
\end{figure}
As a final point, in relation to Figure~\ref{fig:clust_vis}, notice that because the objective of OPGD is to maximise discrimination of clusters under a GMM formulation, it tends to lead to solutions, and hence visualisations, in which the clusters appear as roughly having Gaussian, or at least ellipsoidal shapes. Essentially, even when clusters may not be close to Gaussian in the full dimensional space, it is frequently the case that there are projections upon which the clusters are roughly Gaussian. A clear example of such a case is the cluster in the Texture data labelled ``g'' in the PCA plots. These points betray an elongated tail which would not arise if the data came from a GMM with only eleven components. In the case of the OPGD projection, however, all clusters appear as we would expect in a GMM. The distinction of ellipsoidal, or Gaussian clusters is arguably more easily validated by eye than many other non-Gaussian distributions.
\subsection{Underastanding potential failures of OPGD for clustering}
Of the fifteen data sets considered, the performance only deteriorated substantially as a result of the application of OPGD on one; Seeds. However, in a few other cases (e.g., Image Seg. and Texture) very small decreases in average clustering accuracy were also observed.
It turned out that identifying the potential cause for such a drastic deterioration in clustering performance on the Seeds data set was fairly straightforward. Figure~\ref{fig:seeds} shows plots of (a) the solution obtained by OPGD on the original data; (b) the scree plot of eigenvalues of the covariance matrix; and (c) the solution obtained by OPGD on the reduced data containing only the first 4 principal components. The first plot shows strong collinearity in the data, since despite the fact that the optimal $\mathbf{V}$ is very close to orthonormal we see that the projections $\mathbf{X}\mathbf{v}_1$ and $\mathbf{X}\mathbf{v}_2$ are extremely highly correlated. This collinearity is confirmed by the very small eigenvalues in the covariance matrix of the data. Aside from the standard problems associated with collinearities in clustering, this example highlights a limitation of our approach in this context. Specifically, this can arise from the combination of a reasonable separation of the means in the initial GMM on a projection, $\mathbf{v}_1$, along which the data are highly correlated with their projection on an orthogonal vector, $\mathbf{v}_2$; and the fact that we fit a diagonal covariance matrix to the clusters within the subspace defined by the projection on $\mathbf{V} = [\mathbf{v}_1 \ \mathbf{v}_2]$. This can be most easily intuited by considering the fact that the optimal projection, $\mathbf{V}$, will be characterised by few points which have relatively high density in more than one of the components of the GMM with parameters $\hat \pi_1, ..., \hat \pi_K$; $\mathbf{V}^\top\hat\boldsymbol{\mu}_1, ..., \mathbf{V}^\top\hat\boldsymbol{\mu}_K$; and $\Delta(\mathbf{V}^\top\hat \Sigma_1\mathbf{V}), ..., \Delta(\mathbf{V}^\top\hat \Sigma_K\mathbf{V})$. In other words, there should be relatively few points on or near the boundaries of the clusters. It should be clear that the proportion of such points is far fewer in Figure~\ref{fig:seeds}~(a) than in Figure~\ref{fig:seeds}~(c). The effect of fitting diagonal covariance matrices in the projected space has the effect of artificially decreasing the density at the boundaries of the clusters in Figure~\ref{fig:seeds}~(a) since the density is spread almost equally in the direction along which the data lie and the orthogonal direction. By removing the collinearity in the data the solution is vastly improved, Figure~\ref{fig:seeds}~(c). In this case the performance is similar for the initial solution and the adjusted solution after applying OPGD, with both metrics increasing slightly from the initial values.
\begin{figure}
\caption{Seeds data set. (a) shows the OPGD clustering solution of the data projected into the optimal subspace. There is evidence of high collinearity in the data. This evidence of collinearity is corroborated by the very small eigenvalues of the covariance matrix, shown in (b). Removing collinearity by using only the first four principal components of the data leads to a more satisfying solution, shown in (c).}
\label{fig:seeds}
\end{figure}
We note that it is likely not necessary to always check for such collinearity prior to applying OPGD, as the scenario which led to the poor performance on the Seeds data sets was very specific. However, for completeness, we explored the alternative of applying {\tt mclust} and then OPGD again to all the data sets considered before, except now only the leading principal components which constituted at most 99.9\% of the total data variation were included. The threshold of 99.9\% is arbitrary, and was chosen only for a preliminary investigation into the effect of removing only very small variance components. In none of the other cases did this substantially affect the relative performance of the OPGD enhancement when compared to the GMM initialisation.
\section{Discussion} \label{sec:conclusions}
In this paper we explored the problem of finding optimal projections for discriminant analysis in which each class is endowed with a multivariate Gaussian density. By carefully addressing the differentiation of the objective function we were able to optimise the classification likelihood directly with gradient based optimisation techniques. We found that this approach is very successful in obtaining accurate classification models when compared with other popular Gaussian discriminant models. A simple modification of this objective allowed us to also address the problem of enhancing Gaussian Mixture Models for clustering. We found this approach to consistently offer modest improvements in clustering quality, while simultaneously providing a reduced model formulation through dimension reduction, as well as instructive visualisations for cluster validation and knowledge discovery.
\end{document}
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\begin{document}
\title{Kurosh theorem for certain Koszul Lie algebras}
\newtheorem{thm}{Theorem}[section]
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\newtheorem*{thmB}{Theorem B}
\newtheorem*{thm*}{Theorem}
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\swapnumbers
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\newtheorem*{acknowledgement}{Acknowledgement}
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\newcommand{\invamalg}{\mathbin{\rotatebox[origin=c]{180}{$\amalg$}}}
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\begin{abstract} The Kurosh theorem for groups provides the structure of any subgroup of a free product of groups and its proof relies on Bass-Serre theory of groups acting on trees. In the case of Lie algebras, such a general theory does not exists and the Kurosh theorem is false in general, as it was first noticed by Shirshov in \cite{shir}. However, we prove that, for a class of positively graded Lie algebras satisfying certain local properties in cohomology, such a structure theorem holds true for subalgebras generated in degree 1. Such class consists of Lie algebras, which have all the subalgebras generated in degree $1$ that are Koszul. \end{abstract}
\tableofcontents
\section*{Introduction}
Consider a positively-graded Lie algebra ${\mc L}$ over a field $\mathbb{F}$. This means that ${\mc L}$ is a Lie algebra endowed with a vector space decomposition \[{\mc L}=\displaystyle\bigoplus _{i\geq 1}{\mc L}_i\] such that the Lie bracket is a graded $\mathbb{F}$-linear map $[\hbox to 1.5ex{\hrulefill},\hbox to 1.5ex{\hrulefill}]:{\mc L}\otimes_\mathbb{F} {\mc L}\to {\mc L}$, i.e., $[{\mc L}_n,{\mc L}_m]\subseteq {\mc L}_{n+m}$.
These Lie algebras are much well behaved than their ungraded counterparts, and they share many cohomological features with the class of pro-$p$ groups (cf. \cite{weig},\cite{cmp}).
The cohomology of a graded Lie algebra can be endowed with a second grading, the \textit{internal} one, that makes the cohomology algebra $H^\bullet({\mc L},\mathbb{F})$ into a bigraded algebra.
Among the $\mathbb N$-graded Lie algebras there is the class of the Koszul Lie algebras that are those Lie algebras ${\mc L}$ satisfying the following constraint on the bigraded cohomology algebra:
\begin{center}
$H^{ij}({\mc L},\mathbb{F})=0$ for $i\neq j$.
\end{center}
It is easy to see that Koszul Lie algebras are quadratic, in the sense that they have all of the generators of degree $1$ and the relations have degree $2$.
In this paper we introduce a class of Lie algebras which have a more homogeneous Koszulity property, the Bloch-Kato Lie algebras.
A Lie algebra is \textit{Bloch-Kato} if it is Koszul, as well as any of its subalgebras that are generated in degree $1$.
Since the cohomology ring of a Lie algebra is a graded-commutative algebra, it is natural to try and translate the above property for this class of algebras.
In particular, we see that the Bloch-Kato property of a Lie algebra is closely related with a Koszulity property on its cohomology, which was introduced in the commutative context by A. Conca \cite{conca}.
We say that a graded $\mathbb{F}$-algebra $A$ is \textit{universally Koszul} if every two-sided ideal $I$ of $A$ that is generated in degree $1$ satisfies:
\begin{center}
$\operatorname{Ext}^{ij}_A(A/I,\mathbb{F})=0$ if $i\neq j$.
\end{center} This is equivalent to requiring that the $A$-module $A/I$ has a linear free $A$-resolution.
We prove the following three main results.
\begin{thmA}
Let ${\mc L}$ be an $\mathbb N$-graded $\mathbb{F}$-Lie algebra, with cohomology ring $A=H^\bullet({\mc L},\mathbb{F})$. Then ${\mc L}$ is Bloch-Kato if, and only if, $A$ is universally Koszul.
\end{thmA}
\begin{thmB}
Let $\mc A$ and $\mc B$ be two Bloch-Kato Lie algebras. Then their free product $\mc A\amalg\mc B$ is Bloch-Kato, too.
\end{thmB}
From which follows:
\begin{thm*}[Kurosh’ subalgebra theorem]
Let $\mc A$ and $ \mc B$ be two Bloch-Kato Lie algebras, and let ${\mc L}=\mc A\amalg\mc B$ be their free product. Let $\mc H\leq {\mc L}$ be a Lie subalgebra generated by elements of degree $1$. Then $\mc H$ is a free product of a free Lie algebra and of some subalgebras of $\mc A$ and $\mc B$. More precisely, \[\mc H=\mc F\amalg\gen{\mc H_1\cap \mc A}\amalg\gen{\mc H_1\cap\mc B}\] where $\mc F$ is the free Lie algebra generated by any complement $W\leq \mc A_1\oplus\mc B_1$ with $W\oplus (\mc H_1\cap \mc A)\oplus(\mc H_1\cap\mc B)={\mc L}_1$.
\end{thm*}
In fact, our original purpose was to prove Theorem B by applying some kind of Kurosh subalgebra theorem, just like in the group case, for which it is a standard argument. However, there is no such a result in the general Lie algebra case, as it was proved by Shirshov in \cite{shir}, who showed the existence of Lie algebras such that their free product contain a Lie algebra that is not free and is not isomorphic to the free product of any subalgebra of the factors and a free Lie algebra. Eventually, we managed to prove Theorem B in a different way, and then we realized this could be used to prove the above Kurosh theorem for the distinguished class of Bloch-Kato Lie algebras.
Eventually, notice that the example provided by Shirshov is not $\mathbb N$-graded, so that the main result of this paper sets in a different point of view.
\section{Homological algebra}
In this section we will introduce some tools from homological algebra in the graded context. For a complete treatment of the ungraded homological algebra, consult Weibel's book
By an $\mathbb N_0$-graded (associative) $\mathbb{F}$-algebra $A$ we mean a graded vector space $\bigoplus_{i\geq 0}A_i$ that is locally finite dimensional, i.e., $\dim_\mathbb{F} A_i<\infty$, and that is endowed with a graded $\mathbb{F}$-bilinear and associative product $A\otimes A\to A$ that we will denote by $(a,a')\mapsto aa'$.
All the algebras are assumed to be unital and connected, i.e., $A_0\simeq \mathbb{F}$. Denote by $A_+$ the augmentation ideal $A_+=\bigoplus_{i>0}A_i$, that is the kernel of the augmentation map $\varepsilon:A\to \mathbb{F}$.
The augmentation is graded and it induces a structure of $A$-module on $\mathbb{F}$, that is called the \textit{trivial module}.
A graded (left) $A$-module is a graded vector space $M=\oplus_i M_i$ that is also an $A$-module in the usual sense, for which the $A$-action $A\otimes M\to M$ is a graded $\mathbb{F}$-linear mapping.
Henceforth, all the algebras (resp. modules) will be implicitely assumed to be $\mathbb N_0$-graded and connected (resp. $\mathbb N_0$-graded modules).
If $V$ is a graded $\mathbb{F}$-vector space and $n\in \mathbb Z$, we denote by $V[n]$ the $n$-shifted vector space, i.e., $(V[n])_i=V_{i+n}$.
\subsection{Cohomology of graded algebras}
It is easy to see that the category of locally finite $A$-modules has enough projectives, for any free module is projective.
If $M$ and $N$ are two $A$-modules, we denote by $\operatorname{Ext}^{\bullet j}_A(M,N)$ the right derived functor of \[\operatorname{Hom}^j_A(M,N)=\set{f:M\to N\text{ morphism of $A$-modules}}{f(M_k)\subseteq N_{k-j}}\]
An element in $\operatorname{Hom}_A^j(M,N)$ is called a morphism of degree $-j$.
It is straightforward to check that \begin{equation}\label{shiftExt}
\operatorname{Ext}^{ij}_A(M[m],N[n])=\operatorname{Ext}^{i,j+m-n}_A(M,N)
\end{equation} for graded $A$-modules $M$ and $N$ and integers $m,n$.
The $\operatorname{Ext}$-groups can be computed as follows: Let $P_\bullet\to M$ be a projective $A$-resolution and apply the functor $\operatorname{Hom}_A^j(\hbox to 1.5ex{\hrulefill},N)$ to get the cochain complex \[\dots\to\operatorname{Hom}^j_A(P_{i-1},N)\to\operatorname{Hom}^j_A(P_{i},N)\to\operatorname{Hom}^j_A(P_{i+1},N)\to\dots.\]
Then \[\operatorname{Ext}^{ij}_A(M,N)\simeq H^i(\operatorname{Hom}^j_A(P_\bullet,N))= \frac{\ker(\operatorname{Hom}^j_A(P_{i},N)\to\operatorname{Hom}^j_A(P_{i+1},N))}{\image({\operatorname{Hom}^j_A(P_{i-1},N)\to\operatorname{Hom}^j_A(P_{i},N)})}\]
There is a distinguished projective resolution that one can always use, that is the so-called \textit{Bar complex} \[\dots\to A\otimes A_+^{\otimes n}\otimes M\to \dots \to A\otimes A_+\otimes A_+\otimes M\to A\otimes A_+\otimes M\to A\otimes M\to 0\]
with differential given by \[\partial (a_0\otimes\dots \otimes a_i\otimes m)=\sum_{s=1}^i (-1)^sa_0\otimes\dots\otimes a_{s-1}a_s\otimes \dots\otimes a_i\otimes m+(-1)^{i+1}a_0\otimes \dots\otimes a_im.\]
For an $\mathbb{F}$-algebra $A$ we call $H^\bullet(A):=\operatorname{Ext}^\bullet_A(\mathbb{F},\mathbb{F})$ the \textit{cohomology} of $A$.
The following results give a description of the low degree $\operatorname{Ext}$-groups (see \cite{pp}).
\begin{prop}\label{prop:gen rel}
Let $A$ be an $\mathbb{F}$-algebra, and $M$ an $A$-module. Then the following hold:
\begin{enumerate}
\item $\operatorname{Ext}^{ij}_A(M,\mathbb{F})=0$ for all $i>j$.
\item There exist graded vector spaces $X$ and $Y$ such that $M$ is presented as \[A\otimes Y\to A\otimes X\to M\to 0\]
and \begin{align*}
X_j^\ast\simeq \operatorname{Ext}^{0,j}_A(M,\mathbb{F})\\
Y_j^\ast\simeq \operatorname{Ext}^{1,j}_A(M,\mathbb{F})
\end{align*}
\item There exist graded vector spaces $V$ and $W$ such that $A$ is presented as an algebra as \[{T_\bullet}(V)\otimes W\otimes {T_\bullet}(V)\to {T_\bullet}(V)\to A\to 0\]
and \begin{align*}
V_j^\ast\simeq \operatorname{Ext}^{1,j}_A(\mathbb{F},\mathbb{F})\\
W_j^\ast\simeq \operatorname{Ext}^{2,j}_A(\mathbb{F},\mathbb{F})
\end{align*}
\end{enumerate}
\end{prop} \begin{cor}\label{cor:gen module}
Let $M$ be an $A$-module. If $\operatorname{Ext}^{0,j}_A(M,\mathbb{F})=0$, then $M$ has no relation of degree $j$, i.e., if $m_j\in M_j$ then there are elements $a_k\in A_k$ and $m_k\in M_k$ such that \[m_j=\sum_{i<j}a_im_{j-i}.\] \end{cor}
The vector space $H^\bullet(A,\mathbb{F})=\bigoplus_{i,j}H^{i,j}(A,\mathbb{F})$ can also be endowed with the so-called \textbf{cup-product} $\smile$ that makes it into a bigraded (or $\mathbb N_0\times \mathbb N_0-$) algebra, i.e., \[\smile:H^{ij}(A,\mathbb{F})\otimes H^{pq}(A,\mathbb{F})\to H^{i+p,j+q}(A,\mathbb{F}).\]
The following is the bigraded version of the classical cohomological long exact sequence theorem, and it can be deduced from the general long-exact sequence theorem for (co)chain complexes (cfr. \cite{weib}).
\begin{thm}\label{thm:longext}
Let $A$ be an algebra. If $0\to L\to M\to N\to 0$ is an exact sequence of $A$-modules, then for all $j$ there is an induced long exact sequence \begin{align*}0\to& \operatorname{Ext}^{0,j}_A(N,\mathbb{F})\to \operatorname{Ext}^{0,j}_A(M,\mathbb{F})\to \operatorname{Ext}^{0,j}_A(L,\mathbb{F})\to\operatorname{Ext}^{1,j}_A(N,\mathbb{F})\to\dots\\
\dots\to& \operatorname{Ext}^{i,j}_A(N,\mathbb{F})\to \operatorname{Ext}^{i,j}_A(M,\mathbb{F})\to \operatorname{Ext}^{i,j}_A(L,\mathbb{F})\to\operatorname{Ext}^{i+1,j}_A(N,\mathbb{F})\to\dots
\end{align*}
\end{thm}
Also a version of the Eckmann-Shapiro Lemma holds for the internal degree of cohomology. In order to state such a result it is necessary to endow a tensor product with a grading. Let $A$ and $B$ be two $\mathbb N_0$-graded algebras, $M$ a graded $(A,B)$-bimodule and $N$ a graded left $B$-module. For $M$ being a graded $(A,B)$-bimodule means that $M=\bigoplus M_i$ is a graded vector space and both the action of $A$ and $B$ preserve the grading, i.e., $A_jM_i+M_iB_j\subseteq M_{i+j}$.
Then, the tensor product $M\otimes_B N$ is an $A$-module whose grading is given by \[(M\otimes_B N)_n=\left(\bigoplus_{i+j=n}M_i\otimes_\mathbb{F} N_j\right)\left/S_n\right.\] where $S_n$ is the vector subspace generated by elements of the form $xb\otimes y-x\otimes by$ for $x\in M_i$, $y\in N_j$ and $b\in B_{n-i-j}$.
\begin{thm}[Eckmann-Shapiro Lemma]\label{shap}
Let $A$ be an algebra and let $B$ be a subalgebra of $A$. Suppose that $A$ is projective as a right $B$-module. Let $M$ be a left $B$-module and let $N$ be a left $A$-module.
Then, \[\operatorname{Ext}^{ij}_A(A\otimes _BM,N)\simeq \operatorname{Ext}^{ij}_B(M,N).\]
\begin{proof}
Let $P_\bullet\to M$ be a projective resolution of $M$ over $B$. Since $A$ is a projective right $B$-module, the sequence \[A\otimes_B P_\bullet\to A\otimes_B M\]is a projective resolution over $A$.
Now, $\operatorname{Hom}_A^j(A\otimes_B P_i,N)\simeq \operatorname{Hom}^j_B(P_i,N)$, and hence $H^i(\operatorname{Hom}_A^j(A\otimes_B P_\bullet,N))\simeq H^i(\operatorname{Hom}_B^j(M,N))$.
\end{proof}
\end{thm} \subsection{Cohomology of Lie algebras} Let ${\mc L}$ be an $\mathbb N$-graded $\mathbb{F}$-Lie algebra defined as in the Introduction. Its universal envelope (cfr. \cite{bou}) ${\mc U(\mc L)}$ can be presented as the quotient of the tensor algebra on the vector space ${\mc L}$ by the ideal $J$ generated by the elements $x\otimes y-y\otimes x-[x,y]$ for every homogeneous elements $x,y\in{\mc L}$. By definition, the ideal $J$ is graded, and hence ${\mc U(\mc L)}$ inherits a grading from ${\mc L}$ that makes it into a $\mathbb N_0$-graded associative algebra.
If $M$ is a ${\mc L}$-module (i.e., a graded ${\mc U(\mc L)}$-module) one can define the cohomology of ${\mc L}$ with coefficients in $M$ as \[H^\bullet({\mc L},M):=H^\bullet({\mc U(\mc L)},M)=\operatorname{Ext}^\bullet_{\mc U(\mc L)}(\mathbb{F},M).\] It follows that $H^\bullet({\mc L},M)$ is bigraded. In this case, the cup product makes $H^\bullet({\mc L},\mathbb{F})$ into a graded-commutative algebra, i.e., $ab=(-1)^{nm}ba$ for $a\in H^n({\mc L},\mathbb{F})$ and $b\in H^m({\mc L},\mathbb{F})$.
In order to compute the Lie algebra cohomology one can use the Chevalley-Eilenberg complex $\operatorname{Hom}_\mathbb{F}({\Lambda_\bullet}({\mc L}),M)$ (cf. \cite{weib}), namely, \[H^\bullet({\mc L},M)=H^\bullet(\operatorname{Hom}_\mathbb{F}({\Lambda_\bullet}({\mc L}),M)).\] With respect to such a description, the cup product on $H^\bullet({\mc L},\mathbb{F})$ is induced by the natural map $\Lambda^n ({\mc L})\otimes \Lambda^m({\mc L})\to \Lambda^{n+m}({\mc L})$.
\subsection{Koszul algebras and modules}
For an $\mathbb{F}$-vector space $V$, denote by ${T_\bullet}(V)$ the tensor algebra over $V$, namely the vector space $\bigoplus_{i\geq 0}V^{\otimes i}$ where the product is given by concatenation of tensors.
Let $A$ be an $\mathbb{F}$-algebra that admits a presentation \[A=\frac{{T_\bullet}(V)}{(W)}\]
where $V$ is a graded vector space and $(W)={T_\bullet}(V)\otimes W\otimes {T_\bullet}(V)$ is the ideal generated by some $W\leq{T_\bullet}(V)$.
If $V$ is concentrated in degree $1$, then we say that $A$ is $1$\textbf{-generated}\footnote{Usually, in literature, one says that a certain object is "$n$-generated" whenever it is generated by $n$ of its elements. In this paper we are adopting such an expression for a different meaning.}.
If moreover $W\leq V\otimes V$, one calls $A$ a \textbf{quadratic algebra}. In the latter case, one writes $A=Q(V,W)$.
Henceforth, all the algebras we are dealing with are finitely generated, e.g., for a $1$-generated algebra $A$, $\dim_\mathbb{F} A_1<\infty$.
From Proposition \ref{prop:gen rel}, one can see that $A$ is a $1$-generated algebra if, and only if, $\operatorname{Ext}^{1,j}_A(\mathbb{F},\mathbb{F})=0$ for $j>1$, and $A$ is quadratic if, and only if, $\operatorname{Ext}^{i,j}_A(\mathbb{F},\mathbb{F})=0$ for $i<j$, $i=1,2$. Recall that always $\operatorname{Ext}^{ij}_A(\mathbb{F},\mathbb{F})=0$ for $j<i$.
By extending this cohomological property to higher degrees, one recovers the definiton of a \textbf{Koszul algebra}: $A$ is Koszul if \[\operatorname{Ext}^{i,j}_A(\mathbb{F},\mathbb{F})=0\quad \text{for } i\neq j.\]
In that case, one says that the cohomology of $A$ is concentrated on the diagonal.
Similarly, let $M$ be an $A$-module with a presentation \[M=\frac{A\otimes H}{K}\]with $K$ an $A$-submodule of $A\otimes H$.
If $H$ is concentrated in degree $n$, then we say that $M$ is an $n$-generated $A$-module.
If moreover $A$ is $1$-generated and $K\leq A_1\otimes H$, one calls $M$ a \textbf{quadratic module} generated in degree $n$, or, simply, of degree $n$. In the latter case, one writes $M=Q_A(H,K)$.
If $A$ is a quadratic algebra, from Corollary \ref{cor:gen module}, one has that $M$ is $n$-generated if, and only if, $\operatorname{Ext}^{0,j}_A(M,\mathbb{F})=0$ for $j\neq n$, and moreover $M$ is quadratic if, and only if, $\operatorname{Ext}^{i,j}_A(M,\mathbb{F})=0$ for $i\neq j-n$, $i=0,1$.
Again, by extending this cohomological property to higher degrees, one recovers the definiton of a \textbf{Koszul $A$-module}: $M$ is a Koszul $A$-module if there is some $n\geq 0$ such that \[\operatorname{Ext}^{i,j}_A(M,\mathbb{F})=0\quad \text{for } i+n\neq j.\]
It follows from \ref{shiftExt} that an $n$-generated $A$-module $M$ is Koszul if, and only if, $\operatorname{Ext}^{ij}_A(M[-n],\mathbb{F})=0$ for $j\neq i$, and hence we may only study $0$-generated Koszul $A$-modules.
Such modules are precisely those which admit a linear free resolution over $A$, i.e., a resolution by free $A$-modules $P_\bullet\twoheadrightarrow M$ such that every $P_i=A\cdot (P_i)_i$ is generated in its $i$th degree (\cite{pp}).
\medspace
If $A=Q(V,W)$ is a finitely generated quadratic algebra, there is a kind of dual algebra that one can attach to $A$; this is the \textbf{quadratic dual} of $A$, and it is defined as \[A^!=Q(V^\ast,W^\perp)\] where $W^\perp\leq V^\ast\otimes V^\ast$ is the orthogonal complement of $W$ in $(V\otimes V)^\ast\simeq V^\ast\otimes V^\ast$, i.e.,
\[W^\perp=\set{\alpha\in V^\ast\otimes V^\ast}{\alpha\vert_W\equiv 0}\]
Notice that $(A^!)^!\simeq A$ for any (finitely generated) quadratic algebra $A$.
The motivation for defining such a dual algebra is the following result:
\begin{thm}[cf. \cite{pp}]\label{thm:diagext=dual}
Let $A$ be a quadratic algebra. Then \[A^!\simeq\bigoplus _{i\geq 0} \operatorname{Ext}^{i,i}_A(\mathbb{F}\,\mathbb{F}).\]
In particular, $A$ is Koszul if, and only if, $A^!\simeq H^\bullet(A,\mathbb{F})$.
Moreover, $A$ and $A^!$ are Koszul simultaneously.
\end{thm} Also, one can define a similar object associated with a module $M$ over a graded algebra. Suppose $M$ is a $0$-generated quadratic $A$-module, say $M=Q_A(H,K)$, and $A$ is a quadratic algebra. Then, we define the quadratic dual of $M$ as the $A^!$-module \[M^{!_A}=Q_{A^!}(H^\ast,K^\perp).\] There holds: \begin{thm}[cf. \cite{pp}]
Let $A$ be a quadratic algebra and $M$ a quadratic module of degree $0$. Then \[M^{!_A}\simeq\bigoplus_{i\geq 0}\operatorname{Ext}^{i,i}_A(M,\mathbb{F}).\] \end{thm}
We now introduce a Koszul property for the morphisms of graded algebras. We say that a morphism $f:A\to B$ is a \textbf{(left) Koszul homomorphism} if $B$ is a Koszul left $A$-module, or, equivalently, if the morphism can be completed into a linear free $A$-resolution of $B$
\[\dots\to P_2\to P_1\to A\xrightarrow{f}B\to 0.\]
The definition of right Koszul homomorphism is clear. Notice that, in particular, Koszul homomorphisms need to be surjective.
\begin{prop}[Corollary I.5.4, p. 35, \cite{pp}]\label{koskos}
Let $f:A\to B$ be a Koszul homomorphism. Then $A$ is a Koszul algebra if, and only if, $B$ is.
\end{prop} In fact, an algebra $A$ is Koszul if, and only if, the augmentation map $A\to \mathbb{F}$ is Koszul.
\begin{lem}[Corollary I.5.9, p. 35, \cite{pp}]\label{lem:5.9}
Let $f:A\to B$ be a homomorphism of Koszul algebras. Then $f$ is left-Koszul if, and only if, $A^!$ is a free right $B^!$-module. If this is the case, then the dual morphism $B^!\to A^!$ is injective, and $B$ and $A^!/A^!B_+^!$ are dual Koszul modules over $A$ and $A^!$.
\end{lem}
\begin{prop}{\label{prop:constr ext}}
Let $A$ be a Koszul $\mathbb{F}$-algebra, and let $L\xrightarrow{f}M\xrightarrow{\varepsilon}N\to 0$ be an exact sequence of Koszul $A$-modules of degree $0$.
Then the following constraints hold:
\begin{enumerate}
\item $\operatorname{Ext}^{i,j}_A(\ker\varepsilon,\mathbb{F})=0$ for $j>i+1$,
\item $\operatorname{Ext}^{i,j}_A(\ker f,\mathbb{F})=0$ for $j>i+2$.
\end{enumerate}
In particular, $\ker f$ has no generator of degree $>2$.
\begin{proof}
(1) Consider the exact sequence $0\to \ker\varepsilon\to M\to N\to 0$.
It yields a long exact sequence \[\dots \to\operatorname{Ext}^{ij}_A(M,\mathbb{F})\to\operatorname{Ext}^{ij}_A(\ker\varepsilon,\mathbb{F})\to \operatorname{Ext}^{i+1,j}_A(N,\mathbb{F})\to\dots\]
Since both $M$ and $N$ are Koszul modules, their $\operatorname{Ext}$-groups are concentrated on the diagonal.
Then, for $j>i+1$, there holds $\operatorname{Ext}^{ij}_A(M,\mathbb{F})=\operatorname{Ext}^{i+1,j}_A(N,\mathbb{F})=0$, forcing $\operatorname{Ext}^{ij}_A(\ker\varepsilon,\mathbb{F})$ to vanish.
(2) Similarly, the exact sequence $0\to\ker f\to L\to \ker \varepsilon\to 0$ induces the long exact sequence \[\dots \to\operatorname{Ext}^{ij}_A(L,\mathbb{F})\to\operatorname{Ext}^{ij}_A(\ker f,\mathbb{F})\to \operatorname{Ext}^{i+1,j}_A(\ker\varepsilon,\mathbb{F})\to\dots\]
Let $j>i+2$. For (1), and since $L$ is Koszul, $\operatorname{Ext}^{i+1,j}(\ker\varepsilon,\mathbb{F})=\operatorname{Ext}^{ij}_A(L,\mathbb{F})=0$, forcing $\operatorname{Ext}^{ij}_A(\ker f,\mathbb{F})$ to vanish.
\end{proof}
\end{prop}
One may write the previous result with more generality on the degree of generation of the modules, although for our purpose the latter is enough.
\begin{cor}{\label{inj koszul}}
Let $f:L\to M$ be a morphism of Koszul $A$-modules of degree $0$. Suppose that $f$ is injective in all the degrees up to the second, and that $\text{coker} f$ is Koszul. Then $f:L\to M$ is injective (in every degree).
\begin{proof}
By Lemma \ref{prop:constr ext}, $\ker f$ has no generator of degree $>2$.
But since $\ker f_i$ is trivial for $i=0,1,2$, it must be $0$ for all $i\geq 0$.
\end{proof}
\end{cor} \subsection{Bloch-Kato Lie algebras}
In this subsection we introduce the notion of Bloch-Kato Lie algebra and show some examples. \begin{defin}
Let ${\mc L}$ be an $\mathbb N$-graded $1$-generated $\mathbb{F}$-Lie algebra. We say that ${\mc L}$ is Bloch-Kato if all of its $1$-generated subalgebras are Koszul. \end{defin} Notice that this is equivalent to requiring that the inclusion $\mc M\to {\mc L}$ induce a surjective algebra homomorphism $H^{\bu,\bu}({\mc L})\to H^{\bu,\bu}(\mc M)$, for every $1$-generated subalgebra $\mc M$ of ${\mc L}$.
\medspace
\textbf{Examples.}
(1) Let $\mc F$ be a free Lie algebra generated by elements of degree $1$.
We will see in Theorem \ref{NS} that the $1$-generated subalgebras of $\mc F$ is again a free Lie algebra. For the cohomology of a free Lie algebra over a space $V$ is (isomorphic to) the quotient ${T_\bullet} V/(V\otimes V)=\mathbb{F}\oplus V$, it is clear that $\mc F$ is Bloch-Kato.
(2) Also, every $1$-generated abelian Lie algebra is Bloch-Kato, as its universal envelope -- as well as that of any of its $1$-generated subalgebras -- is a polynomial ring. However, these are not the only examples of Bloch-Kato Lie algebras, as the example shows.
(3) Let ${\mc L}$ be the Lie algebra generated by $2d$ elements $x_1,y_1,\dots,x_d,y_d$ with the single relation $\sum_i[x_i,y_i]=0$. Put $A={\mc U(\mc L)}$.
The following is a \textit{free resolution} of $\mathbb{F}$ over $A$: \[ [F_\bullet\to \mathbb{F}]=\dots\to 0\to A\overset{d_1}{\to} A^{2d}\overset{d_0}{\to} A\overset{\varepsilon}{\to} \mathbb{F}\to 0\] where \[d_0:(a_1,b_1,\dots,a_d,b_d)\mapsto \sum a_ix_i+b_iy_i,\] \[d_1:a\mapsto a(y_1,-x_1,\dots,y_d,-x_d)\] and \[\varepsilon:x_i,y_i\mapsto 0,\ 1\mapsto 1.\] That resolution is minimal and linear, whence, $\mc L$ is Koszul with cohomological dimension $cd \mc L=2$.
This also shows that \[\operatorname{Ext}^2_A(\mathbb{F},A)\simeq \mathbb{F}\] with the right action of $A$ induced by the augmentation map $\varepsilon:A\to \mathbb{F}$. Moreover, $\operatorname{Ext}^i_A(\mathbb{F},A)=0$ for $i\neq 2$, whence $A$ is a \textit{Poincaré-duality algebra} of dimension $2$. \begin{prop}[Poincaré duality algebras]
Let $A\neq 0$ be an associative $\mathbb{F}$-algebra of type FP
satisfying \[\operatorname{Ext}^i_A(\mathbb{F},A)=H^i(A,A)=0,\ i\neq n.\]
Let $D$ be the right $A$-module $H^n(A,A)$.
Then for every left $A$-module $M$ there holds the following homological duality \[H^i(A,M)\simeq \operatorname{Tor}^A_{n-i}(D,M).\]
\begin{proof}
Notice that one has $cd A=n$ since $A$ is of type FP.
Let $P_\bullet=(P_i)_{0\leq i\leq n}$ be a finite projective resolution of $\mathbb{F}$ over $A$, i.e. the sequence \[0\to P_n\to P_{n-1}\to\dots\to P_0\to \mathbb{F}\to 0\] is exact and $P_\bullet$ is a finitely generated projective $A$-module.
Let $\bar P^{-i}=\operatorname{Hom} _A(P_i,A)$ be the dual complex. Thus, $\bar P^\bullet$ is projective.
The sequence $\bar P^0\to\dots\to \bar P^{-n}$ is exact, as $H_{-i}(\bar P^\bullet)=H^i(A,A)=0$, for $i\neq n$.
Moreover, $\operatorname{Ext}^n_A(\mathbb{F},A)=H_{-n}(\bar P^\bullet)=\bar P^{-n}/im(\bar P^{-n+1}\to \bar P^{-n})$, and thus we have a projective $A$-resolution for $D=H^n(A,A)=\operatorname{Ext}_A^n(\mathbb{F},A)$ \[\dots\to \bar P^{-(n-1)}\to \bar P^{-n}\to D;\] more precisely, $\bar P^{n+\bullet}$ is the projective resolution.
For every $i$ and every $A$-module $M$, $\operatorname{Hom}_A(P_i,M)\simeq \bar P^{-i}\otimes_A M$, for $P_\bullet$ is finitely generated and projective.
Finally, \begin{align*}H^i(A,M)&=\operatorname{Ext}^i_A(\mathbb{F},M)=H_{-i}(\operatorname{Hom}_A(P_\bullet,M))\\
&\simeq H_{-i}(\bar P^\bullet\otimes_A M)=H_{n-i}(\bar P^{n+\bullet}\otimes_A M)\\
&=\mbox{Tor}_{n-i}^A(D,M)\end{align*}
i.e. \[H^i(A,M)=\operatorname{Ext}^i_A(\mathbb{F},M)\simeq\mbox{Tor}_{n-i}^A(D,M)\]
\end{proof} \end{prop}
It follows from the latter result that for our Lie algebra ${\mc L}$, it holds \begin{align*}
&\operatorname{Ext}^0_A(\mathbb{F},\hbox to 1.5ex{\hrulefill})\simeq \mbox{Tor}_2^A(\mathbb{F},\hbox to 1.5ex{\hrulefill})\\
&\operatorname{Ext}^1_A(\mathbb{F},\hbox to 1.5ex{\hrulefill})\simeq \mbox{Tor}_1^A(\mathbb{F},\hbox to 1.5ex{\hrulefill})\\
&\operatorname{Ext}^2_A(\mathbb{F},\hbox to 1.5ex{\hrulefill})\simeq \mbox{Tor}_0^A(\mathbb{F},\hbox to 1.5ex{\hrulefill})=\mathbb{F}\otimes_A \hbox to 1.5ex{\hrulefill} \end{align*}
Now, since the vector space dual functor $\hbox to 1.5ex{\hrulefill}^\ast=\operatorname{Hom}_\mathbb{F}(\hbox to 1.5ex{\hrulefill},\mathbb{F})$ is exact on finite dimensional vector spaces, one has \begin{align}\label{torext}\mbox{Tor}_2^A(\mathbb{F},\mathbb{F})\simeq \operatorname{Ext}_A^2(\mathbb{F},\mathbb{F})^\ast.\end{align}\\ Let now $\mc M$ be a proper Lie subalgebra of $\mc L$, and we show that $\mc M$ has cohomological dimension at most $1$, and thus it is free. This finally proves that $\mc L$ is Bloch-Kato.
In order to do that, we compute $H^2(\mc M,\mathbb{F})$. For the isomorphism \ref{torext} holds, it is enough to compute $H_2(\mc M, \mathbb{F})$. By applying the homological version of Eckmann-Shapiro Lemma (cf. \cite{weib}) to the subalgebra $B=\mc U(\mc M)$ of $A$, we get \[\mbox{Tor}_2^B(\mathbb{F},\mathbb{F})\simeq \mbox{Tor}_2^A(\mathbb{F},A\otimes _B \mathbb{F}).\] Therefore, for the duality relations of $A$, \[\mbox{Tor}_2^B(\mathbb{F},\mathbb{F})\simeq \operatorname{Ext}^0_A(\mathbb{F},A\otimes_B \mathbb{F})=\operatorname{Hom}_A(\mathbb{F},A\otimes_B \mathbb{F}).\] It follows from PBW theorem that the latter space is trivial, namely, the $A$-module $A\otimes _B\mathbb{F}$ has no $A$-fixed points. Indeed, let $\alpha:\mathbb{F}\to A\otimes _B \mathbb{F}$ be an $A$-linear map. For $\mathbb{F}$ is a simple $A$-module, the map $\alpha$ is determined by $\alpha(1)$. Complete an $\mathbb{F}$-basis $\{y_i,\ i\in I\}$ of $\mc M$ to an $\mathbb{F}$-basis $\{y_i,z_j,\ i\in I,\ j\in J\}$ for $\mc L$. By PBW theorem, $B$ has $\mathbb{F}$-basis $\{y_{i_1}\cdots y_{i_n}\vert\ i_1,\dots, i_n\in I,\ n\in\mathbb N_0\}$ and it is infinite-codimensional in $A$, whose $\mathbb{F}$-basis can be suitably chosen to be $\{z_{j_1}\cdots z_{j_m}y_{i_1}\cdots y_{i_n}\vert\ i_1,\dots, i_n\in I,\ j_1\dots,j_m\in J,\ m,n\in\mathbb N_0\}$. Note now that the induced module can be written as the following quotient of $A$: \[A\otimes_B \mathbb{F}\simeq A/AB_+.\] With respect to the above isomorphism, if $\alpha(1)=u+AB_+$, one can choose $u\in A$ to be of the form \[ u= \sum r_{j_1,\dots,j_r}z_{j_1}\cdots z_{j_r}\] for some $r_{j_1,\dots,j_r}\in \mathbb{F}$. Finally, \[0=\alpha(z_j\cdot 1)=z_j\alpha(1),\ j\in J\] implies $\sum r_{j_1,\dots,j_r}z_jz_{j_1}\cdots z_{j_r}\in AB^+$, and thus the $r_{j_1,\dots,j_r}$'s need to vanish. \\
We conclude that $H^2(\mc M,\mathbb{F})=0$, and thus $\mc M$ is free by Theorem \ref{freecoh}.
\section{Free products} In this section, we recall the notion of free products of Lie algebras, and we state the graded version of the PBW Theorem. For a more detailed description, see \cite{bou}.
\begin{defin}
Let $\mc A$ and $\mc B$ be two $\mathbb N$-graded Lie algebras over a field $\mathbb F$. Their free product is an $\mathbb N$-graded Lie algebra $\mc F$ endowed with two morphisms of graded Lie algebras $\iota_{\mc A}:\mc A\to\mc F$ and $\iota_{\mc B}:\mc B\to \mc F$, which satisfy the following universal property. For any pair of morphisms $\alpha:\mc A\to {\mc L}$ and $\beta:\mc B\to {\mc L}$ of graded Lie algebras, there is a unique morphism $\phi: \mc F\to {\mc L}$ of graded Lie algebras such that $\alpha=\phi\circ\iota_{\mc A}$ and $\beta=\phi\circ\iota_{\mc B}$. \end{defin}
We also have a notion of free product of connected $\mathbb{F}$-algebras. In particular, if $A$ and $B$ are connected $\mathbb{F}$-algebras, we may define their free product $F=A\amalg B$ as follows: \begin{enumerate}
\item $F_0\simeq \mathbb{F}$, namely, $F$ is connected;
\item $F_n=A_n\oplus [(A_{n-1}\otimes B_1)\oplus (B_1\otimes A_{n-1}) ]\oplus[(A_{n-2}\otimes B_2)\oplus (B_1\otimes A_{n-2}\otimes B_1)\oplus( B_2\otimes A_{n-2})]\oplus\dots\oplus B_n$ for $n>0$. \end{enumerate} The multiplication is given by concatenation followed, possibly, by the composition of elements that do belong to the same factor.
Eventually, there is a description for the free product of $\mathbb N$-graded Lie algebras $\mc A$ and $\mc B$. Let $\mc U(\mc A)$ and $\mc U(\mc B)$ be the universal envelopes of $\mc A$ and $\mc B$ respectively. Now, the free product of $\mc A$ and $\mc B$ can be described as the Lie subalgebra of $(\mc U(\mc A)\amalg\mc U(\mc B))_L$ generated by the images of $\mc A$ and $\mc B$.
Recall the following important result. \begin{thm}[Poincaré-Birkhoff-Witt]
Let ${\mc L}$ be a $\mathbb N$-graded $\mathbb{F}$-Lie algebra. Let $\mathcal B$ be an ordered graded basis of ${\mc L}$ as a vector space. Then \[\mc\mathcal B=\set{v_1v_2\dots v_k}{v_i\in\mathcal B,\ v_i\leq v_{i+1}, k\geq 0}\] is a basis of ${\mc U(\mc L)}$ as an $\mathbb{F}$-vector space.
Moreover, if $\mc M$ is a graded subalgebra of ${\mc L}$, then ${\mc U(\mc L)}$ is a free $\mathbb N_0$-graded $\mc U(\mc M)$-bimodule. \end{thm}
Let ${\mc L}$ be an $\mathbb N$-graded $\mathbb{F}$-Lie algebra, and denote by $\mc U({\mc L})$ its universal envelope. If $\mc H\subseteq {\mc L}$ is an $\mathbb N$-graded $\mathbb{F}$-Lie subalgebra, then by the Poincar\'e-Birkhoff-Witt theorem, $\mc U({\mc L})$ is a $\mathbb N_0$-graded free left $\mc U(\mc H)$-module. Let $M$ be a $\mathbb N_0$-graded left $\mc H$-module, and define \[\ind_{\mc H}^{{\mc L}}(M):=\mc U({\mc L})\otimes_{\mc U(\mc H)} M.\] The functor $\ind_{\mc H}^{{\mc L}}(\hbox to 1.5ex{\hrulefill})$ is called the \textbf{induction functor}. It is a covariant additive exact functor which is left adjoint to the restriction functor $\res_{\mc H}^{{\mc L}}(\hbox to 1.5ex{\hrulefill})$, i.e. one has natural isomorphisms of bifunctors \begin{equation}{\label{ind}}
\operatorname{Hom}_{{\mc L}}(\ind_{\mc H}^{{\mc L}}(M),Q)\simeq \operatorname{Hom}_{\mc H}(M,\res_{\mc H}^{{\mc L}}(Q)). \end{equation} The natural isomorphisms (\ref{ind}) are also called the Nakayama relations. It follows that the identity map $\mathbb{F}\to \mathbb{F}$ of the trivial $\mc H$-module $\mathbb{F}$ induces a homomorphism of graded left ${\mc L}$-modules \begin{equation}{\label{canmap}}
\varepsilon^{\mc H}:\ind_{\mc H}^{{\mc L}}(\mathbb{F})\longrightarrow \mathbb{F}. \end{equation} \begin{thm}{\label{thm:mayervietoris}}
Let ${\mc L}$ be an $\mathbb N$-graded $1$-generated $\mathbb{F}$-Lie algebra containing two $\mathbb N$-graded $\mathbb{F}$-subalgebras $\mc A$ and $\mc B$ satisfying ${\mc L}=\gen{\mc A,\mc B}$.
Then the following are equivalent:
\begin{itemize}
\item[(i)] ${\mc L}\simeq \mc A\amalg\mc B$,
\item[(ii)] $\ker\left(\varepsilon^{\mc A}-\varepsilon^{\mc B}:\ind_{\mc A}^{{\mc L}}(\mathbb{F})\oplus\ind_{\mc B}^{{\mc L}}(\mathbb{F})\to \mathbb{F}\right)\simeq \mc U(\mc {\mc L})$.
\end{itemize}
\begin{proof}
(i)$\Rightarrow$(ii) is Bass-Serre Theory for $\mathbb N$-graded Lie algebras (cf. \cite{cmp}).
Suppose now that (ii) holds.
Note that the canonical homomorphism $\phi:\mc F\to\mc L$ is surjective, where $\mc F=\mc A\amalg\mc B$.
By (ii), the sequence
\begin{equation}
\label{mayervietoris}
0\to \mc U({\mc L})\to \ind_{\mc A}^{{\mc L}}(\mathbb{F})\oplus\ind_{\mc B}^{{\mc L}}(\mathbb{F})\to \mathbb{F}\to 0
\end{equation}
is exact.
The same short exact sequence holds for $\mc F$, by Bass-Serre theory for graded Lie algebras.
By the Eckmann-Shapiro Lemma,
\begin{align*}
\operatorname{Ext}_{\mc U({\mc L})}^\bullet(\mc U({\mc L})\otimes_{\mc U(\mc A)}\mathbb{F},\mathbb{F})\simeq \operatorname{Ext}^\bullet_{\mc U(\mc A)}(\mathbb{F},\mathbb{F}),\\
\operatorname{Ext}_{\mc U({\mc L})}^\bullet(\mc U({\mc L})\otimes_{\mc U(\mc B)}\mathbb{F},\mathbb{F})\simeq \operatorname{Ext}^\bullet_{\mc U(\mc B)}(\mathbb{F},\mathbb{F}).
\end{align*}
Thus we get a long exact sequences from Theorem \ref{thm:longext} induced by (\ref{mayervietoris})
\begin{align*}
\operatorname{Ext}^1_{\mc U({\mc L})}(\mc U({\mc L}),\mathbb{F})\to \operatorname{Ext}^2_{\mc U(\mc A)}(\mathbb{F},\mathbb{F})\oplus \operatorname{Ext}^2_{\mc U(\mc B)}(\mathbb{F},\mathbb{F})\to \\
\to\operatorname{Ext}^2_{\mc U(\mc L)}(\mathbb{F},\mathbb{F})\to \operatorname{Ext}^2_{\mc U(\mc L)}(\mc U(\mc L),\mathbb{F})\to\dots
\end{align*}
But $\operatorname{Ext}^i_{\mc U(\mc L)}({\mc U(\mc L)},\mathbb{F})=0$ for $i>0$, and thus the inflation $\inf:H^2({\mc L},\mathbb{F})\to H^2(\mc F,\mathbb{F})$ is an isomorphism $H^2(\mc L,\mathbb{F})\simeq H^2(\mc A,\mathbb{F})\oplus H^2(\mc B,\mathbb{F})\simeq H^2(\mc U(\mc F),\mathbb{F})$.
Now, consider the short exact sequence associated with the surjection $\phi:\mc F\to \mc L$, \[0\to I\to\mc F\overset{\phi}{\to}{\mc L}\to 0.\]
This yields the $5$-term exact sequence in cohomology
\begin{equation*}
\xymatrix{
0\ar[r]&H^1({\mc L},\mathbb{F})\ar[r]^{H^1(\phi)}&H^1(\mc F,\mathbb{F})\ar[r]^\alpha&
H^1(I,\mathbb{F})^{{\mc L}}\ar[d]\\
&&H^2(\mc F,\mathbb{F})&H^2({\mc L},\mathbb{F})\ar[l]\\}
\end{equation*}
But $H^i({\mc L},\mathbb{F})\to H^i(\mc F,\mathbb{F})$, for $i=1,2$, are the isomorphisms $H^1(\phi)$ and $\inf$, and thus $H^1(I,\mathbb{F})^{{\mc L}}=0$. As $I $ is an $\mathbb N$-graded ideal of $\mc F $, one has $H^1(I ,\mathbb{F})=\operatorname{Hom}_{\mbox{\small{Lie}}}(I ,\mathbb{F})=\operatorname{Hom}_{\mathbb{F}}(I /[I ,I ],\mathbb{F})$.
Thus, $H^1(I ,\mathbb{F})^{{\mc L} }=0$ implies $I /[\mc F ,I ]=0$, and therefore $I =[\mc F ,I ]$.
Now, suppose $n=\min\set{m\geq 0}{I_m\neq 0}$ is finite. One has $I_n=[I ,\mc F ]_n=\sum_{1\leq j<n}[I_j,\mc F_{n-j}]=0$, since $I $ is a graded ideal, whence $I =0$, proving that $\ker(\phi)=I=0$.
\end{proof} \end{thm} \begin{lem}{\label{cohom free prod}}
Let $\mc A$ and $\mc B$ be two $\mathbb N$-graded finitely generated $\mathbb{F}$-Lie algebras. Then there is an isomorphism of graded algebras \[H^\bullet(\mc A\amalg\mc B,\mathbb{F})\simeq H^\bullet(\mc A,\mathbb{F})\sqcap H^\bullet(\mc B,\mathbb{F}),\] where $\sqcap$ denotes the product in the category of graded connected algebras.
\begin{proof}
Let ${\mc L}=\mc A\amalg B$. Then, by Theorem \ref{thm:mayervietoris}, there is a short exact sequence of $\mc U({\mc L})$-modules \[0\to \mc U({\mc L})\to \ind_{\mc A}^{{\mc L}}(\mathbb{F})\oplus\ind_{\mc B}^{{\mc L}}(\mathbb{F})\to \mathbb{F}\to 0.\]
The latter and the Eckmann-Shapiro Lemma induce a long exact sequence
\begin{align*}
\dots\to\operatorname{Ext}^i_{\mc U({\mc L})}(\mc U({\mc L}),\mathbb{F})\to \operatorname{Ext}^{i+1}_{\mc U(\mc A)}(\mathbb{F},\mathbb{F})\oplus \operatorname{Ext}^{i+1}_{\mc U(\mc B)}(\mathbb{F},\mathbb{F})\to \\
\to\operatorname{Ext}^{i+1}_{\mc U(\mc L)}(\mathbb{F},\mathbb{F})\to \operatorname{Ext}^{i+1}_{\mc U(\mc L)}(\mc U(\mc L),\mathbb{F})\to\dots
\end{align*}
and it follows that for $i\geq 1$,
\begin{equation}\label{prod coho}
\operatorname{Ext}_{\mc U({\mc L})}^i(\mathbb{F},\mathbb{F})\simeq \operatorname{Ext}_{\mc U(\mc A)}^i(\mathbb{F},\mathbb{F})\oplus \operatorname{Ext}_{\mc U(\mc B)}^i(\mathbb{F},\mathbb{F}),
\end{equation}
or $H^i(\mc L,\mathbb{F})\simeq H^i(\mc A,\mathbb{F})\oplus H^i(\mc B,\mathbb{F})$.
Now, the natural inclusions of $\mc A$ and $\mc B$ into $\mc A\amalg\mc B$ induce the algebra homomorphisms $H^\bullet(\mc A\amalg\mc B)\to H^\bullet(\mc A)$ and $H^\bullet(\mc A\amalg\mc B)\to H^\bullet(\mc B)$. By the universal property of the direct product, we get an algebra homomorphism $H^\bullet(\mc A\amalg\mc B)\to H^\bullet(\mc A)\sqcap H^\bullet(\mc B)$, that is an isomorphism, in the light of (\ref{prod coho}).
\end{proof} \end{lem}
\section{Universally Koszul algebras}
In this section we focus on the graded-commutative algebras, whose main example is the cohomology algebra of a Lie algebra. Recall that a graded algebra $A=\oplus_{i\geq 0}A_i$ is said to be \textbf{graded-commutative} if $ab=(-1)^{nm}ba$ for any homogenous elements $a\in A_{n},\ b\in A_m$.
For a graded-commutative algebra $A$, define the \textbf{elementary skew-extension ring} $A[x]$ as the vector space $A\oplus Ax$, where $x$ has degree $1$, i.e., $A[x]\simeq A\oplus A[-1]$ as graded $\mathbb{F}$-vector spaces, endowed with the following product:
\[(a_1+a_2x)(a_1'+a_2'x)=a_1a_1'+(a_1a_2'+(-1)^{\abs{a_1'}}a_2a_1')x.\]
The elementary skew-extension algebra coincides thus with the free product in the category of the graded-commutative algebras of $A$ and ${\Lambda_\bullet}(x)$, the free graded-commutative algebra on a single element $x$.
Also, for a graded-commutative algebra $B$, define the \textbf{direct sum} $A\sqcap B$ as the categorical product of $A$ and $B$ in the category of graded-commutative connected algebras.
Explicitely, it is the connected algebra such that $(A\sqcap B)_i=A_i\oplus B_i$, $i>0$, and the multiplication is given component-wise.
\begin{defin}
A graded-commutative algebra $A$ is said to be \textbf{universally Koszul} if $A\to A/I$ is a Koszul homomorphism for any ideal $I$ that is generated by elements of degree $1$. \end{defin} In particular, since $A_+$ is a $1$-generated ideal, the fact that $A\to A/A_+=\mathbb{F}$ is a Koszul homomorphism implies that $A$ is Koszul (cfr. Proposition \ref{koskos}).
\begin{lem}{\label{lem:pol ring}}
Let $A$ and $B$ be universally Koszul graded-commutative algebras. Then,
\begin{enumerate}
\item $A[x]$ is universally Koszul
\item The direct sum $A\sqcap B$ is universally Koszul
\end{enumerate}
\begin{proof}
See Proposition 30 in \cite{enhanced}.
\end{proof}\end{lem}
\section{Proofs of Theorems A and B}
The first result of this section allows one to pass from Lie algebras to graded-commutative algebras and vice versa.
\begin{lem}\label{dual ue}
\begin{enumerate}
\item If $A$ is a quadratic graded-commutative algebra, then $A^!$ is the universal envelope of a quadratic Lie algebra.
\item Conversely, if ${\mc L}$ is a quadratic Lie algebra, then ${\mc U(\mc L)}^!$ is a (quadratic) graded-commutative algebra.
\end{enumerate}
\end{lem}
\begin{proof}
(1) Let $A={\Lambda_\bullet}(V^\ast)/(\Omega)$, where $\Omega\leq \Lambda_2(V^\ast)$. Choose a basis $(\omega_i)$ for $\Omega$, and elements $x_i,y_i\in V$ such that $\omega_i(x_j\otimes y_j)=\delta_{ij}$.
Define the Lie algebra given by the following presentation:
\[{\mc L}=\pres{V}{[v,v']-\sum_i\omega_i(v,v')[x_i,y_i]:\ v,v'\in V}.\]
We want to show that ${\mc U(\mc L)}$ is isomorphic to $A^!$.
To this end, we present $A$ as a quadratic algebra $A=Q(V^\ast,R)$, where $R=\tilde\Omega\oplus \text{Span}_\mathbb{F}(\alpha\otimes \alpha:\alpha\in V)\leq V^\ast\otimes V^\ast$, and $\tilde \Omega$ is a lifting of $\Omega$ to a subspace of $V^\ast\otimes V^\ast$ with respect to the canonical projection $V^\ast\otimes V^\ast\to V^\ast\wedge V^\ast$.
It is easy to see that $\ker({T_\bullet}(V)\to {\mc U(\mc L)})\subseteq\ker({T_\bullet}(V)\to A^!)$, as \[\omega_j\left( (v\otimes v'-v'\otimes v)-\sum_i\omega_i(v,v')(x_i\otimes y_i-y_i\otimes x_i)\right)=0\]
Also the opposite inclusion holds true.
(2) Since ${\mc U(\mc L)}$ is quadratic, by Theorem \ref{thm:diagext=dual}, its dual is isomorphic to the diagonal subalgebra $\bigoplus_i\operatorname{Ext}^{ii}_{\mc U(\mc L)}(\mathbb{F},\mathbb{F})$ of the cohomology algebra $\operatorname{Ext}^{\bu,\bu}_{\mc U(\mc L)}(\mathbb{F},\mathbb{F})=H^\bullet({\mc L},\mathbb{F})$, which is well-known to be graded-commutative.
\end{proof}
There follows Theorem A.
\begin{thm}{\label{bk uk}}
Let ${\mc L}$ be a graded Lie algebra with cohomology algebra $A=H^\bullet({\mc L},\mathbb{F})$.
Then ${\mc L}$ is Bloch-Kato if, and only if, $A$ is universally Koszul.
\begin{proof}
(1) Let ${\mc L}$ be a Bloch-Kato Lie algebra. If $I_1\leq A_1={\mc L}_1^\ast$, consider the $1$-generated Lie subalgebra \[\mc M=\pres{m\in{\mc L}_1}{i(m)=0,\ \forall i\in I_1}=\gen{I_1^\perp}.\]
Since ${\mc L}$ is Bloch-Kato, the Lie algebra $\mc M$ is Koszul, and $H^\bullet(\mc M,\mathbb{F})\simeq \mc U(\mc M)^!$.
By definition, $\res:H^\bullet({\mc L},\mathbb{F})\to H^\bullet(\mc M,\mathbb{F})$ is surjective and \[\ker \res_1=\set{\alpha\in{\mc L}_1^\ast}{\alpha\vert_{\mc M_1}=0}=I_1.\]
Therefore, there is an ideal $J=I(I_1)\lhd A$ such that \[H^\bullet({\mc L},\mathbb{F})/I(I_1)\simeq B=H^\bullet(\mc M,\mathbb{F})\] is Koszul and $I(I_1)_1=I_1$. We want to show that $J$ is generated by $I_1$ as an ideal of $A$.
Now, the projection $A\to B$ is Koszul, by Proposition \ref{lem:5.9}, since $A^!={\mc U(\mc L)}$ is a free right $B^!$-module, for $B^!=\mc U(\mc M)$ and the PBW Theorem. Thus, $B$ is a Koszul $A$-module.
The exact sequence of $A$-modules $0\to J\to A\to B\to 0$ induces a long exact sequence involving the $\operatorname{Ext}$ functor, that is,
\[\operatorname{Ext}^{0,j}_A(A,\mathbb{F})\to\operatorname{Ext}^{0,j}_A(J,\mathbb{F})\to\operatorname{Ext}^{1,j}_A(B,\mathbb{F})\to\operatorname{Ext}^{1,j}_A(A,\mathbb{F})\]
Notice that $J_0=0$, for $B_0\simeq A_0$.
For $B$ being a Koszul $A$-module it means that $\operatorname{Ext}^{\bullet,\bullet}_A(B,\mathbb{F})$ is concentrated on the diagonal, and hence the long exact sequence shows that $J$ is generated in degree $1$, namely $J=AI_1$.
(More precisely $\operatorname{Ext}^{i,j}_A(J,\mathbb{F})=0$ for $j\neq i+1$, and $J$ is a Koszul $A$-module.)
(2) Suppose $A$ is a universally Koszul graded-commutative algebra.
Then $A$ is Koszul and $A^!=\mc U(\mc A)$ is the universal envelope of a Koszul Lie algebra $\mc A$ generated by $A_1^\ast$, by Lemma \ref{dual ue}.
Let $V\leq A_1^\ast$, and $\mc B=\gen{V}$ be the Lie subalgebra of $\mc A$ generated in degree $1$ by $V$.
Let $I$ be the ideal of $A$ generated by $V^\perp\leq A_1$ in degree $1$.
Thus $A/I$ is Koszul since $A$ is universally Koszul, and $(A/I)^!$ is a quadratic subalgebra of $A^!$ by Lemma \ref{lem:5.9}.
It follows that $(A/I)^!=\mc U(\mc N)$ for some (quadratic) Lie algebra $\mc N$ generated by $(A_1/I_1)^\ast=V$.
But, by Lemma \ref{lem:5.9}, $\mc N$ is a Lie subalgebra of $\mc A$, and hence, $\mc N=\mc B$, as they have the same generating space.
Thus, $\mc B$ is Koszul and \[H^\bullet(\mc B,\mathbb{F})\simeq A/I.\]
Eventually, notice that since $A=H^\bullet({\mc L},\mathbb{F})$ is $1$-generated, $\mc U({\mc L})$ is a Koszul algebra, and hence $A^!=\mc U({\mc L})$, namely $\mc A={\mc L}$.
\end{proof}
\end{thm}
Since the direct sum of universally Koszul algebra is universally Koszul, by Lemma \ref{lem:pol ring}, the proof of Theorem B follows immediately.
\begin{thm}\label{et=>bk}
The class of Bloch-Kato Lie algebras is closed under taking free products.
\begin{proof}
Let $\mc A$ and $\mc B$ be two Bloch-Kato $\mathbb{F}$-Lie algebras. Then their cohomology rings $A=H^\bullet(\mc A,\mathbb{F})$ and $B=H^\bullet(\mc B,\mathbb{F})$ are universally Koszul by Theorem A.
Now, $H^\bullet(\mc A\amalg\mc B,\mathbb{F})=A\sqcap B$ by Lemma \ref{cohom free prod}, and hence it is universally Koszul by Lemma \ref{lem:pol ring}.
Therefore, $\mc A\amalg \mc B$ is Bloch-Kato by Theorem A.\end{proof}
\end{thm}
\section{Kurosh subalgebra theorem}
In the realm of (pro-$p$) groups, the Kurosh theorem gives the structure for an arbitrary subgroup of the free product of groups. This result is an immediate consequence of the Bass-Serre theory for groups acting on trees, and bacause of this geometric setting, its proof appears very elegant.
Unfortunately, for the (ungraded) Lie algebras such a result is not available, and indeed it is false (see \cite{shir}.)
However, for the positively-graded case, things behave better, and more similar to the group case, even though a complete Bass-Serre theory is not available; in fact, there is no action of a Lie algebra on graphs.
To this end we begin by proving a Nielsen-Schreier theorem for Lie algebras. It can be seen as a Kurosh' theorem for free $\mathbb N$-graded Lie algebras freely generated in degree $1$.
\begin{thm}\label{NS}
Let $\mc M \subseteq {\mc L}\langle V\rangle$ be a $1$-generated subalgebra of the free $\mathbb{F}$-Lie algebra ${\mc L}\langle V\rangle$, $V$ an $\mathbb{F}$-vector space. Then $\mc M \simeq {\mc L}\gen{\mc M_1}$ is a free Lie algebra.
\begin{proof}
It suffices to show that the canonical $\mathbb{F}$-Lie algebra homomorphism $\phi:{\mc L}\gen{\mc M_1}\to \mc M $ is injective.
Let $\mc X$ be a basis for $V$ such that $\mc X\cap\mc M_1$ is a basis for $\mc M_1$.
Let $\mc B $ be the Hall-basis made with respect to an ordering on $\mc X$, and let $\mc B ^{\mc M}$ be the subset of $\mc B $ made up from the basis $\mc M_1\cap \mc X$ with the induced ordering.
Then $\text{Span}_\mathbb{F}(\mc B ^{\mc M})=\text{im}(\phi)$, and as $\mc B ^{\mc M}\subseteq \mc B $, one concludes that $\mc B ^{\mc M}$ is a linearly independent set.
Thus $\phi$ must be injective.
\end{proof}
\end{thm}
We also prove the following characterization of the free Lie algebras, that the analogue of Stallings-Swan theorem for groups of cohomological dimension $1$.
\begin{thm}{\label{freecoh}}
Let ${\mc L}$ be a $1$-generated $\mathbb N$-graded $\mathbb{F}$-Lie algebra. Then $H^2({\mc L} ,\mathbb{F})=0$ if, and only if, ${\mc L} \simeq {\mc L}\gen{{\mc L}_1}$ is a free $\mathbb{F}$-Lie algebra.
\begin{proof}
If ${\mc L} $ is free, then $H^2({\mc L} ,\mathbb{F})=0$, since $0\to \mc U ({\mc L} )\otimes {\mc L}_1\to \mc U ({\mc L} )\to \mathbb{F}\to 0$ is a projective resolution of the trivial $\mc U ({\mc L} )$-module $\mathbb{F}$. \\
Suppose $H^2({\mc L} ,\mathbb{F})=0$. It suffices to show that the canonical map $\phi :{\mc L} \gen{{\mc L}_1}\to {\mc L} $ is injective. Let $I =\ker(\phi )$, and put $\mc F ={\mc L} \gen{{\mc L}_1}$. Then the inflation-restriction sequence (namely, the $5$-term sequence induced by the Hochschild-Serre spectral sequence) associated with the exact sequence of $\mathbb N$-graded $\mathbb{F}$-Lie algebras $0\to I \to\mc F \to {\mc L} \to 0$ yields
\begin{equation}
\label{5term}
\xymatrix{
0\ar[r]&H^1({\mc L} ,\mathbb{F})\ar[r]^{H^1(\phi )}&H^1(\mc F ,\mathbb{F})\ar[r]^\alpha&
H^1(I ,\mathbb{F})^{{\mc L} }\ar[d]\\
&&H^2(\mc F ,\mathbb{F})&H^2({\mc L} ,\mathbb{F})\ar[l]\\}
\end{equation}
where the action of ${\mc L} $ on the $1$-cocycles $\alpha:I \to \mathbb{F}$ is given by $x\cdot\alpha(i)=\alpha([\tilde x,i])$, $\tilde x$ being any lift of $x\in {\mc L} $ to $\mc F $.
By construction, $H^1(\phi )$ is an isomorphism, and $H^2({\mc L} ,\mathbb{F})=0$, by hypothesis.
This implies that $H^1(I ,\mathbb{F})^{{\mc L} }=0$.
As $I $ is an $\mathbb N$-graded ideal of $\mc F $, one has $H^1(I ,\mathbb{F})=\operatorname{Hom}_{\mbox{\small{Lie}}}(I ,\mathbb{F})=\operatorname{Hom}_{\mathbb{F}}(I /[I ,I ],\mathbb{F})$.
Thus, $H^1(I ,\mathbb{F})^{{\mc L} }=0$ implies $I /[\mc F ,I ]=0$, and therefore $I =[\mc F ,I ]$.
Now, suppose $n=\min\set{m\geq 0}{I_m\neq 0}$ is finite. One has $I_n=[I ,\mc F ]_n=\sum_{1\leq j<n}[I_j,\mc F_{n-j}]=0$, since $I $ is a graded ideal, whence $I =0$, and $\phi $ is injective.
\end{proof}
\end{thm}
Notice that the cohomological dimension of subagebras does not exceed that of the Lie algebra. In particular, one may see Theorem \ref{NS} as a direct consequence of Theorem \ref{freecoh}. However, we decided to give the above proof that is purely combinatorial and it does not involve any cohomological arguement.
In order to prove the Kurosh Theorem we first need to understand the algebra generated by some subalgebras of the factors $\mc A$ and $\mc B$ in the free product $\mc A\amalg\mc B$. The following
\begin{prop}{\label{subfreeproduct}}
Let $\mc A$ and $\mc B$ be $\mathbb N$-graded $1$-generated $\mathbb{F}$-Lie algebras, and let $\mc H\subseteq \mc A\amalg\mc B$ be an $\mathbb N$-graded $1$-generated subalgebra of their free product, such that $\mc H=\gen{\mc H_1\cap\mc A,\mc H_1\cap \mc B}$. Then \[\mc H=\gen{\mc H_1\cap\mc A}\amalg\gen{\mc H_1\cap \mc B}.\]
\begin{proof}
Let $\mf X$ be a graded basis for $\mc H\cap\mc A$, and $\mf Y$ be a graded basis for $\mc H\cap\mc B$. Extend $\mf X$ and $\mf Y$ to graded bases $\mf X'$ for $\mc A$ and $\mf Y'$ for $\mc B$, respectively. Now, for every $x_1,x_2\in \mf X'$ there exist scalars $c_{x_1,x_2,z}\in \mathbb{F}$ such that \[[x_1,x_2]=\sum_{z\in \mf X'}c_{x_1,x_2,z}z.\]
Similarly, for every $y_1,y_2\in\mf Y'$, one can write \[[y_1,y_2]=\sum_{w\in \mf Y'}d_{y_1,y_2,w}w.\]
Define
\begin{equation}
J=\left([x_1,x_2]-\sum_{z\in \mf X}c_{x_1,x_2,z}z,\ [y_1,y_2]-\sum_{w\in \mf Y}d_{y_1,y_2,w}w\ \vert\ x_i\in \mf X,\ y_j\in \mf Y\right)\end{equation}
as an ideal of $ {\mc L}(\mf X\sqcup\mf Y)$, and
\begin{equation}
J'=\left([x_1,x_2]-\sum_{z\in \mf X'}c_{x_1,x_2,z}z,\ [y_1,y_2]=\sum_{w\in \mf Y'}d_{y_1,y_2,w}w\vert\ x_i\in \mf X',\ y_j\in \mf Y'\right)\end{equation} as an ideal of ${\mc L}(\mf X'\sqcup\mc Y')$.
It is clear that $J={\mc L}(\mf X\sqcup\mc Y)\cap J'$, where one considers the free Lie algebra as a subset of the free associative algebra.
Now, the canonical epimorphism ${\mc L}(\mf X'\sqcup\mf Y')\to \mc A\amalg\mc B$ has kernel $J'$, and thus its restriction to ${\mc L}(\mf X\sqcup\mf Y)$ has kernel $J$, whence we get an isomorphism $\mc H\overset{\sim}{\to} \gen{\mc H_1\cap\mc A}\amalg\gen{\mc H_1\cap \mc B}.$
\end{proof}
\end{prop}
The latter two results are the main ingredients for the Kurosh subalgebra theorem.
Let $V$ be a $\mathbb{F}$-subspace of the direct sum $A\oplus B$ of two finite dimensional $\mathbb{F}$-vector spaces $A$ and $B$.
Then one can find a distinguished basis $\mathcal B=\mathcal B_A\cup\mathcal B_B\cup\tilde{\mathcal B}$ for $V$, such that $\mathcal B_A$ is a basis for $V\cap A$, $\mathcal B_B$ is a basis for $V\cap B$, and $\tilde{\mathcal B}$ satisfies the following property:
The projections $\pi_A(x)$, $x\in \tilde{\mathcal B}$, and the set $\mathcal B_A$ form a linearly independent set in $A$ (and similarly for $B$).
Now, since $\pi_A(\tilde{\mathcal B})\cup \mathcal B_A$ is a linearly independent set in $A$, one can thus complete this set to a basis $\mathcal B'_A$ for the whole $A$.
Indeed, let the set $\tilde{\mathcal B}=\{c_i=a_i'+b_i'\}$ ($a_i'\in A\setminus\{0\},b_i'\in B\setminus\{0\}$) complete the set $\mathcal B_A\cup \mathcal B_B$ to a basis for $V$. Suppose that there are scalars $\alpha_i,\beta_j\in\mathbb{F}$, such that $\sum_i\alpha_ia_i'+\sum_j\beta_j a_j=0$. Hence the element $\sum_i\alpha_i c_i+\sum_j\beta_j a_j=\sum_i\alpha_i b_i$ lies in $V\cap B$. Since $\mbox{Span}(\mathcal B_B,\mathcal B_A)\cap \mbox{Span}(\tilde{\mathcal B})=0$, one has $\alpha_i=0$, and thus $\beta_j=0$. In case $V\cap A=0$, one can set $\mathcal B_A=\emptyset$.\\
The above discussion also holds for $B$.
Let us now specialize the previous consideration for the $1$-generated subalgebras of the free product $\mc A\amalg\mc B$ of two $1$-generated $\mathbb N$-graded $\mathbb{F}$-Lie algebras $\mc A$ and $\mc B$.
Let $W\leq \mc A_1\oplus\mc B_1$ be a subspace such that $W\cap\mc A_1=W\cap\mc B_1=0$. Let $(c_i)$ be a basis for $W$. One can write $c_i=a_i+b_i$ in a unique way, such that $a_i\in\mc A_1$ and $b_i\in \mc B_1$. Moreover, $(a_i)$ is a linearly independent set in $\mc A_1$, and $(b_i)$ is a linearly independent set in $\mc B_1$. Consider the envelopes $A=\mc U(\mc A)$ and $ B=\mc U(\mc B)$. Therefore, \[\mc U(\mc A\amalg\mc B)=A\amalg B=\mathbb{F}\oplus\overbrace{ A_1\oplus B_1}^{\text{degree}\ 1}\oplus \overbrace{A_2\oplus (A_1\otimes B_1)\oplus (B_1\otimes A_1)\oplus B_2}^{\text{degree}\ 2}\oplus \dots\]
If an arbitrary element $\sum_{i,j}\alpha_{ij}c_ic_j$ of $\gen{W}_2$ lies in $A$, then $\sum_{i,j}\alpha_{i,j}(a_ib_j+b_ia_j)\in A_2$. But the latter belongs to a complementary space of $A_2$ in $(A\amalg B)_2$, namely $A_1\otimes B_1\oplus B_1\otimes A_1$, and thus $\alpha_{ij}=0$.
The same holds for greater degrees.
Let $x=\sum_{\vert I\vert=n}\alpha_Ic_I\in \gen{W}_{\mathbb{F}-\text{alg}}\cap A_n$, where $c_I=c_{i_1}\cdots c_{i_n}$, for $I=(i_1,\dots,i_n)$, and $\alpha_I\in\mathbb{F}$.
Note that, if $n$ is even, then the elements $a_{i_1}b_{i_2}\dots a_{i_{n-1}}b_{i_n}$ $(I=(i_1,\dots,i_n))$ are linearly independent in \[\overbrace{A_1\otimes B_1\otimes \dots \otimes A_1\otimes B_1}^{n\text{ terms}}\] and thus $\alpha_I=0$, for all $I$. For $n$ odd, the same holds.
We have thus proven
\begin{prop}{\label{nullinters}}
Let $\mc A$ and $\mc B$ be $1$-generated $\mathbb N$- graded $\mathbb{F}$-Lie algebras, and let $\mc H\leq \mc A\amalg \mc B$ be a $1$-generated Lie subalgebra.
Then the following are equivalent:
\begin{enumerate}
\item $\mc H_1\cap\mc A_1=0$, and
\item $\mc H\cap \mc A=0$.
\end{enumerate}
\end{prop}
\begin{lem}{\label{freeres}}
Let ${\mc L}$ be an $\mathbb N$-graded $\mathbb{F}$-Lie algebra, and let $\mc A,\mc H\leq {\mc L}$ be $\mathbb N$-graded Lie subalgebras of ${\mc L}$ satisfying $\mc A\cap \mc H=\{0\}$.
Then $\mc U({\mc L})$ is a free $(\mc U(\mc H),\mc U(\mc A))$-bimodule.
In particular, $\res^{{\mc L}}_{\mc H}\ind^{{\mc L}}_{\mc A}(\mathbb{F})$ is a free left $\mc U(\mc H)$-module.
\begin{proof}
Set $H=\mc U(\mc H)$ and $A=\mc U(\mc A)$.\\
The first assertion follows by the PBW theorem.
Indeed, if $\{a_i\}$ and $\{h_j\}$ are basis for $\mc A$ and $\mc H$, respectively, then there is a basis $\{a_i,h_j,l_k\}$ for $\mc L$.
Therefore, $\mc U({\mc L})$ has basis $\{\prod_j h_i\prod_k l_k\prod_i a_i\}$ where $i,j,k$ range over ordered sets of indices.
This means that \[\mc U({\mc L})=\bigoplus_{K'} (H\otimes A^{op})\prod_{k\in K'} l_k,\] where $K'$ ranges over some finite sets of indices $k$, is a free $(H\otimes A^{op})$-module, i.e. a free $(H,A)$-bimodule.
\end{proof}
\end{lem}
\begin{thm}{\label{subfree}}
Let $\mc A$ and $\mc B$ be $\mathbb N$-graded $1$-generated $\mathbb{F}$-Lie algebras, and let $W\subseteq \mc A_1\oplus\mc B_1$ be a subspace satisfying ${W}\cap \mc A={W}\cap \mc B=0$. Then the $1$-generated Lie subalgebra $\mc F=\gen{W}\subseteq \mc A\amalg\mc B$ is a free $\mathbb{F}$-Lie algebra.
\begin{proof}
Put ${\mc L}=\mc A\amalg\mc B$.
By Theorem \ref{thm:mayervietoris}, one has an exact sequence \begin{equation*}
0\to \mc U({\mc L})\to \ind_{\mc A}^{{\mc L}}(\mathbb{F})\oplus\ind_{\mc B}^{{\mc L}}(\mathbb{F})\to \mathbb{F}\to 0
\end{equation*} of $\mathbb N_0$-graded left $\mc U({\mc L})$-modules.
Now, by Proposition \ref{nullinters}, $\mc F\cap \mc A=\mc F\cap\mc B=0$, and thus, by Lemma \ref{freeres}, $\res_{\mc F}^{{\mc L}}\ind_{\mc A}^{{\mc L}}(\mathbb{F})$ and $\res_{\mc F}^{{\mc L}}\ind_{\mc B}^{{\mc L}}(\mathbb{F})$ are free $\mc U(\mc F)$-modules.
Therefore,
\begin{equation*}
0\to \res_{\mc F}^{{\mc L}}\mc U({\mc L})\to \res_{\mc F}^{{\mc L}}\ind_{\mc A}^{{\mc L}}(\mathbb{F})\oplus\res_{\mc F}^{{\mc L}}\ind_{\mc B}^{{\mc L}}(\mathbb{F})\to \res_{\mc F}^{{\mc L}}\mathbb{F}\to 0
\end{equation*}
is a free resolution of $\mathbb{F}$ over $\mc U(\mc F)$, and hence $\text{cd}(\mc F)\leq 1$. This proves that $\mc F$ is free, by Theorem \ref{freecoh}.
\end{proof}
\end{thm}
\begin{lem}\label{indep sum}
Let $\mc A$ and $\mc B$ be two quadratic Lie algebras. Let $a_i\in \mc A_1$ and let $\{b_i\}\subset \mc B_1$ be an independent set. If $\sum_i a_ib_i=0$ in the free product $\mc U(\mc A\amalg \mc B)$, then $a_i=0$, $\forall i$.
\begin{proof}
This follows from the fact that there is no non-trivial defining relation for $\mc U(\mc A\amalg\mc B)$ involving elements in $\mc A_1\mc B_1$.
Indeed, since both the Lie algebras are quadratic, there are relations $r_{\mc A}\in \mc A_1^{\otimes 2},\ r_{\mc B}\in \mc B_1^{\otimes 2}$ such that $\sum_i a_ib_i=r_{\mc A}+r_{\mc B}$.
\end{proof}
\end{lem} We are finally ready to prove the main result of this paper.
\begin{thm}[Kurosh' subalgebra theorem]
Let $\mc A$ and $\mc B$ be two Bloch-Kato $\mathbb{F}$-Lie algebras, and let $\mc H \subseteq \mc A\amalg \mc B$ be a $1$-generated subalgebra. Then \begin{equation}
\mc H\simeq \gen{\mc H_1\cap \mc A}\amalg \gen{\mc H_1\cap \mc B}\amalg \mc F
\end{equation} where $\mc F$ is a free Lie algebra generated by any distinguished subspace $W\subseteq \mc A_1\oplus\mc B_1$ such that $\mc H_1=W\oplus (\mc H_1\cap \mc A)\oplus(\mc H_1\cap \mc B)$.
\begin{proof}
Notice first that $\mc H$ is a Koszul Lie algebra, for ${\mc L}$ is Bloch-Kato by Theorem \ref{et=>bk}.
Decompose $\mc H_1$ into the direct sum $(\mc H_1\cap \mc A)\oplus(\mc H_1\cap \mc B)\oplus W$.
Then, $\mc F=\gen{W}$ is a free Lie subalgebra of $\mc H$ by Theorem \ref{subfree}, and $\mc H=\gen{W,\mc H_1\cap \mc A,\mc H_1\cap \mc B}$.
By Proposition \ref{subfreeproduct}, one has \[\mc H=\gen{\mc F,\gen{\mc H_1\cap \mc A}\amalg\gen{\mc H_1\cap \mc B}},\] where $\mc F$ is the free Lie algebra on $W$.
Let $\mc P=\gen{\mc H_1\cap \mc A}\amalg\gen{\mc H_1\cap \mc B}$. Thus, in light of Theorem \ref{mayervietoris}, it suffices to prove that $\mc U(\mc H)$ is the kernel of the canonical mapping \[\varepsilon=\varepsilon^{\mc F}-\varepsilon^{\mc P}:\ind_{\mc F}^{\mc H}\mathbb{F}\oplus\ind_{\mc P}^{\mc H}\mathbb{F}\to \mathbb{F}.\]
We first prove that the image of the natural map $\mc U(\mc H)\to\ind_{\mc F}^{\mc H}\mathbb{F}\oplus\ind_{\mc P}^{\mc H}\mathbb{F}$ is the kernel of $\varepsilon$.
Let $(x\otimes 1,y\otimes 1)\in \ind_{\mc F}^{\mc H}\mathbb{F}\oplus\ind_{\mc P}^{\mc H}\mathbb{F}$ be a homogeneous element in the kernel of $\varepsilon$.
This means that $x-y$ has positive degree in $\mc U(\mc H)$.
Since $\mc H$ is generated by $\mc P$ and $\mc F$, there are elements $p,f\in\mc U(\mc H)$ such that $x-y=p-f$, and $p$ and $f$ can be written as combinations of products ending with elements in $\mc P_1$, and $\mc F_1$ respectively.
Thus, \[x+f\in\mc U(\mc H)\longmapsto ((x+f)\otimes 1,(y+p)\otimes 1)=(x\otimes 1,y\otimes 1).\]
Now, it remains to prove that the map $\mc U(\mc H)\to \ind_{\mc F}^{\mc H}\mathbb{F}\oplus\ind_{\mc P}^{\mc H}\mathbb{F}$ is injective.
Let $x\in \mc U(\mc H)_2$ such that $f(x)=(x\otimes 1,x\otimes 1)=0$.
It follows that $x\in \mc U(\mc H)\mc P_1\cap\mc U(\mc H)\mc F_1$, namely there are elements $ x_i,y_j,z_k\in\mc H_1$ such that \[x=\sum x_ia_i+\sum y_jb_j=\sum z_k (a'_k+b'_k)\] where $(a_i,b_j)$ and $(a'_k+b'_k)$ are $\mathbb{F}$-basis respectively of $\mc P_1$ and $\mc F_1=W$.
Moreover, the set $(a_i,b_j,a'_k,b'_k)$ is linearly independent.
Let $(\hbox to 1.5ex{\hrulefill})^{\mc A}:\mc H_1\to\mc A_1$ and $(\hbox to 1.5ex{\hrulefill})^{\mc B}:\mc H_1\to\mc B_1$ be the projections ($\mc H_1\subseteq \mc A_1\oplus\mc B_1$).
Since ${\mc U(\mc L)}_2=\mc U(\mc A)_2\oplus \mc A_1\mc B_1\oplus\mc B_1\mc A_1\oplus\mc U(\mc B)_2$, it follows that \begin{align}
\sum x_i^{\mc A}a_i=\sum z_k^{\mc A}a'_k\\
\label{second} \sum x_i^{\mc B}a_i=\sum z_k^{\mc B}a'_k\\
\label{third} \sum y_j^{\mc A}b_j=\sum z_k^{\mc A}b'_k\\
\sum y_j^{\mc B}b_j=\sum z_k^{\mc B}b'_k
\end{align}
Consider the equations (\ref{second}), (\ref{third}).
From (\ref{second}), we recover a relation for ${\mc U(\mc L)}$ involving elements in $\mc B_1\mc A_1$. Since the set $\{a_i,a'_k\}$ is linearly independent, the relation is not trivially satisfied but when $x_i^{\mc B}=z_k^{\mc B}=0$.
For the equation (\ref{third}), the same holds, proving that $z_k=0$. Therefore, $x=0$.
This shows that the map $\mc U(\mc H)\to\ind_{\mc F}^{\mc H}\mathbb{F}\oplus\ind_{\mc P}^{\mc H}\mathbb{F}$ is injective in degree $2$.
By construction, it is also injective in degree $< 2$.
By Corollary \ref{inj koszul}, the map is injective in all the degrees.
\end{proof}
\end{thm} \begin{rem}
If $\mc U(\mc H)\to \ind_{\mc F}^{\mc H}\mathbb{F}\oplus\ind_{\mc P}^{\mc H}\mathbb{F}$ is injective, since both the modules are Koszul, also the cokernel is Koszul, and generated in degree $0$. But the cokernel is $\mathbb{F}$ in degree $0$ and it is $0$ in degree $1$, proving that it must be the trivial module.
This argument should give another proof for the first part of the proof.
Notice that the latter proof may be adapted to give a less combinatorial proof for Proposition \ref{subfreeproduct}. \end{rem}
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\end{document}
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\begin{document}
\title{\textbf{A finite theorem for Ahlfors' covering surface theory}} \author{Tian~Run Lin \& Yun~Ling Chen \& Guang~Yuan Zhang} \address{Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China.} \address{\slshape Email: [email protected]} \address{\slshape Email: [email protected]} \address{\slshape Email: [email protected]} \thanks{Project 10971112 and 12171264 supported by NSFC}
\begin{abstract} Ahlfors' theory of covering surfaces is one of the major mathematical achievement of last century. The most important part of his theory is the Second Fundamental Theorem (SFT). We are interested in the relation of errors of Ahlfors' SFT with the same boundary curve.
In this paper we will prove a result which is used to establish the best bound of the constant in Ahlfors' SFT (in \cite{Zh}).
Precisely speaking, we will prove that for any surface $\Sigma\in\mathcal{F} _r(L,m)$, a new surface $\Sigma_1$ can be constructed based on it, such that $R(\Sigma_1)\ge R(\Sigma)$ and $L(\partial\Sigma_1)\le L(\partial\Sigma)$, where $R(\Sigma)$ is Ahlfors' error term and $L(\partial\Sigma)$ is the boundary length of the surface $\Sigma$, and the covering degree of $ \Sigma_1 $ has an upper bound independent of surfaces. Meanwhile, this conclusion suggests that the supremum of $H(\Sigma)=R(\Sigma)/L(\partial \Sigma)$ can be achieved by surfaces in the space $\mathcal{F}_r^{\prime }(L,m)$. \end{abstract}
\subjclass[2020]{30D35, 30D45, 52B60} \maketitle \tableofcontents
\section{Introduction}
\label{sec:intro}
In the study of the value distribution theory, Nevanlinna's theory plays a major role as we know (\cite{MR0164038,MR3751331,MR1555200,MR1301781}). And Ahlfors' theory of covering surfaces, which is parallel to the theory of Nevanlinna, can also give us useful methods to discuss meromorphic functions (\cite{MR1555403,MR994468,MR0012669,MR1786560,MR0164038}). Ahlfors' theory builds upon geometrical views and gives the Second Fundamental Theorem parallel to, but different from Nevanlinna's theory, whose form is similar to a type of isoperimetric inequality. For this reason Ahlfors' theory leads to the interesting geometrical constant called Ahlfors constant, which have been discussed in \cite{MR4125732,MR3004780}.
We need recall some basic definitions and notations. The extended complex plane $\overline{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ can be naturally identified with the unit sphere $S$ in $\mathbb{R}^3$ via the stereographic projection as introduced in \cite{MR510197}. Following the Euclidean metric on $\mathbb{R}^3$, the sphere $S$ has the standard metric and the corresponding K\"{a}hler form. We can represent them with coordinate in $ \mathbb{C}$: \begin{align*} ds^2&=\frac{4}{(1+\lvert z\rvert^2)^2}\lvert dz\rvert^2, & \omega&=\frac{2 \sqrt{-1}}{\pi}\frac{dz\wedge d\bar{z}}{(1+\lvert z\rvert^2)^2}, & &z\in \mathbb{C}. \end{align*} Then the spherical length and the spherical area can be defined. We will use $L$ and $A$ to denote them respectively.
Let $U$ be $\overline{\mathbb{C}}$, or be a domain in $\overline{\mathbb{C}}$ enclosed by a finite number of Jordan curves, and let $f$ be a meromorphic function defined on $\overline{U}$. Then $f$ can be regard as a holomorphic mapping from $\overline{U}\subset \overline{\mathbb{C}}$ into $S$, via stereographic projection. The pair $\Sigma =(f,\overline{U})$ can be regarded as a holomorphic covering surface spread over the sphere $S$, and the boundary $\partial \Sigma $ of $\Sigma $ is defined to be the pair $ \partial \Sigma =(f,\partial U),$ where $\partial U$ is the boundary of $U$. The length $L(\partial \Sigma )=L(f,\partial U)$ and area $A(\Sigma )=A(f,U)$ can be given by \begin{align*} L(f,\partial U)& =\int_{\partial U}f^{\ast }ds=\int_{\partial U}\frac{ 2\lvert f^{\prime }(z)\rvert }{1+\lvert f(z)\rvert ^{2}}\lvert dz\rvert , \\ A(f,U)& =\int_{U}f^{\ast }\omega =\frac{2\sqrt{-1}}{\pi }\int_{U}\frac{
\lvert f^{\prime }(z)|^{2}}{(1+\lvert f(z)\rvert ^{2})^{2}}dz\wedge d\bar{z}, \end{align*} where $f^{\ast }$ means the pull-back by $f$. Note that the length $ L(f,\partial U)$ may be infinite.
Compared to holomorphic covering surfaces, we define covering surfaces as in Ahlfors' paper \cite{MR1555403}: a covering surface over $S$ is built up by a finite number of one sheeted closed topological triangular domains. Equivalently to say, it can be defined as
\begin{definition} \label{def:surface} A \emph{covering surface} over $S$ is a pair $(f, \overline{U})$, where $U$ is $\overline{\mathbb{C}}$ or a domain in $ \overline{\mathbb{C}}$ enclosed by a finite number of Jordan curves and $f: \overline{U}\to S$ is an orientation-preserving, continuous, open and finite-to-one mapping (OPCOFOM), which in fact means that $f$ can be extended to be OPCOFO on a neighborhood of $\overline{U}$. Here orientation-preserving means that when $S$ is identified with $\overline{ \mathbb{C}}$ via stereograpchic projection, $f$ is orientation-preserving on $\mathbb{C}\cap(\overline{U}\setminus f^{-1}(\infty))$.
We denote by $\mathbf{F}$ the set of all covering surfaces such that for each $(f,\overline{U})\in\mathbf{F}$, $U$ is a Jordan domain. Then we denote by $\mathbf{F}(L)$ the subspace of $\mathbf{F}$ such that for each $\Sigma\in \mathbf{F}(L)$, $L(\partial\Sigma)\le L$. \end{definition}
By Stoilow's theorem, a covering surface can be regarded as a holomorphic covering surface, up to an orientation-preserving homeomorphic (OPH) transform of the domain of definition. That is,
\begin{theorem}[Sto\"{\i}low, see \protect\cite{MR0082545} pp.120--121] \label{thr:stoilow} Let $U$ be a domain on $\overline{\mathbb{C}}$ and $ f:U\to S$ be an OPCOFOM. Then there exists a domain $U_1$ in $\overline{ \mathbb{C}}$ and an OPH $\varphi:U_1\to U$, such that $f_1=f\circ\varphi:U_1 \to S$ is a non-constant holomorphic mapping on $U_1$. \end{theorem}
By the compactness of $\overline{\mathbb{C}}$, if $U=\overline{\mathbb{C}}$ we must have $U_1=\overline{\mathbb{C}}$ as well.
Now we can state Ahlfors' famous theorem published in \cite{MR1555403} in 1935, parallel to Nevanlinna's Second Fundamental Theorem.
\begin{theorem}[Ahlfors' Second Fundamental Theorem] Let $q\ge3$, and $E_q=\{a_1,a_2,\dots,a_q\}$ be a set consisting of distinct $q$ points on $S$. Then there exists a constant $h>0$ that depends only on $ E_q$, such that \begin{equation*} (q-2)A(\Sigma)\le 4\pi\sum_{j=1}^{q}\overline{n}(\Sigma,a_j)+hL(\partial \Sigma), \end{equation*} for any $\Sigma=(f,\overline{U})\in\mathbf{F}$, where $\overline{n} (\Sigma,a_j)$ is the cardinality of $f^{-1}(a_j)\cap U$ (ignoring multiplicity). \end{theorem}
We will assume $E_q=\{a_1,a_2,\dots,a_q\}$ be the set of arbitrarily given distinct $q$ points, $q\ge3$, and for any surface $\Sigma=(f,\overline{U})$, write \begin{equation*} \overline{n}(\Sigma,E_q)=\sum_{j=1}^q\overline{n}(\Sigma,a_j). \end{equation*} We call $E_q$ the \emph{special set} and the points in $E_q$ the \emph{ special points}.
The inequality in Ahlfors' Second Fundamental Theorem, which makes a relation between geometric quantity and the counting function $\overline{n} (\Sigma,a_j)$, works effectively in research of the value distribution theory of meromorphic functions obviously. For example \cite{MR0164038}, if $ q=3$ and $f$ is a meromorphic function on $\mathbb{C}$ that omits $a_1$, $ a_2 $, $a_3$, then this inequality tells us $A(R)\le hL(R)$, where $ A(R)=A(f,\Delta_{R})$, $L(R)=L(f,\partial\Delta_{R})$ and $\Delta_{R}=\{z\in \mathbb{C}:\lvert z\rvert<R\}$. On the other hand, by the expression of $L$ and $A$ we can easily deduce that $L^2(R)\le2\pi^2RA^{\prime }(R)$. Then $ A^2(R)\le2\pi^2h^2RA^{\prime }(R)$, which is a type of Gronwall inequality and finally gives $A(R)=0$. It proves Picard's theorem, which asserts that any meromorphic function on $\mathbb{C}$ that omits three distinct values must be constant.
As we see, the existence of constant $h$ in Ahlfors' Second Fundamental Theorem has played a big role in the value distribution theory. However, the minimal possible value of this constant has not been sufficiently studied up to now. This minimal value is called \emph{Ahlfors constant} in \cite {MR4125732}.
This problem can be traced back to the early 1940s, when J. Dufresnoy first gave a numerical estimate of $h$ in \cite{MR0012669} as follows.
\begin{theorem}[Dufresnoy \protect\cite{MR0012669}] For any $\Sigma=(f,\overline{U})\in\mathbf{F}$, \begin{equation*} (q-2)A(\Sigma)\le4\pi\overline{n}(\Sigma,E_q)+(q-2)\frac{6\pi}{\delta_{E_q}} L(\partial\Sigma), \end{equation*} where $\delta_{E_q}=\min_{1\le i<j\le q}d(a_i,a_j)$. \end{theorem}
In 2009, the precise bound for $h$ have been identified in a special case as follows.
\begin{theorem}[Zhang \protect\cite{MR3004780}] For any $\Sigma\in\mathbf{F}$ with $\overline{n}(\Sigma,\{0,1,\infty\})=0$, \begin{equation*} A(\Sigma)\le h_0L(\partial\Sigma), \end{equation*} where \begin{equation*} h_0=\max_{\theta\in\lbrack0,\pi/2]}\left(\frac{(\pi+\theta)\sqrt{ 1+\sin^2\theta}}{\arctan\frac{\sqrt{1+\sin^2\theta}}{\cos\theta}} -\sin\theta\right), \end{equation*} and there exists a sequence $\{\Sigma_{n}\}=\{(f_n,\overline{\Delta})\}$ in $ \mathbf{F}$ with $\overline{n}(\Sigma_n,\{0,1,\infty\})=0$ such that $ A(\Sigma_{n})/L(\partial\Sigma_{n})\to h_0$ as $n\to\infty$. \end{theorem}
\begin{definition} We will write \begin{equation*} R(\Sigma,E_q)=(q-2)A(\Sigma)-4\pi\overline{n}(\Sigma,E_q) \end{equation*} and \begin{equation*} H(\Sigma,E_q)=\frac{R(\Sigma,E_q)}{L(\partial\Sigma)}. \end{equation*}
When $E_q$ is fixed, we can write $\overline{n}(\Sigma)=\overline{n} (\Sigma,E_q)$, $R(\Sigma)=R(\Sigma,E_q)$, and $H(\Sigma)=H(\Sigma,E_q)$ for convenience. \end{definition}
Then, computing the minimal value $h$ in Ahlfors' Second Fundamental Theorem is equivalent to find the supremum \begin{equation*} H_0=\sup\{H(\Sigma):\Sigma\in\mathbf{F}\}. \end{equation*}
It is proved in \cite{Zh1} that the supremum $H_0$ of $H(\Sigma)$ cannot be realized by any surface of $\mathbf{F}$. But as pointed in \cite{MR3004780}, to study $H_0$, one has to study the supremum \begin{equation*} H_L=\sup\{R(\Sigma)/L(\partial\Sigma):\Sigma\in\mathbf{F}(L)\} \end{equation*} for given $L>0,$ and has to find an extremal surface $\Sigma_L$ in $\mathbf{F }(L)$ which assume $H_L$: $H(\Sigma_L)=H_L$.
The inequality in Ahlfors' Second Fundamental Theorem can be regarded as a type of isoperimetric inequality. Let us recall that how could we prove the most basic isoperimetric inequality on Euclidean plane. Any closed curve can be approximated by polygons, and for fixed $m$, the space composed of all $m$ -polygons possesses certain compactness so we can easily show that regular polygons have the largest area when the perimeter and edge number are fixed. Then all the work was finished.
To study the existence of $\Sigma_L$, it is natural to consider the ``$m$ -polygons'' on the Riemann sphere $S$, whose boundary can be divided into $m$ simple circular arcs, as in \cite{Zh}. On the other hand, any Jordan domain $ U$ is conformally equivalent to $\Delta=\{z\in\mathbb{C}:\lvert z\rvert<1\}$ , the unit disk centered at the origin. This leads to a corresponding between surfaces defined on $\overline{U}$ and $\overline{\Delta}$. Of course it will be convenient if we only discuss surfaces defined on the closed unit disk $\overline{\Delta}$. Now we give the definition of ``$m$ -polygon'' as in \cite{Zh}:
\begin{definition} \label{def:FLM} We denote by $\mathcal{F}(L,m)$ the subset of $\mathbf{F}(L)$ , which consists of all surfaces $\Sigma=(f,\overline{\Delta})\in\mathbf{F} (L)$ satisfying the following conditions:
(1) $\partial \Delta $ and $\partial \Sigma $ have partitions \begin{equation} \partial \Delta =\alpha _{1}(\mathfrak{p}_{1},\mathfrak{p}_{2})+\alpha _{2}( \mathfrak{p}_{2},\mathfrak{p}_{3})+\dots +\alpha _{m}(\mathfrak{p}_{m}, \mathfrak{p}_{1}) \label{def:FLM-1} \end{equation} and \begin{equation} \partial \Sigma =c_{1}(p_{1},p_{2})+c_{2}(p_{2},p_{3})+\dots +c_{m}(p_{m},p_{1}), \label{def:FLM-2} \end{equation} such that for each $\alpha _{j}$, $c_{j}=(f,\alpha _{j})$ is a convex simple circular arc;
(2) for each $\alpha _{j}$, there exists a neighborhood $U_{j}$ of $\alpha _{j}^{\circ }=\alpha _{j}\setminus \{\mathfrak{p}_{j-1},\mathfrak{p}_{j}\}$ (means the interior of $\alpha _{j}$) in $\overline{\Delta }$, such that $f$ restricted to $U_{j}$ is an OPH. \end{definition}
We call (\ref{def:FLM-1}) and (\ref{def:FLM-2}) $\mathcal{F}(L,m)$ -partitions of $\partial\Sigma$.
\begin{remark} \label{rm:FLMconvex} In this definition, we say that the circular arcs $c_j$ is convex, which means that for any $x\in\alpha_j^{\circ}$, there exists a neighborhood $U$ in $\overline{\Delta}$, and a closed hemisphere $\overline{ S^{\prime }}$ on $S$, such that $f(x)\in\partial\overline{S^{\prime }}$, and $f(U)\subset\overline{S^{\prime }}$. More detailed discussion of convex curves will be mentioned in Definition \ref{def:oriconv}.
Note that $\mathcal{F}(L,m)$-partitions are not only defined for the boundary $\partial\Sigma$, but also involve the information of $f$ near the boundary $\partial\Delta$. \end{remark}
The space $\mathcal{F}(L,m)$ consists of some relatively simple surfaces, but actually this space is still too large to study. Many surfaces in $ \mathcal{F}(L,m)$ have bad performance and it is difficult to talk about the convergence. Our target in this paper is to reduce the space $\mathcal{F} (L,m)$ further, that means we need find the subspace such that $H(\Sigma)$ has the same supremum as $\mathcal{F}(L,m)$.
It is easy to see \begin{equation*} H_L=\sup\{H(\Sigma):\Sigma\in\mathbf{F}(L)\}=\sup\{H(\Sigma:\Sigma\in \mathcal{F}(L,m))\} \end{equation*} as in \cite{Zh}. Then to find extremal surfaces of $\mathbf{F}(L)$ is to find extremal surfaces of $\mathcal{F}(L,m)$. Surfaces in $\mathcal{F}(L,m)$ may not be one sheeted over $S$ and the defining function may have branch values. Then we define for each $\Sigma=(f,\overline{U})\in\mathbf{F}(L)$: \begin{align*} \deg_{\min}(\Sigma)&=\min\{\#f^{-1}(y):y\in S\setminus\partial\Sigma\text{ and }y\text{ is not a branch value}\}, \\ \deg_{\max}(\Sigma)&=\max\{\#f^{-1}(y):y\in S\setminus\partial\Sigma\text{ and }y\text{ is not a branch value}\}. \end{align*} For convenience, they will be also denoted by $\deg_{\min}(f)$ and $ \deg_{\max}(f)$.
For a surface $(f,\overline{\Delta})\in\mathcal{F}(L,m)$, a point $ p\in\Delta $ is a branch point if $f$ is not injective in any neighborhood of $p$, and a point $p\in\partial\Delta$ is called a branch point of $f$ if $ f$ is not injective on $\Delta\cap D$ for any neighborhood $D$ of $p$ in $ \overline{\Delta}$. A point $q\in S$ is a branch value of $f$ if $q$ is the image of some branch point.
In the study of the existence of extremal surfaces of $\mathcal{F}(L,m)$, the subspaces $\mathcal{F}_r^{\prime }(L,m)$ and $\mathcal{F}_r(L,m)$ play important roles, which is defined in \cite{Zh} as follows.
\begin{definition} We define $\mathcal{F}_r^{\prime }(L,m)\subset\mathcal{F}(L,m)$, such that for each $\Sigma=(f,\overline{\Delta})\in\mathcal{F}_r^{\prime }(L,m)$, the following hold:
(1) $f$ has no branch value outside $E_{q}$;
(2) $\deg _{\max }(f)<d^{\ast }$, where $d^{\ast }=d^{\ast }(m,q)$ is a constant which depends only on $m$ and $q$.
Moreover, we use $\mathcal{F}_{r}(L,m)$ to denote the subspace of $\mathcal{F }(L,m)$ consisting of all surfaces satisfying (1). \end{definition}
In \cite{Zh}, the last author has proved the existence of extremal surfaces of $\mathbf{F}(L)$. The starting point of the proof is the following two theorems, which are only asserted there.
\begin{theorem} \label{thr:branch} Let $\Sigma=(f,\overline{\Delta})\in\mathcal{F}(L,m)$ and assume $H(\Sigma)\ge H_L-\frac{2\pi}{L(\partial\Sigma)}$. Then there exists a surface $\Sigma_1=(f_1,\overline{\Delta})\in\mathcal{F}_r(L,m)$, such that \begin{equation*} L(\partial\Sigma_1)\le L(\partial\Sigma) \end{equation*} and \begin{equation*} H(\Sigma_1)\ge H(\Sigma). \end{equation*} \end{theorem}
\begin{theorem} \label{thr:main} There exists a constant $d^{\ast}=d^{\ast}(q,m)$ depending only on $m$ and $q$, such that for any $\Sigma\in\mathcal{F}_r(L,m)$, there exists a surface $\Sigma_1=(f_1,\overline{\Delta})\in\mathcal{F}_r^{\prime }(L,m)$ satisfying \begin{equation*} \partial\Sigma_1=\partial\Sigma \end{equation*} and \begin{equation*} H(\Sigma_1)=H(\Sigma). \end{equation*} \end{theorem}
The importance of Theorem \ref{thr:branch} and Theorem \ref{thr:main}, is that they respectively imply
\begin{corollary} \begin{equation*} \sup_{\Sigma\in\mathcal{F}(L,m)}H(\Sigma)=\sup_{\Sigma\in\mathcal{F} _r(L,m)}H(\Sigma)=\sup_{\Sigma\in\mathcal{F}_r^{\prime }(L,m)}H(\Sigma). \end{equation*} \end{corollary}
Theorem \ref{thr:branch} has been proved in \cite{CYL}. The goal of this paper is to prove Theorem \ref{thr:main}.
\section{Elementary properties of surfaces}
In this section, we will introduce some concepts about surfaces, and basic methods, some of which have been mentioned in Section \ref{sec:intro}. For rigorousness, let us review the geometry on the sphere firstly.
\begin{definition} On the Riemann sphere $S$, a simple arc on a circle is called a \emph{simple circular arc}, and a simple circular arc on a great circle is called a \emph{ line segment}. \end{definition}
The following result is obvious.
\begin{lemma} For any two non-antipodal points on $S$, there exists exactly one great circle passing through them. Any two distinct great circles on $S$ intersect at exactly two points. \end{lemma}
\begin{definition} A \emph{curve} $\gamma$ on $S$ is a continuous mapping $\gamma:[0,1]\to S$. We call $\gamma(0)$, $\gamma(1)$ the initial point and the terminal point respectively.
The interior of $\gamma$ is denoted by $\gamma^{\circ}$ which is the restriction of $\gamma$ to $(0,1)$. An arc of $\gamma$ is a curve given by $ t\mapsto\gamma(s+lt)$ for some $0\le s<s+l\le1$.
$\gamma$ is called closed if $\gamma(0)=\gamma(1)$, and called simple if $ \gamma(s)\ne\gamma(t)$ for any $0\le s<t\le1$ except $s=0$, $t=1$. \end{definition}
\begin{remark} \label{rm:curveeq} Sometimes we only consider the image of curves, and we directly use set operators with them.
For two curves $\gamma_1$, $\gamma_2$, we say that $\gamma_1$ and $\gamma_2$ are equivalent if there is an increasing homeomorphism $\varphi:[0,1]\to[0,1] $ such that $\gamma_1=\gamma_2\circ\varphi$. In this case, without causing misunderstanding, we will not distinguish them and directly write $ \gamma_1=\gamma_2$ for convenience.
Sometimes for emphasis, we use $\gamma(x_0,x_1)$ to represent a curve $ \gamma $ with the initial point $x_0$ and the terminal point $x_1$. \end{remark}
\begin{definition} For a curve $\gamma$ on $S$, $-\gamma$ is defined by \begin{align*} (-\gamma)(t)&=\gamma(1-t), & &0\le t\le1, \end{align*} which represents the opposite curve of $\gamma$ from $\gamma(1)$ to $ \gamma(0)$.
For two curves $\gamma_1$ and $\gamma_2$ with $\gamma_1(1)=\gamma_2(0)$, their sum $\gamma_1+\gamma_2$ is defined by \begin{align*} (\gamma_1+\gamma_2)(t)&=\gamma_1(2t), & &0\le t\le1/2, \\ (\gamma_1+\gamma_2)(t)&=\gamma_2(2t-1), & &1/2\le t\le1, \end{align*} which represents the joined curve. \end{definition}
According to Remark \ref{rm:curveeq}, the addition of curves satisfies the associative law. However, $\gamma+(-\gamma)$ is nonvanishing, which shows a point moving along the curve $\gamma$ and then returning back.
A class of curves which we often discuss consists of the boundaries of Jordan domains. For this situation, it is necessary to define its orientation and convexity.
\begin{definition} \label{def:oriconv}(1) A domain $D$ on $S$ is called a \emph{Jordan domain}, when $\partial D$ is a \emph{Jordan curve}, say, $\partial D$ is a simple and closed curve.
(2) For a Jordan domain $D$ on $S$, let $h$ be a M\"{o}bius transformation with $h(D)\subset \Delta $. Then $\partial D$ is oriented by the anticlockwise orientation of $\partial h(D)$. We also say $D$ is a domain enclosed by $\partial D$ with this orientation.
(3) A domain $D$ on $S$ is called \emph{convex} if for any two non-antipodal points $x_{1}$, $x_{2}$ in $D$, the shorter line segment connecting them is contained in $D$. The closure of a convex Jordan domain is called a convex closed domain. A Jordan curve on $S$ is called \emph{convex} if it encloses a convex Jordan domain.
(4) Let $\gamma $ be a curve on $S$ and $t_{0}\in (0,1)$, $\gamma $ is called (locally) \emph{convex} at $\gamma (t_{0})$, if $\gamma $ restricted to a neighborhood of $t_{0}$ is an arc of some convex Jordan curve. \end{definition}
For instance, the disk $\{z\in\overline{\mathbb{C}}:\lvert z\rvert>2\}$ is viewed as a convex domain on $S$, and thus the circle $\lvert z\rvert=2$ oriented clockwise is also convex on $S$, conversely this circle oriented anticlockwise is not convex. It is easy to see the definition here is consistent with the previous one in Remark \ref{rm:FLMconvex}.
Now we introduce the concept of surface equivalence, where surfaces is described in Definition \ref{def:surface}.
\begin{definition} For two surfaces $\Sigma_1=(f_1,\overline{U_1})$ and $\Sigma_2=(f_2, \overline{U_2})$, we say $\Sigma_1$ and $\Sigma_2$ are equivalent, and write \begin{equation*} (f_1,\overline{U_1})\sim(f_2,\overline{U_2}), \end{equation*} when there is an orientation-preserving homeomorphism $\varphi:\overline{U_1} \to\overline{U_2}$ such that $f_1=f_2\circ\varphi$.
Moreover, for two pairs $(f_1,\gamma_1)$ and $(f_2,\gamma_2)$, where $ \gamma_1$ and $\gamma_2$ are curves in $\overline{U_1}$ and $\overline{U_2}$ respectively, we write \begin{equation*} (f_1,\gamma_1)\sim(f_2,\gamma_2), \end{equation*} when $f_1\circ\gamma_1$ and $f_2\circ\gamma_2$ are equivalent curves. \end{definition}
Obviously, if two surfaces $(f_1,\overline{U_1})$ and $(f_2,\overline{U_2})$ satisfy $(f_1,\overline{U_1})\sim(f_2,\overline{U_2})$, then $(f_1,\partial U_1)\sim(f_2,\partial U_2)$. For equivalent surfaces $\Sigma_1$ and $ \Sigma_2 $, we have $A(\Sigma_1)=A(\Sigma_2)$, $\overline{n}(\Sigma_1)= \overline{n}(\Sigma_2)$, and $L(\partial\Sigma_1)=L(\partial\Sigma_2)$.
By the theorem of Sto\"ilow (Theorem \ref{thr:stoilow}), for each surface $ \Sigma=(f,\overline{U})$, there is a homeomorphism $\varphi:V_1\to V\supset \overline{U}$, where $V$ and $V_1$ are domains on $\overline{\mathbb{C}}$, such that $f\circ\varphi$ is a meromorphic function on $V_1$. So any surface is given by a meromorphic function in the sense of equivalence. $A(\Sigma)$, $\overline{n}(\Sigma)$ and $L(\partial\Sigma)$ can be determined in this way. With such equivalence, the definition of branch points can also be determined.
The following conclusion is contained in Sto\"ilow's theorem.
\begin{lemma} \label{lm:stoilowloc} Let $(f,\overline{U})$ be a surface and $x\in\overline{ U}$. Then there exist homeomorphisms $\varphi:\Delta\to\varphi(\Delta)\subset \overline{\mathbb{C}}$ and $\psi:\Delta\to\psi(\Delta)\subset\overline{ \mathbb{C}}$, such that $\varphi(0)=x$, $\psi(0)=f(x)$, $f$ can be extended to $\varphi(\Delta)\cup\overline{U}$ with $f(\varphi(\Delta))=\psi(\Delta)$, and $(\psi^{-1}\circ f\circ\varphi)(\zeta)=\zeta^{\omega}$ on $\Delta$, where $\omega$ is a positive integer. \end{lemma}
In other words, $f$ is locally a branched covering mapping. But its mapping behavior near an interior point and a boundary point are quite different.
Let $(f,\overline{U})$ be a surface, where $U$ is $\overline{\mathbb{C}}$ or a domain in $\overline{\mathbb{C}}$ enclosed by a finite number of Jordan curves by Definition \ref{def:surface}. Then for any $x\in\overline{U}$, by Lemma \ref{lm:stoilowloc} we can find the corresponding homeomorphism $ \varphi$, $\psi$. If $x$ is an interior point say, $x\in U$, we can require $ \varphi(\Delta)\subset U$ for extra, and for each $y\in\psi(\Delta)\setminus \{f(x)\}$, the pre-image $f^{-1}(y)$ has exactly $\omega$ points in $ \varphi(\Delta)$.
If $x$ is a boundary point, say, $x\in \partial U$, we can require that $ \varphi $ maps the diameter interval $(-1,1)\subset \Delta $ onto $\varphi ((-1,1))=\varphi (\Delta )\cap \partial U$ for extra, and maps the upper half disk $\Delta ^{+}$ onto $\varphi (\Delta ^{+})=\varphi (\Delta )\cap U$ . For $y\in \psi (\Delta )\setminus \{f(x)\}$, $f^{-1}(y)$ still has $\omega $ points in $\varphi (\Delta )$, however, some of those points may be located outside $\overline{U}$. Thus there are two possibilities need to be discussed.
When $\omega$ is odd, $\zeta\mapsto\zeta^{\omega}$ maps the interval $(-1,1)$ still onto $(-1,1)$. So $f(\varphi(\Delta)\cap\partial U)=\psi((-1,1))$ and $ f(x)=\psi(0)$ is an interior point of $f(\varphi(\Delta)\cap\partial U)$. On the other hand, for $y\in\psi(\Delta)$, the number of its pre-image points in $\varphi(\Delta)\cap U$ is equal to the number of pre-image points of $ \psi^{-1}(y)$ by mapping $\zeta\mapsto\zeta^{\omega}$ in $\Delta^+$. Thus, if $y$ lies in $\psi(\Delta^+)$, this number is $(\omega+1)/2$ and if $y$ lies on $\psi(\Delta^-)$, this number is $(\omega-1)/2$.
When $\omega$ is even, $\zeta\mapsto\zeta^{\omega}$ maps the interval $ (-1,1) $ onto $[0,1)$. Similarly we know that $f(x)$ is an endpoint of $ f(\varphi(\Delta)\cap\partial U)$. In this case, $(f,\partial U)$ is folded at $f(x)$. Then for either $y\in\psi(\Delta^+)$ or $y\in\psi(\Delta^-)$, the number of its pre-image points in $\varphi(\Delta)\cap U$ is equal to $ \omega/2$.
Summarizing above discussion, we give the definition of order.
\begin{definition} Let $(f,\overline{U})$ be a surface and $x\in\overline{U}$, $\omega$ be the positive integer determined by Lemma \ref{lm:stoilowloc}. If $x\in U$, the \emph{order} $v_f(x)$ of $(f,\overline{U})$ at $x$ equals $\omega$, and if $ x\in\partial U$, the \emph{order} $v_f(x)$ is the minimum integer not less than $\omega/2$. \end{definition}
\begin{remark} \label{rm:liftcount} The order defined here comes from the number of pre-image points, it can be also used to count path lifts. For a surface $(f, \overline{U})$, $x\in\overline{U}$, and a curve $\gamma$ on $S$ starting from $f(x)$, we consider the $f$-lift of $\gamma$ at $x$, which means a curve $\widetilde{\gamma}$ with initial point $x$ satisfying $f\circ \widetilde{\gamma}=\gamma$ and $\widetilde{\gamma}^{\circ}\subset U$. The existence of lifts and their number depend on the position of $\gamma$.
However, whether $x\in U$ or $x\in\partial U$, the number of $f$-lifts at $x$ is not greater than $v_f(x)$. In fact, when $\gamma$ is chosen properly, the number of such lifts is exactly $v_f(x)$. \end{remark}
\begin{definition} Let $\Sigma=(f,\overline{U})$ be a surface, a point $x\in\overline{U}$ is called a \emph{branch point} of $f$ (or $\Sigma$) if $v_f(x)>1$, otherwise called a \emph{regular point}. $y\in S$ is called a \emph{branch value} if there exists a branch point $x$ such that $f(x)=y$. \end{definition}
\begin{corollary} \label{cr:reghomeo} Let $\Sigma =(f,\overline{U})$ be a surface, $x\in \overline{U}$.
(1) If $f$ restricted to some neighborhood of $x$ in $\overline{U}$ is a homeomorphism, then $x\in \overline{U}$ is a regular point of $f$.
(2) If $x\in U$ and it is a regular point of $f$, then $f$ restricted to some neighborhood of $x$ in $U$ is a homeomorphism. \end{corollary}
At the end of this section, we will introduce a method of surface reconstruction. For this purpose, let us see a simple example. Consider a surface $\Sigma=(f,\overline{\Delta})$, and two paths $\tau^+$, $\tau^-$ contained in $\overline{\Delta}$ whose image are \begin{align*} \tau^+&=\{z\in\overline{\Delta^+}:\lvert z-1/2\rvert=1/2\}, & \tau^-&=\{z\in \overline{\Delta^-}:\lvert z-1/2\rvert=1/2\}, \end{align*} both of which have initial point $1\in\overline{\Delta}$ and terminal point $ 0\in\overline{\Delta}$. The curve $\tau^+-\tau^-$ enclose the domain $ D=\{z\in\mathbb{C}:\lvert z-1/2\rvert<1/2\}$.
If $f$ maps $\tau^+$, $\tau^-$ onto the same simple curve, for example, $f$ satisfies $f(z)=f(\bar{z})$ on $\partial D$, then the two ``edges'' $\tau^+$ , $\tau^-$ can be topologically pasted. In this way, two new surfaces can be obtained by sewing along $(f,\tau^+)\sim(f,\tau^-)$.
For $(f,\overline{D})$, we can find a homeomorphism $\varphi_0:D\to\overline{ \mathbb{C}}\setminus[0,1]$, which can be continuously extended to $\overline{ D}$ and maps both $\tau^+$, $\tau^-$ onto $[0,1]$ homeomorphically, such that there exists a continuous mapping $f_0:\overline{\mathbb{C}}\to S$ which satisfies $f=f_0\circ\varphi_0$. Since $f$ is OPCOFO, so is $f_0$, and we get a new surface $\Sigma_0=(f_0,\overline{\mathbb{C}})$. Similarly, for $ (f,\overline{\Delta\setminus D})$, we can find a homeomorphism $ \varphi_1:\Delta\setminus\overline{D}\to\Delta\setminus[0,1]$, which can be continuously extended to $\overline{\Delta\setminus D}$, maps $ \partial\Delta $ as identity and maps both $\tau^+$ and $\tau^-$ onto $[0,1]$ homeomorphically, to get another surface $\Sigma_1=(f_1,\overline{\Delta})$ where $f=f_1\circ\varphi_1$.
This also shows how we construct new surfaces by sewing in general. We can expect to get ``simpler'' surfaces in this way. That is why we introduce this method.
\begin{lemma} \label{lm:sew} Let $\Sigma=(f,\overline{\Delta})$ be a surface, $ \tau^+(x_0,x_1)$, $\tau^-(x_0,x_1)$ be two paths from $x_0$ to $x_1$ in $ \overline{\Delta}$, such that $f\circ\tau^+$, $f\circ\tau^-$ are equivalent paths consisting of line segments, in other word, \begin{equation*} (f,\tau^+)\sim(f,\tau^-), \end{equation*} which means there exists an increasing homeomorphism $\varphi:[0,1]\to[0,1]$ such that $f\circ\tau^+=f\circ\tau^-\circ\varphi$.
Assume $\tau ^{+}-\tau ^{-}$ is a Jordan curve enclosing a Jordan domain $ D\subset \Delta $, such that one of the following conditions hold:
(1) $\#(\partial D)\cap (\partial \Delta )<2$.
(2) As in Figure \ref{lm:sew-fig}.\subref{lm:sew-fig1}, $\#(\partial D)\cap (\partial \Delta )=2$, and $\tau ^{+}(t)$ and $\tau ^{-}(\varphi (t))$ can not be contained in $\partial \Delta $ simultaneously for any $t\in (0,1)$.
Then there exists a neighborhood $V$ of $(\partial \Delta )\setminus (\partial D)$ in $\overline{\Delta }\setminus ((\partial D)\cap (\partial \Delta ))$, and a simple curve $\tau ^{\ast }(x_{0},x_{1})$ contained in $ \overline{\Delta }$, as in Figure \ref{lm:sew-fig}.\subref{lm:sew-fig4}, such that two surfaces can be constructed:
(i) $\Sigma _{0}=(f_{0},\overline{\mathbb{C}})$ which satisfies $ f=f_{0}\circ \varphi _{0}$ for a continuous mapping $\varphi _{0}:\overline{D }\rightarrow \overline{\mathbb{C}}$;
(ii) $\Sigma _{1}=(f_{1},\overline{\Delta })$ which satisfies $f=f_{1}\circ \varphi _{1}$ for a continuous mapping $\varphi _{1}:\overline{\Delta \setminus D}\rightarrow \overline{\Delta }$.
Moreover, $\varphi _{0}$ restricted to $D$ is a homeomorphism onto $ \overline{\mathbb{C}}\setminus \tau ^{\ast }$, $\varphi _{1}$ restricted to $ \Delta \setminus D$ is a homeomorphism onto $\Delta \setminus \tau ^{\ast }$ . Four restrictions $\varphi _{0}:\tau ^{+}\rightarrow \tau ^{\ast }$, $ \varphi _{0}:\tau ^{-}\rightarrow \tau ^{\ast }$, $\varphi _{1}:\tau ^{+}\rightarrow \tau ^{\ast }$, $\varphi _{1}:\tau ^{-}\rightarrow \tau ^{\ast }$ are all homeomorphism. Finally, $\varphi _{1}$ restricted to $V$ is identical. \end{lemma}
\begin{proof} Without losing generality, we may assume that $\varphi:[0,1]\to[0,1]$ is identical, for otherwise we may replace $\tau^-$ by $\tau^-\circ\varphi$.
Firstly, $\tau^+(t)\mapsto\tau^-(t)$ gives the homeomorphism $\widetilde{ \varphi}:\tau^+\to\tau^-$. What we need to do is suitably choose the ``suture line'' $\tau^{\ast}$, to give homeomorphism $\varphi_0$, $\varphi_1$ from $\tau^+$, $\tau^-$ onto $\tau^{\ast}$, and then they are defined on $ \partial D=\tau^+-\tau^-$ and can be further extended to the whole $ \overline{D}$ and $\overline{\Delta\setminus D}$ respectively.
(1) When $(\partial D)\cap (\partial \Delta )$ is contained in $\tau ^{+}$ or $\tau ^{-}$, without losing generality, assume $\partial D\cap \partial \Delta \subset \tau ^{+}$, and let $\varphi _{1}(\tau ^{+}(t))=\varphi _{1}(\tau ^{-}(t))=\tau ^{+}(t)$ for $t\in \lbrack 0,1]$. By this way $ \varphi _{1}$ is defined on $\partial D$, and restricted to $\tau ^{+}$ and $ \tau ^{-}$ are both homeomorphism onto $\tau ^{\ast }$, where $\tau ^{\ast }=\tau ^{+}$ actually.
(2) Assume $(\partial D)\cap (\partial \Delta )\nsubseteq \tau ^{+}$ and $ (\partial D)\cap (\partial \Delta )\nsubseteq \tau ^{-}$, says, $(\partial D)\cap (\partial \Delta )$ consists of two points lied in $(\tau ^{+})^{\circ }$ and $(\tau ^{-})^{\circ }$ respectively. Then we can define a mapping $\varphi _{1}:\partial \Delta \rightarrow $ Let $t^{+}$, $t^{-}$ be numbers in $(0,1)$ such that $\tau ^{+}(t^{+})\in \partial \Delta $, $ \tau ^{-}(t^{-})\in \partial \Delta $. Then $t^{+}\neq t^{-}$, and without loss of generality, assume $t^{+}<t^{-}$. Let $\widetilde{\tau } :[0,1]\rightarrow \overline{D}$ be a simple path from $\tau ^{+}(t^{+})$ to $ \tau ^{-}(t^{-})$ with $\widetilde{\tau }\cap \partial D=\{\tau ^{+}(t^{+}),\tau ^{-}(t^{-})\}$ and $\widetilde{\tau }^{\circ }\subset D$. Then define \begin{align*} \varphi _{1}(\tau ^{+}(t))=\varphi _{1}(\tau ^{-}(t))& =\tau ^{+}(t), & & t\in \lbrack 0,t^{+}]; \\ \varphi _{1}(\tau ^{+}(t))=\varphi _{1}(\tau ^{-}(t))& =\widetilde{\tau } ((t-t^{+})/(t^{-}-t^{+})), & & t\in \lbrack t^{+},t^{-}]; \\ \varphi _{1}(\tau ^{+}(t))=\varphi _{1}(\tau ^{-}(t))& =\tau ^{-}(t), & & t\in \lbrack t^{-},1]. \end{align*} By this way $\varphi _{1}$ is defined on $\partial D$, and restricted to $ \tau ^{+}$ and $\tau ^{-}$ are both homeomorphism onto $\tau ^{\ast }=\tau ^{+}(x_{0},\widetilde{\tau }(0))+\widetilde{\tau }+\tau ^{-}(\widetilde{\tau }(1),x_{1})$, where $\tau ^{+}(x_{0},\widetilde{\tau }(0))$ means the arc of $\tau ^{+}$ from $x_{0}$ to $\widetilde{\tau }(0)$ and $\tau ^{-}(\widetilde{ \tau }(1),x_{1})$ means the arc of $\tau ^{-}$ from $\widetilde{\tau }(1)$ to $x_{1}$.
\begin{figure}
\caption{ }
\label{lm:sew-fig1}
\label{lm:sew-fig4}
\label{lm:sew-fig}
\end{figure}
After $\varphi _{1}$ is defined on $\partial D$, let $\varphi _{0}=\varphi _{1}$ on $\partial D$, and extend $\varphi _{0}$ to a continuous mapping on $ \overline{D}$ which restricted to $D$ is a homeomorphism onto $\overline{ \mathbb{C}}\setminus \tau ^{\ast }$. Then $\varphi _{0}$ satisfies our conclusion.
Further, choose the neighborhood $V$ of $(\partial \Delta )\setminus (\partial D)$ such that $V$ has no intersection with a neighborhood of $ \overline{D}\setminus ((\partial D)\cap (\partial \Delta ))$ in $\overline{ \Delta }$. Then extend $\varphi _{1}$ to a continuous mapping on $\overline{ \Delta \setminus D}$ which restricted to $\Delta \setminus D$ is a homeomorphism onto $\Delta \setminus \tau ^{\ast }$, and restricted to $V$ is identical. Similarly $\varphi _{1}$ satisfies our conclusion. \end{proof}
When $\tau^+(t)$ and $\tau^-(\varphi(t))$ are contained in $\partial\Delta$ simultaneously for some $t\in(0,1)$, this lemma can not be used. In this case, the quotient space of $\overline{\Delta\setminus D}$ obtained by identifying $\tau^+(t)$ and $\tau^-(\varphi(t))$ for $t\in(0,1)$ is topologically not one disk anymore.
\begin{remark} Let $\Sigma $ be a surface, and $\Sigma _{0}$, $\Sigma _{1}$ be surfaces obtained by Lemma \ref{lm:sew}. It is obviously that \begin{equation*} \deg _{\min }(\Sigma )=\deg _{\min }(\Sigma _{0})+\deg _{\min }(\Sigma _{1}) \mathrm{\ and\ }\deg _{\min }(\Sigma _{1})<\deg _{\min }(\Sigma ). \end{equation*} \end{remark}
\section{Reducing surfaces and proof of the main theorem}
Now we will complete the proof of our main result (Theorem \ref{thr:main}). It is $\deg_{\max}(f)$ that this theorem refers, but actually:
\begin{lemma} \label{lm:arg} We have $\deg_{\max}(\Sigma)-\deg_{\min}(\Sigma)\le m$ for any $\Sigma\in\mathcal{F}_r(L,m)$. \end{lemma}
\begin{proof} Assume $\Sigma=(f,\overline{\Delta})\in\mathcal{F}_r(L,m)$. By Sto\"{\i} low's theorem, $f$ is meromorphic up to a homeomorphic transformation, and thus the argument principle applies to $f$.
Let points $y$ and $y_0$ in $S\setminus\partial\Sigma$ satisfy $ \#f^{-1}(y)=\deg_{\min}(f)$ and $\#f^{-1}(y_0)=\deg_{\max}(f)$. Up to a fractional linear mapping on $S$, without loss of generality, we may assume that $y=\infty$ and $y_0=0$ in $S$. Recall the $\mathcal{F}(L,m)$-partition of $\Sigma$ in \eqref{def:FLM-1}, \begin{equation*} \partial\Delta=\alpha_1(\mathfrak{p}_1,\mathfrak{p}_2)+\alpha_2(\mathfrak{p} _2,\mathfrak{p}_3)+\dots+\alpha_m(\mathfrak{p}_m,\mathfrak{p}_1). \end{equation*} By the argument principle, we have \begin{equation*} 2\pi\deg_{\max}(f)-2\pi\deg_{\min}(f)=\theta_1+\theta_2+\dots+\theta_m, \end{equation*} where $\theta_j$ is the change in the argument of $f$ over $\alpha_j$. Since $f$ maps each $\alpha_j$ to a simple circular arc, $\lvert\theta_j\rvert\le2 \pi$. Thus we get $\deg_{\max}(f)-\deg_{\min}(f)\le m$. \end{proof}
To prove Theorem \ref{thr:main}, our aim is to find a way to decrease the degree of surfaces. The method of sewing introduced in Lemma \ref{lm:sew} can help us. For a surface $\Sigma=(f,\overline{\Delta})$, we hope to take an inverse of $f$ on $S\setminus\beta$, where $\beta$ is a simple path on $S$ , and hope that the $f$-lifts of $\beta$ may give the two paths that satisfy the conditions of Lemma \ref{lm:sew}.
Because we need to calculate $\overline{n}(\Sigma)=\overline{n}(\Sigma,E_q)$ , the path $\beta$ can be chosen as follows.
\begin{lemma} For the special set $E_q$ on $S$, we can order the $q$ points of $E_q$ as $ a_1$, $a_2$, $\dots$, $a_q$, such that \begin{equation*} \beta=\beta_{E_q}=\beta(a_1,a_2)+\beta(a_2,a_3)+\dots+\beta(a_{q-1},a_q) \end{equation*} is a simple path, where each $\beta(a_j,a_{j+1})$ is a line segment from $ a_j $ to $a_{j+1}$. \end{lemma}
\begin{proof} Let $b$ and $b^{\ast}$ be fixed antipodal points on $S\setminus E_q$, such that any great circle passing through $b$ and $b^{\ast}$ contains at most one point of $E_q$. Consider a line segment $l$ on $S$ from $b$ to $b^{\ast}$ . Let $l$ fix the endpoints and rotate continuously on $S$. Assume that it encounters $a_1$, $a_2$, $\dots$, $a_q$ in $E_q$ successively. In this way, we can take $\beta=\beta(a_1,a_2)+\beta(a_2,a_3)+\dots+\beta(a_{q-1},a_q)$ as the path consisting of $q-1$ line segments which satisfies requirements. \end{proof}
Next, our task is to find the inverse of $f$ on $S\setminus\beta$ for a surface $\Sigma=(f,\overline{\Delta})$. This can be achieved locally and so that the standard concept of triangulated surfaces is useful in our study.
\begin{definition} A \emph{topological triangle} $F$ on $S$ is a closed Jordan domain with three distinguished points on its boundary, which are called \emph{vertices} of $F$. The vertices divide $\partial F$ into three edges, each of which is called an \emph{edge} of $F$.
Let $\mathcal{T}$ be a collection of topological triangles. The vertices and the edges of the topological triangles of $\mathcal{T}$ are called vertices and edges of $\mathcal{T}$, and the topological triangles of $\mathcal{T}$ are called \emph{faces} of $\mathcal{T}$.
Let $U\subset S$ be a domain. A finite collection $\mathcal{T}$ of topological triangles in $\overline{U}$ is called a \emph{triangulation} of $ \overline{U}$ if the following hold:
(1) $\overline{U}=\cup _{F\in \mathcal{T}}F$;
(2) for any two distinct edges $e_{1}$ and $e_{2}$ of $\mathcal{T}$, $ e_{1}\cap e_{2}$ is empty or is a singleton which is the common endpoint of them;
(3) for any two distinct faces $F_{1}$ and $F_{2}$ of $\mathcal{T}$, $ F_{1}\cap F_{2}$ is empty, or is a singleton which is a common vertex of $ F_{1}$ and $F_{2}$, or is a common edge of $F_{1}$ and $F_{2}$. \end{definition}
\begin{lemma} \label{lm:trian} Let $k$ be a natural number. Then there exists a constant $ C_{0}(k)$ which depends only on $k$, such that for any $k$ simple circular arcs $c_{1}(x_{1},y_{1})$, $c_{2}(x_{2},y_{2})$, $\dots $, $ c_{k}(x_{k},y_{k})$ on $S$, there exists a triangulation $\mathcal{T}$ of $S$ satisfying the following conditions:
(1) for each simple circular arc $c_{j}(x_{j},y_{j})$, its initial point $ x_{j}$ and terminal point $y_{j}$ are vertices of $\mathcal{T}$;
(2) for each $j=1,2,\dots ,k$, $c_{j}$ is the union of some edges of $ \mathcal{T}$;
(3) the number of faces of $\mathcal{T}$ is not greater than $C_{0}(k)$. \end{lemma}
\begin{proof} Let $E=c_1\cup c_2\cup\dots\cup c_k$. Without losing generality, we can assume that $E$ is connected, for otherwise we can add at most $k$ simple circular arcs to $E$. It is clear that we have a collection of edges \begin{equation*} \mathcal{E}=\{e_j\}_{j=1}^{k_1}, \end{equation*} such that $E=e_1\cup e_2\cup\dots\cup e_{k_1}$, where each $e_j$ is a simple circular arc which is a arc of some $c_{j^{\prime }}$, $1\le j^{\prime }\le k $. Moreover, for any two distinct $e_{j_1}$ and $e_{j_2}$, $e_{j_1}\cap e_{j_2}$ is empty or is a singleton which is the common endpoint of them. Noticing that any two distinct circles on $S$ have at most two common points, we can assume that $k_1\le C_1(k)$ for some constant $C_1(k)$ which depends only on $k$.
Let $\mathcal{V}$ be the set of all endpoints of edges in $\mathcal{E}$. Since $E$ is connected, we can assume that \begin{equation*} S\setminus E=U_1\cup U_2\cup\dots\cup U_{k_2}, \end{equation*} where $U_1$, $U_2$, $\dots$, $U_{k_2}$ are all connected components of $ S\setminus E$, and $k_2\le C_2(k)$ for some constant $C_2(k)$ which depends only on $k$. In fact we have $C_2(k)\le2C_1(k)$, since each edge of $ \mathcal{E}$ is shared by at most two components of $S\setminus E$. Then each $U_j$ is a simply connected domain with boundary consisting of at most $ 2k_1$ edges in $\mathcal{E}$, and without losing generality, we can assume that the number of these edges is at least three for each $U_j$, for otherwise we can replace the edges in $\mathcal{E}$ by some arcs. Each $U_j$ is conformally equivalent to $\Delta$, and so $\overline{U_j}$ has a triangulation $\mathcal{T}_j$, whose vertex set is $(\mathcal{V}\cap\partial U_j)\cup\{v_j\}$, where $v_j\in U_j$. Then $\mathcal{T}=\cup_{j=1}^{k_2} \mathcal{T}_j$ is the desired triangulation with at most $C_0(k)\le 2C_1(k)C_2(k)$ faces. \end{proof}
Now, we show the following.
\begin{lemma} \label{lm:triansur} Let $\Sigma=(f,\overline{\Delta})\in\mathcal{F}_r(L,m)$ with $\mathcal{F}(L,m)$-partitions \eqref{def:FLM-1} and \eqref{def:FLM-2}. Then there exists a triangulation $\mathcal{T}$ of $S$ such that:
(1) every point of $E_{q}\cup \{p_{j}\}_{j=1}^{m}$ is a vertex of $\mathcal{T }$, where $\{p_{j}\}_{j=1}^{m}$ is the set of points appeared in \eqref{def:FLM-2};
(2) for each $j=1,2,\dots ,m$, $c_{j}$ in \eqref{def:FLM-2} is a union of some edges of $\mathcal{T}$, and so is the arc $\beta (a_{j},a_{j+1})$ of $ \beta $ for each $j=1,2,\dots ,q-1$;
(3) all faces of $\mathcal{T}$ can be arranged as $F_{1}$, $F_{2}$, $\dots $ , $F_{k_{0}}$, where $k_{0}\leq C_{0}(m+q)$ for the constant $C_{0}(m+q)$ appeared in Lemma \ref{lm:trian}, such that for each $j$, \begin{equation*} F_{j}^{\ast }=(F_{1}\cup F_{2}\cup \dots \cup F_{j})^{\circ }\setminus \beta \end{equation*} is a simply connected domain, and \begin{equation*} F_{1}^{\ast }\subset F_{2}^{\ast }\subset \dots \subset F_{k_{0}}^{\ast }=S\setminus \beta . \end{equation*} \end{lemma}
For such a triangulation $\mathcal{T}$, by the definition of triangulation it is easy to see $(F_j)^{\circ}\cap(f(\partial\Delta)\cup\beta)=\varnothing$ for each face $F_j$.
\begin{proof} Let $\mathcal{T}$ be the triangulation obtained by Lemma \ref{lm:trian} applied to $c_{1}(p_{1},p_{2})$, $c_{2}(p_{2},p_{3})$, $\dots $, $ c_{m}(p_{m},p_{1})$, $\beta (a_{1},a_{2})$, $\beta (a_{2},a_{3})$, $\dots $, $\beta (a_{q-1},a_{q})$. It is clear that we only need to prove the existence of the arrangement in (3).
Since $\beta$ is a simple polygonal path and $S\setminus\beta$ is a simply connected domain, by Riemann mapping theorem, there exists a continuous mapping $\psi:\overline{\Delta}\to S$, which is conformal from $\Delta$ onto $S\setminus\beta$, and $\partial\Delta$ has a partition $\partial\Delta= \tau^+-\tau^-$, such that $\psi$ maps both $\tau^+$ and $\tau^-$ homeomorphically onto $\beta$. Then we can get the induced triangulation of $ \overline{\Delta}$ by the pullback of $\psi$, denoted by $\psi^{\ast}( \mathcal{T})$, which consists of every face $F$ such that $\psi(F)$ is a face of $\mathcal{T}$. Because $\psi$ is conformal on $\Delta$, there is a one-to-one correspondence between the faces of $\mathcal{T}$ and $ \psi^{\ast}(\mathcal{T})$ by $\psi$.
It is clear that the last face $F_{k_{0}}^{\prime }$ of $\psi ^{\ast }( \mathcal{T})$ can be chosen so that $\Delta \setminus F_{k_{0}}^{\prime }$ is also a Jordan domain, since each face $F$ of $\mathcal{T}$ is either contained in $S\setminus \beta $, or intersects $\beta $ on an edge or at a vertex of $\mathcal{T}$. For the same reason, the order of faces of $\psi ^{\ast }(\mathcal{T})$ can be so arranged as $F_{1}^{\prime }$, $ F_{2}^{\prime }$, $\dots $, $F_{k_{0}}^{\prime }$, such that $\Delta $, $ \Delta \setminus F_{k_{0}}^{\prime }$, $\Delta \setminus (F_{k_{0}-1}^{\prime }\cup F_{k_{0}}^{\prime })$, $\dots $, $\Delta \setminus (F_{2}^{\prime }\cup \dots \cup F_{k_{0}}^{\prime })$ are all Jordan domains. Then for each $k\leq k_{0}$, $(\cup _{j=1}^{k}F_{j}^{\prime })^{\circ }$ is a Jordan domain. Then $F_{j}=\psi (F_{j}^{\prime })$ and $ F_{j}^{\ast }=\psi ((F_{1}^{\prime }\cup F_{2}^{\prime }\cup \dots \cup F_{j}^{\prime })^{\circ })$ satisfy (3). \end{proof}
For a surface $(f,\overline{\Delta})$ and such a triangulation $\mathcal{T}$ of $S$, we can consider the triangulation of $\overline{\Delta}$ induced by the pullback of $f$. So we can prove:
\begin{lemma} \label{lm:GF} Let $\Sigma=(f,\overline{\Delta})\in\mathcal{F}_r(L,m)$, $ \mathcal{G}_{\infty}(f)$ be the set consisting of all univalent branches of $ f^{-1}$ defined on $S\setminus\beta$, such that for each $g\in\mathcal{G} _{\infty}(f)$, \begin{equation*} \#(\overline{g(S\setminus\beta)}\cap\partial\Delta)<+\infty. \end{equation*} Then we have \begin{equation*} \#\mathcal{G}_{\infty}(f)\ge \deg_{\min}(f)-d_{\infty}, \end{equation*} where $d_{\infty}$ is a constant depends only on $m$ and $q$. \end{lemma}
\begin{proof} Let $\mathcal{T}$ be the triangulation of $S$ defined in Lemma \ref {lm:triansur}. Each branch value of $f$ is a point in $E_q$ and thus is a vertex of $\mathcal{T}$. We can get the induced triangulation of $\overline{ \Delta}$ by the pullback of $f$, denoted by $f^{\ast}(\mathcal{T})$. The vertices, edges and faces of $f^{\ast}(\mathcal{T})$ are mapped homeomorphically onto vertices, edges and faces of $\mathcal{T}$ respectively. Then $\partial\Delta$ has a partition \begin{equation*} \partial\Delta=\gamma_1+\gamma_2+\dots+\gamma_{d_{\infty}}, \end{equation*} where each $\gamma_j$ is an edge of $f^{\ast}(\mathcal{T})$. Because $ f(\partial\Delta)$ consists of $m$ simple circular arcs, each point of $ f(\partial\Delta)$ has at most $m$ inverses on $\partial\Delta$. Thus \begin{equation*} d_{\infty}\le3mC_0(m+q), \end{equation*} where $C_0(m+q)$ is the constant mentioned in Lemma \ref{lm:triansur}.
It is clear that $f^{-1}(F_1)$ consists of at least $\deg_{\min}(f)$ faces of $f^{\ast}(\mathcal{T})$. There are also at least $\deg_{\min}(f)$ distinct univalent branches of $f^{-1}$ defined on $F_1=\overline{F_1^{\ast}} $. Let $G_1$ be the set of all these univalent branches of $f^{-1}$ defined on $\overline{F_1^{\ast}}\setminus\beta$, such that each $g\in G_1$ satisfies $\#(\overline{g(F_1^{\ast})}\cap\partial\Delta)<\infty$.
For each univalent branch $g$ of $f^{-1}$ defined on $\overline{F_1^{\ast}} \setminus\beta$, $g\notin G_1$ can only happen in the case that one edge of $ \overline{g(F_1^{\ast})}$ is contained in $\partial\Delta$, and there is only one face in $f^{\ast}(\mathcal{T})$ containing this edge. Then we have \begin{equation*} \#G_1\ge \deg_{\min}(f)-\#\mathcal{E}_1, \end{equation*} where $\mathcal{E}_1$ is the subset of $\{\gamma_j\}_{j=1}^{d_{\infty}}$, consisting of such $\gamma_j$ with $f(\gamma_j)\subset\partial F_1$.
Generally, let $G_j$ be the set of all univalent branches of $f^{-1}$ defined on $\overline{F_j^{\ast}}\setminus\beta$, such that each $g\in G_j$ satisfies $\#(\overline{g(F_j^{\ast})}\cap\partial\Delta)<\infty$. By this definition, for each fixed $j<k_0$ and each $g\in G_j$, $g$ can be definitely extended to $\overline{F_{j+1}^{\ast}}\setminus\beta$ across $ \Gamma_j$, where $\Gamma_j$ is not contained in $\beta$ and is the common edge of $\overline{F_j^{\ast}}$ and $F_{j+1}$, which is a simple arc consisting of one or two edges of $\mathcal{T}$. In fact $f$ has no branch value on $S\setminus\beta\supset\Gamma_j\setminus\beta\supset\Gamma_j^{ \circ} $ and $g(\Gamma_j^{\circ})\subset\Delta$, which implies the existence of extension.
For this extension of $g$ defined on $\overline{F_{j+1}^{\ast}} \setminus\beta $, it will certainly belong to $G_{j+1}$ when there is no edge of $F_{j+1}$ mapped into $\partial\Delta$ by $g$. So we have \begin{equation*} \#G_{j+1}\ge\#G_j-\#\mathcal{E}_j, \end{equation*} where $\mathcal{E}_j\subset\{\gamma_j\}_{j=1}^{d_{\infty}}$ consists of $ \gamma_j$ with $f(\gamma_j)\subset(\partial F_{j+1})\setminus\Gamma_j^{\circ} $.
Inductively, we get $\mathcal{G}_{\infty}(f)=G_{k_0}$ with \begin{equation*} \#G_{k_0}\ge \deg_{\min}(f)-\#\mathcal{E}_1-\#\mathcal{E}_2-\dots-\#\mathcal{ E}_{d_{\infty}-1}. \end{equation*} Because $\partial F_1$, $(\partial F_2)\setminus\Gamma_1^{\circ}$, $\dots$, $ (\partial F_{k_0})\setminus\Gamma_{k_0-1}^{\circ}$ consist of some distinct edges of $\mathcal{T}$, we know \begin{equation*} \#\mathcal{E}_1+\#\mathcal{E}_2+\dots+\#\mathcal{E}_{d_{\infty}-1}\le d_{\infty}, \end{equation*} which proves the conclusion. \end{proof}
We obtain the lower bound of the number of elements in $\mathcal{G} _{\infty}(f)$, and now need to have a basic discussion on the properties of these elements.
\begin{lemma} \label{lm:GFin} Let $\Sigma =(f,\overline{\Delta })\in \mathcal{F}_{r}(L,m)$ , $\mathcal{G}_{\infty }(f)$ be the set given by Lemma \ref{lm:GF}. If there are distinct $g_{1}$ and $g_{2}$ in $\mathcal{G}_{\infty }(f)$, then $ g_{1}(S\setminus \beta )\cap g_{2}(S\setminus \beta )=\varnothing $. \end{lemma}
\begin{proof} Let $x$ be a points in $g_{1}(S\setminus \beta )\cap g_{2}(S\setminus \beta ) $, which implies $g_{1}(f(x))=g_{2}(f(x))$. Because $g_{1}\neq g_{2}$, there exists $y\in S\setminus \beta $ such that $g_{1}(y)\neq g_{2}(y)$. Let $\gamma $ be a path from $f(x)$ to $y$ with $\gamma ^{\circ }\subset S\setminus \beta $. Then $g_{1}\circ \gamma $ and $g_{2}\circ \gamma $ give two different $f$-lifts of $\gamma $, but $f$ has no branch value outside $ E_{q}\subset \beta $, which implies a contradiction. \end{proof}
\begin{lemma} \label{lm:GFbound} Let $\Sigma=(f,\overline{\Delta})\in\mathcal{F}_r(L,m)$ and $g\in\mathcal{G}_{\infty}(f)$. Then the following hold.
(1) \label{lm:GFbound-1}$\partial g(S\setminus \beta )$ has the partition \begin{equation*} \partial g(S\setminus \beta )=(\tau _{1}^{+}+\tau _{2}^{+}+\dots +\tau _{q-1}^{+})-(\tau _{1}^{-}+\tau _{2}^{-}+\dots +\tau _{q-1}^{-}), \end{equation*} where $f$ restricted to $\tau ^{+}=\tau _{1}^{+}+\tau _{2}^{+}+\dots +\tau _{q-1}^{+}$ and $\tau ^{-}=\tau _{1}^{-}+\tau _{2}^{-}+\dots +\tau _{q-1}^{-} $ are both homeomorphisms onto $\beta $, and $f$ maps both $\tau _{j}^{+}$ and $\tau _{j}^{-}$ onto $\beta (a_{j},a_{j+1})$ homeomorphically for each $j $.
(2) \label{lm:GFbound-2}If $\tau ^{+}$ and $\tau ^{-}$ have a common point $ x $, then there exists $j$ such that $x$ is a common point of $\tau _{j}^{+}$ and $\tau _{j}^{-}$. Moreover, if $x\in (\tau _{j}^{+})^{\circ }$ or $x\in (\tau _{j}^{-})^{\circ }$, then $\tau _{j}^{+}=\tau _{j}^{-}$.
(3) \label{lm:GFbound-3}There exist arcs of $\tau ^{+}$ and $\tau ^{-}$, respectively given by \begin{equation*} (\tau ^{\ast })^{+}=\tau _{j_{0}}^{+}+\tau _{j_{0}+1}^{+}+\dots +\tau _{j_{0}+k}^{+}, (\tau ^{\ast })^{-}=\tau _{j_{0}}^{-}+\tau _{j_{0}+1}^{-}+\dots +\tau _{j_{0}+k}^{-}, \end{equation*} such that $\tau ^{\ast }=(\tau ^{\ast })^{+}-(\tau ^{\ast })^{-}$ is a Jordan curve. Moreover, $g(S\setminus \beta )$ is contained in the domain enclosed by $\tau ^{\ast }$. \end{lemma}
\begin{proof} Because $S\setminus \beta $ is conformally equivalent to $\Delta $, it is clear that $g$ can be continuously extended to either side of $\beta $. From this we obtain two $f$-lifts $\tau ^{+}$ and $\tau ^{-}$ of $\beta $ which are defined by the extension of $g$ on different sides of $\beta $. Obviously $f$ restricted to $\tau ^{+}$ and $\tau ^{-}$ are both homeomorphisms onto $\beta $. Let $\tau _{j}^{+}$ and $\tau _{j}^{-}$ be arcs of $\tau ^{+}$ and $\tau ^{-}$ respectively, which are homeomorphically mapped onto $\beta (a_{j},a_{j+1})$ by $f$. Now we get the partition of $ \partial g(S\setminus \beta )=\tau ^{+}-\tau ^{-}$, and (1) is proved.
To prove (2), let $x\in \tau ^{+}\cap \tau ^{-}$. We have $f(x)\in \beta $. If $f(x)\in \beta (a_{j},a_{j+1})$, we know $x\in \tau _{j}^{+}\cap \tau _{j}^{-}$. If $x\in (\tau _{j}^{+})^{\circ }\cup (\tau _{j}^{-})^{\circ }$, say, $f(x)\in (\beta (a_{j},a_{j+1}))^{\circ }$, then $f(x)\notin E_{q}$, and then $x$ is a regular point of $f$, and then by the uniqueness of $f$ -lift, we must have $\tau _{j}^{+}=\tau _{j}^{-}$, and (2) is proved.
Now we prove (3). Because $g(S\setminus \beta )\subset \Delta $, $\tau ^{+}\neq \tau ^{-}$. For any $j$ with $\tau _{j}^{+}\neq \tau _{j}^{-}$, we take $j^{\prime }\leq j$ as the maximum integer such that $\tau _{j^{\prime }}^{+}$ and $\tau _{j^{\prime }}^{-}$ have the same initial point, and take $ j^{\prime \prime }\geq j$ as the minimum integer such that $\tau _{j^{\prime \prime }}^{+}$ and $\tau _{j^{\prime \prime }}^{-}$ have the same terminal point. We denote $(\tau _{j}^{\ast })^{+}=\tau _{j^{\prime }}^{+}+\tau _{j^{\prime }+1}^{+}+\dots +\tau _{j^{\prime \prime }}^{+}$ and $(\tau _{j}^{\ast })^{-}=\tau _{j^{\prime }}^{-}+\tau _{j^{\prime }+1}^{-}+\dots +\tau _{j^{\prime \prime }}^{-}$. Then by (1) and (2), $\tau _{j}^{\ast }=(\tau _{j}^{\ast })^{+}-(\tau _{j}^{\ast })^{-}$ must be a Jordan curve. Assume that $U_{j}\subset \Delta $ is the domain enclosed by $\tau _{j}^{\ast }$ (when $\tau _{j}^{+}=\tau _{j}^{-}$, we can assume $ U_{j}=\varnothing $).
For $j_{1}$ and $j_{2}$ such that $\tau _{j_{1}}^{\ast }\neq \tau _{j_{2}}^{\ast }$, we may assume $j_{1}<j_{2}$. If $\tau _{j_{1}}^{\ast }$ and $\tau _{j_{2}}^{\ast }$ have a common point $x$, then by (1) and (2), $x$ must be the terminal point of $(\tau _{j_{1}}^{\ast })^{+}$ and $(\tau _{j_{1}}^{\ast })^{-}$, and meanwhile $x$ is the initial point of $(\tau _{j_{2}}^{\ast })^{+}$ and $(\tau _{j_{2}}^{\ast })^{-}$, and thus $x$ is the unique common point of $\tau _{j_{1}}^{\ast }$ and $\tau _{j_{2}}^{\ast } $. In fact, $f(x)$ is the unique common point of $f((\tau _{j_{1}}^{\ast })^{+})$ and $f((\tau _{j_{2}}^{\ast })^{+})$, where both $f((\tau _{j_{1}}^{\ast })^{+})=f((\tau _{j_{1}}^{\ast })^{-})$ and $f((\tau _{j_{2}}^{\ast })^{+})=f((\tau _{j_{2}}^{\ast })^{-})$ are arcs of $\beta $. Then for any $U_{j_{1}}$ and $U_{j_{2}}$, we have $U_{j_{1}}\subset U_{j_{2}} $, or $U_{j_{1}}\supset U_{j_{2}}$, or $U_{j_{1}}\cap U_{j_{2}}=\varnothing $.
Now let $U_{j_{0}}$ be \textquotedblleft maximal\textquotedblright , which means that there is no $U_{j}$ such that $U_{j}\neq U_{j_{0}}$ and $ U_{j}\supset U_{j_{0}}$. Since $g(S\setminus \beta )$ is connected, we get $ g(S\setminus \beta )\subset U_{j_{0}}$. Thus $\tau ^{\ast }=\tau _{j_{0}}^{\ast }$, $(\tau ^{\ast })^{+}=(\tau _{j_{0}}^{\ast })^{+}$ and $ (\tau ^{\ast })^{-}=(\tau _{j_{0}}^{\ast })^{-}$ satisfy (3). \end{proof}
The two curves $(\tau^{\ast})^+$ and $(\tau^{\ast})^-$ we find are as that described in Lemma \ref{lm:sew}. If some other conditions can be also satisfied, we can use Lemma \ref{lm:sew} to construct a new surface by sewing, whose degree is smaller.
\begin{lemma} \label{lm:sewdeg} Let $\Sigma=(f,\overline{\Delta})\in\mathcal{F}_r(L,m)$ be a surface with $\mathcal{F}(L,m)$-partitions \eqref{def:FLM-1} and \eqref{def:FLM-2}, $\tau^+$ and $\tau^-$ be two curves satisfying all conditions in Lemma \ref{lm:sew}, and moreover, $(\tau^+-\tau^-)\cap\partial \Delta\subset f^{-1}(E_q)\cap\{\mathfrak{p}_j\}_{j=1}^m$. Then there exists a surface $\Sigma_1=(f_1,\overline{\Delta})\in\mathcal{F}_r(L,m)$, such that \begin{align*} \partial\Sigma_1&=\partial\Sigma, \\ H(\Sigma_1)&=H(\Sigma), \end{align*} and moreover, \begin{equation*} \deg_{\min}(f_1)<\deg_{\min}(f). \end{equation*} \end{lemma}
\begin{proof} Let $D\subset\Delta$ be the Jordan domain with $\partial D=\tau^+-\tau^-$. Since $(f,\tau^+)\sim(f,\tau^-)$, which means that the $f$-lifts near the endpoints of $\tau^+$ and $\tau^-$ are not unique, these two endpoints are branch points of $f$ (see Remark \ref{rm:liftcount}, where $f$-lifts are also defined).
Let $\Sigma_0=(f_0,\overline{\mathbb{C}})$ and $\Sigma_1=(f_1,\overline{ \Delta})$ be the surfaces, and $\varphi_0:\overline{D}\to\overline{\mathbb{C} }$ and $\varphi_1:\overline{\Delta\setminus D}\to\overline{\Delta}$ be the continuous mappings obtained by Lemma \ref{lm:sew}. It is obviously that $ \deg_{\min}(f_1)<\deg_{\min}(f)$ and $\partial\Sigma_1=\partial\Sigma$. We prove $\Sigma_1\in\mathcal{F}_r(L,m)$ at first. By Lemma \ref{lm:sew}, there exists a neighborhood $V$ of $(\partial\Delta)\setminus(\partial D)$ in $ \overline{\Delta}\setminus((\partial D)\cap(\partial\Delta))$ such that $ \varphi_1$ restricted to $V$ is identical. We get $\Sigma_1\in\mathcal{F} (L,m)$, and $\partial\Sigma_1=(f_1,\partial\Delta)=(f,\Delta)=\partial\Sigma$ , since $(\partial D)\cap(\partial\Delta)\subset\{\mathfrak{p}_j\}_{j=1}^m$.
Assume $y\in\overline{\Delta}$ and $f_1(y)\notin E_q$. For each $ x\in\varphi_1^{-1}(y)$, we have $f(x)=f_1(y)\notin E_q$, and $x$ is a regular point of $f$. By Corollary \ref{cr:reghomeo}, if $x\in\Delta$, $f$ must be locally homeomorphic near $x$. So when $y\in\Delta$, $ \varphi_1^{-1}(y)\subset\Delta$ and $y$ is also a regular point of $f_1$. On the other hand, $(\partial D)\cap(\partial\Delta)\subset f^{-1}(E_q)$ and $ \varphi_1$ restricted to $V$ is identical. $f_1$ has no branch points on $ \partial\Delta$ outside $f^{-1}(E_q)$. Thus $\Sigma_1\in\mathcal{F}_r(L,m)$.
\begin{figure}
\caption{Lemma \protect\ref{lm:sewdeg}}
\end{figure}
Next we just need to prove $H(\Sigma_1)=H(\Sigma)$, which is equivalent to \begin{equation*} (q-2)A(\Sigma_1)-4\pi\overline{n}(\Sigma_1)=(q-2)A(\Sigma)-4\pi\overline{n} (\Sigma). \end{equation*}
By Sto\"ilow's theorem (Theorem \ref{thr:stoilow}), the surface $ \Sigma_0=(f_0,\overline{\mathbb{C}})$ is equivalent to a surface given by a meromorphic function. That is, $f_0$ is a rational function via a homeomorphic transformation of coordinates, and Riemann-Hurwitz formula applies to $f_0$. We denote $d=\deg_{\max}(f_0)=\deg_{\min}(f_0)$. The total order of ramification of $f_0$ is equal to \begin{equation*} \sum_{x\in\overline{\mathbb{C}}}(v_{f_0}(x)-1)=2d-2. \end{equation*} Similarly to $f_1$, $f_0$ has also no branch value outside $E_q$. For each $ a_j\in E_q$, consider the set $f_0^{-1}(a_j)$. We have $\#f_0^{-1}(a_j)= \overline{n}(\Sigma_0,a_j)$, and for each $j=1,2,\dots,q$, \begin{equation*} \sum_{x\in f_0^{-1}(a_j)}(v_{f_0}(x)-1)=\sum_{x\in f_0^{-1}(a_j)}v_{f_0}(x)- \overline{n}(\Sigma_0,a_j)=d-\overline{n}(\Sigma_0,a_j). \end{equation*} Summing this we get \begin{equation*} 2d-2=\sum_{j=1}^q\sum_{x\in f_0^{-1}(a_j)}(v_{f_0}(x)-1)=\sum_{j=1}^q(d- \overline{n}(\Sigma_0,a_j))=qd-\overline{n}(\Sigma_0). \end{equation*} Thus we finally get $\overline{n}(\Sigma_0)=(q-2)d+2$.
It is clearly that $A(\Sigma)=A(\Sigma_0)+A(\Sigma_1)$, and $ A(\Sigma_0)=4\pi d$ actually. Next, we will compute $\overline{n}(\Sigma)$ and $\overline{n}(\Sigma_0)+\overline{n}(\Sigma_1)$. For this purpose, consider the contribution of each $x\in\overline{\Delta}$ with $f(x)\in E_q$.
\begin{enumerate} \item If $x\in (\partial \Delta )\setminus (\tau ^{+}-\tau ^{-})$, it has no contribution to $\overline{n}(\Sigma )$, $\overline{n}(\Sigma _{0})$ and $ \overline{n}(\Sigma _{1})$.
\item If $x\in D$, then its contribution to $\overline{n}(\Sigma )$ is $1$, $ \varphi _{0}(x)$ gives contribution $1$ to $\overline{n}(\Sigma _{0})$, but $ \varphi _{1}(x)$ is undefined and has no contribution to $\overline{n} (\Sigma _{1})$.
\item If $x\in \Delta \setminus \overline{D}$, then its contribution to $ \overline{n}(\Sigma )$ is $1$, $\varphi _{1}(x)$ gives contribution $1$ to $ \overline{n}(\Sigma _{1})$, but $\varphi _{0}(x)$ is undefined and has no contribution to $\overline{n}(\Sigma _{0})$.
\item If $x\in (\tau ^{+})^{\circ }$, then there exists exactly one point $ x^{\ast }\in (\tau ^{-})^{\circ }$ as its corresponding point, say, $\varphi _{0}(x)=\varphi _{0}(x^{\ast })$, $\varphi _{1}(x)=\varphi _{1}(x^{\ast })$, and vice versa. For such a pair of points $x$, $x^{\ast }$, if none of them lies on $\partial \Delta $, their contribution to $\overline{n}(\Sigma )$ is $2$, and $\varphi _{0}(x)=\varphi _{0}(x^{\ast })$, $\varphi _{1}(x)=\varphi _{1}(x^{\ast })$ give contribution $1$ to $\overline{n}(\Sigma _{0})$ and $ \overline{n}(\Sigma _{1})$ respectively; if only one point of $x$ and $ x^{\ast }$ lies on $\partial \Delta $, their contribution to $\overline{n} (\Sigma )$ is $1$, $\varphi _{0}(x)=\varphi _{0}(x^{\ast })$ gives contribution $1$ to $\overline{n}(\Sigma _{0})$, and $\varphi _{1}(x)\in \partial \Delta $ has no contribution to $\overline{n}(\Sigma _{1})$.
\item If $x$ is an endpoint of $\tau ^{+}$ and $x\notin \partial \Delta $, its contribution to $\overline{n}(\Sigma )$ is $1$, and $\varphi _{0}(x)$, $ \varphi _{1}(x)$ give contribution $1$ to $\overline{n}(\Sigma _{0})$ and $ \overline{n}(\Sigma _{1})$ respectively. if If $x$ is an endpoint of $\tau ^{+}$ and $x\in \partial \Delta $, it has no contribution to $\overline{n} (\Sigma )$ and $\overline{n}(\Sigma _{1})$, and $\varphi _{1}(x)$ gives contribution $1$ to $\overline{n}(\Sigma _{0})$. \end{enumerate}
Thus, except in the case that $x$ is the endpoint of $\tau ^{+}$, $x$ (or its corresponding pair) always gives same contribution to $\overline{n} (\Sigma )$ and $\overline{n}(\Sigma _{0})+\overline{n}(\Sigma _{1})$. Therefore \begin{equation*} \overline{n}(\Sigma )=\overline{n}(\Sigma _{0})+\overline{n}(\Sigma _{1})-2. \end{equation*} Because $A(\Sigma _{0})=4\pi d$ and $\overline{n}(\Sigma _{0})=(q-2)d+2$, we finally get $H(\Sigma _{1})=H(\Sigma )$. \end{proof}
Apparently, if we can always find such $\tau^+$ and $\tau^-$ satisfying related conditions for $\Sigma\in\mathcal{F}_r(L,m)$ with sufficiently large $\deg_{\min}(\Sigma)$, then Lemma \ref{lm:sewdeg} implies Theorem \ref {thr:main} inductively. At present, Lemma \ref{lm:GF} tells us the existence of ``pullback'' $g$, and Lemma \ref{lm:GFbound} tells us that we can take a Jordan domain based on $\partial g(S\setminus\beta)$. Next we just need to find $g\in\mathcal{G}_{\infty}(f)$ such that the corresponding boundary satisfies other conditions in Lemma \ref{lm:sewdeg}.
Our first consideration is the intersections of $\overline{g(S\setminus\beta) }$ and $\partial\Delta$.
\begin{definition} Let $\Sigma=(f,\overline{\Delta})\in\mathcal{F}_r(L,m)$, $\mathcal{G} _{\infty}(f)$ be the set introduced in Lemma \ref{lm:GF}. For a natural number $d$, denote $\mathcal{G}_d(f)$ the subset of $\mathcal{G}_{\infty}(f)$ , which consists of all $g\in\mathcal{G}_{\infty}(f)$ such that \begin{equation*} \#(\overline{g(S\setminus\beta)}\cap\partial\Delta)\le d. \end{equation*} \end{definition}
\begin{lemma} \label{lm:GFvertex} Let $\Sigma=(f,\overline{\Delta})\in\mathcal{F}_r(L,m)$ with $\mathcal{F}(L,m)$-partitions \eqref{def:FLM-1} and \eqref{def:FLM-2}. We have $\mathcal{G}_{\infty}(f)=\mathcal{G}_{mq+m}(f)$. Moreover, for each $ g\in\mathcal{G}_{\infty}(f)$, we have \begin{equation*} \overline{g(S\setminus\beta)}\cap\partial\Delta\subset f^{-1}(E_q)\cup\{ \mathfrak{p}_j\}_{j=1}^m, \end{equation*} where $\mathfrak{p}_1$, $\mathfrak{p}_2$, $\dots$, $\mathfrak{p}_m$ are appeared in \eqref{def:FLM-2}. \end{lemma}
\begin{proof} Let $g\in\mathcal{G}_{\infty}(f)$, and $x\in\overline{g(S\setminus\beta)}$ be a point in $\partial\Delta\setminus\{\mathfrak{p}_j\}_{j=1}^m$. Then $ x\in\partial g(S\setminus\beta)$. Because $\overline{g(S\setminus\beta)} \cap\partial\Delta<\infty$, $x$ is an isolated point of $\partial g(S\setminus\beta)\cap\partial\Delta$. Assume $x\in\alpha_j^{\circ}$ for some $j=1,2,\dots,q$. By Definition \ref{def:FLM}, $c_j=(f,\alpha_j)$ is a convex simple circular arc, and $f$ restricted to a neighborhood of $ \alpha_j^{\circ}$ in $\overline{\Delta}$ is homeomorphic.
Assume $f(x)\notin E_{q}$. Then we can assume $f(x)\in l_{j}^{\circ }$ for some $j=1,2,\dots ,q$, where $l_{j}=\beta (a_{j},a_{j+1})$. Thus there exists a neighborhood $U$ of $x$ in $\overline{\Delta }$, where $U^{\circ }$ is a Jordan domain, such that the following hold.
(1) $f|_{\overline{U}}:\overline{U}\rightarrow f(\overline{U})$ is a homeomorphism;
(2) \label{lm:GFvertex-ct2}$x$ is the unique point of $U\cap \overline{ g(S\setminus \beta )}\cap \partial \Delta $;
(3) \label{lm:GFvertex-ct3}$f(\overline{U})$ is a convex closed domain;
(4) $c_{j}$ has an arc $c_{j}^{\prime }\subset \partial f(\overline{U})$, such that $f(x)\in c_{j}^{\circ }$ and $(f|_{\overline{U}})^{-1}$ maps $ c_{j}^{\prime }$ into $\alpha _{j}$ homeomorphically;
(5) $l_{j}$ has an arc $l_{j}^{\prime }\subset f(\overline{U})$, such that $
f(x)\in l_{j}^{\circ }$ and $(f|_{\overline{U}})^{-1}$ maps $l_{j}^{\prime }$ into $\partial g(S\setminus \beta )$ homeomorphically.
But $c_{j}^{\prime }$ is a simple circular arc and $l_{j}^{\prime }$ is a line segment, which implies a contradiction with (2) and (3)\footnote{ If $D$ is a convex Jordan domain on $S$ and $l^{\circ }$ is an open line segment in $\overline{D}$, and $l^{\circ }\cap \partial D\neq \varnothing $, then each component of $l^{\circ }\cap \partial D$ is not a point.}. Thus $ f(x)\in E_{q}$. We get $\overline{g(S\setminus \beta )}\cap \partial \Delta \subset f^{-1}(E_{q})\cup \{\mathfrak{p}_{j}\}_{j=1}^{m}$.
On the other hand, $f(\partial \Delta )$ consists of $m$ circular arcs, each point of $f(\partial \Delta )$ has at most $m$ inverses in $\partial \Delta $ . So $\#(f^{-1}(E_{q})\cap \partial \Delta )\leq mq$ and $g\in \mathcal{G} _{mq+m}(f)$. \end{proof}
\begin{lemma} \label{lm:GF2} Let $\Sigma=(f,\overline{\Delta})\in\mathcal{F}_r(L,m)$. We have \begin{equation*} \#(\mathcal{G}_{\infty}(f)\setminus\mathcal{G}_2(f))\le d_2, \end{equation*} where $d_2=mq+m-2$. \end{lemma}
\begin{proof} For each $g\in \mathcal{G}_{\infty }(f)\setminus \mathcal{G}_{2}(f)$, let $ \mathcal{V}_{g}=\overline{g(S\setminus \beta )}\cap \partial \Delta \subset \mathcal{V}$, and $P_{g}$ be the polygonal domain in the Euclidean space, which is composed of points in $\mathcal{V}_{g}$ (noticing $\#\mathcal{V} _{g}\geq 3$).
By Lemma \ref{lm:GFin}, for distinct $g_1,g_2\in\mathcal{G} _{\infty}(f)\setminus\mathcal{G}_2(f)$, there is no common point of $ g_1(S\setminus\beta)$ and $g_2(S\setminus\beta)$. $g_2(S\setminus\beta)$ must be contained in a connected component of $\Delta\setminus\overline{ g_1(S\setminus\beta)}$, and $\mathcal{V}_{g_2}$ is contained in the closure of a connected component of $(\partial\Delta)\setminus\mathcal{V}_{g_1}$. Then $P_{g_1}\cap P_{g_2}=\varnothing$.
Let $\mathcal{V}=(f^{-1}(E_q)\cap\partial\Delta)\cup\{\mathfrak{p} _j\}_{j=1}^m$. By Lemma \ref{lm:GFvertex}, for all $g\in\mathcal{G} _{\infty}(f)\setminus\mathcal{G}_2(f)$, these $P_g$ are pairwise disjoint polygonal domains with vertices in $\mathcal{V}$. We have \begin{equation*} \#\mathcal{V}\ge\#(\mathcal{G}_{\infty}(f)\setminus\mathcal{G}_2(f))+2. \end{equation*} By Lemma \ref{lm:GFvertex}, $\#\mathcal{V}\le mq+m$ and we complete the proof. \end{proof}
According to our discussion on branch points and path lifts, the lift near a regular point must be unique, and we get:
\begin{lemma} \label{lm:GF2nreg} Let $\Sigma=(f,\overline{\Delta})\in\mathcal{F}_r(L,m)$ with $\mathcal{F}(L,m)$-partitions \eqref{def:FLM-1} and \eqref{def:FLM-2}, $ \mathcal{G}_2^{\prime }(f)$ be the subset of $\mathcal{G}_2(f)$ such that for each $g\in\mathcal{G}_2^{\prime }(f)$, \begin{equation*} \partial\overline{g(S\setminus\beta)}\cap\partial\Delta\subset f^{-1}(E_q)\cap\{\mathfrak{p}_j\}_{j=1}^m. \end{equation*} Then \begin{equation*} \#(\mathcal{G}_2(f)\setminus\mathcal{G}_2^{\prime }(f))\le d_2^{\prime }, \end{equation*} where $d_2^{\prime }=mq+m$. \end{lemma}
\begin{proof} Similar to Lemma \ref{lm:GF2}, let $\mathcal{V}=(f^{-1}(E_q)\cap\partial \Delta)\cup\{\mathfrak{p}_j\}_{j=1}^m$. If in $\mathcal{G}_2(f)$ there exist distinct $g_1$ and $g_2$, such that \begin{equation*} \overline{g_1(S\setminus\beta)}\cap\overline{g_2(S\setminus\beta)} \cap\partial\Delta \end{equation*} owns a point $x\in\mathcal{V}$, then $\beta$ gives two different $f$-lifts with the same initial point $x$, which makes $x$ a branch point and $f(x)\in E_q$. By Corollary \ref{cr:reghomeo} and the definition of $\mathcal{F}(L,m)$ , we know that $x\in\{\mathfrak{p}_j\}_{j=1}^m$.
For any point $x\in\mathcal{V}$, there is at most one $g\in\mathcal{G} _2(f)\setminus\mathcal{G}_2^{\prime }(f)$ such that $x\in\overline{ g(S\setminus\beta)}$. Then $\#(\mathcal{G}_2(f)\setminus\mathcal{G} _2^{\prime }(f))\le\#\mathcal{V}$. We have known that $\#\mathcal{V}\le mq+m$ in Lemma \ref{lm:GFvertex}, which implies the result. \end{proof}
So far, for a surface $\Sigma=(f,\overline{\Delta})\in\mathcal{F}_r(L,m)$ and $g\in\mathcal{G}_2^{\prime }(f)$, let $(\tau^{\ast})^+$ and $ (\tau^{\ast})^-$ be two arcs of $\partial g(S\setminus\beta)$ described in Lemma \ref{lm:GFbound}. By comparing with the conditions in Lemma \ref {lm:sew} and Lemma \ref{lm:sewdeg}, the only case in which we cannot use Lemma \ref{lm:sew} is that $((\tau^{\ast})^+\cup(\tau^{\ast})^-)\cap\partial \Delta$ contains two points, and $f$ maps them to the same point.
\begin{definition} For a surface $\Sigma=(f,\overline{\Delta})\in\mathcal{F}_r(L,m)$, let $ \mathcal{G}_2^{\prime \prime }(f)$ be the subset of $\mathcal{G}_2^{\prime }(f)$, which consists of all $g\in\mathcal{G}_2^{\prime }(f)$ such that \begin{equation*} \#(\overline{g(S\setminus\beta)}\cap\partial\Delta)\ne\#f(\overline{ g(S\setminus\beta)}\cap\partial\Delta), \end{equation*} say, $\overline{g(S\setminus\beta)}\cap\partial\Delta$ has two distinct points $x_1$ and $x_2$, and $f(x_1)=f(x_2)$. \end{definition}
According to our previous discussion, when there exists a $g\in\mathcal{G} _2^{\prime }(f)\setminus\mathcal{G}_2^{\prime \prime }(f)$ for $\Sigma=(f, \overline{\Delta})\in\mathcal{F}_r(L,m)$, by Lemma \ref{lm:sewdeg} we can find a surface $\Sigma_1=(f_1,\overline{\Delta})\in\mathcal{F}_r(L,m)$, such that $\partial\Sigma_1=\partial\Sigma$, $H(\Sigma_1)=H(\Sigma)$, and $ \deg_{\min}(f_1)<\deg_{\min}(f)$. However, this will fail when $g\in\mathcal{ G}_2^{\prime \prime }(f)$. We need to find a workaround.
\begin{lemma} \label{lm:GF2coin} Let $\Sigma=(f,\overline{\Delta})\in\mathcal{F}_r(L,m)$ with $\mathcal{F}(L,m)$-partitions \eqref{def:FLM-1} and \eqref{def:FLM-2}, and $d_2^{\prime \prime }=m(m-1)/2$. If \begin{equation*} \#\mathcal{G}_2^{\prime \prime }(f)>d_2^{\prime \prime }, \end{equation*} then there exist two paths $\tau^+(\mathfrak{p}_{j_1},\mathfrak{p}_{j_2})$, $ \tau^-(\mathfrak{p}_{j_1},\mathfrak{p}_{j_2})$, where $\mathfrak{p}_{j_1}$, $ \mathfrak{p}_{j_2}$ are in the $\mathcal{F}(L,m)$-partition \eqref{def:FLM-1} and $f(\mathfrak{p}_{j_1})=f(\mathfrak{p}_{j_1})\in E_q$, such that
(1) $\tau ^{+}-\tau ^{-}$ is a Jordan curve, and $(\tau ^{+}-\tau ^{-})\cap \partial \Delta =\{\mathfrak{p}_{j_{1}},\mathfrak{p}_{j_{2}}\}$;
(2) $f(\tau ^{+})=f(\tau ^{-})=\beta ^{\ast }$, which is a Jordan path on $S$ , and $f$ restricted to $(\tau ^{+})^{\circ }$ and $(\tau ^{-})^{\circ }$ are both homeomorphisms onto $(\beta ^{\ast })^{\circ }$. \end{lemma}
\begin{proof} For distinct $j_1$ and $j_2$, consider the pair of points $\{\mathfrak{p} _{j_1},\mathfrak{p}_{j_2}\}$, and let $G_{j_1,j_2}$ be the subset of $ \mathcal{G}_2^{\prime \prime }(f)$, which consists of all $g\in\mathcal{G} _2^{\prime \prime }(f)$ such that \begin{equation*} \overline{g(S\setminus\beta)}\cap\partial\Delta=\{\mathfrak{p}_{j_1}, \mathfrak{p}_{j_2}\}. \end{equation*} For $g\in G_{j_1,j_2}$, we have $f(\mathfrak{p}_{j_1})=f(\mathfrak{p}_{j_2})$ . Then $\mathcal{G}_2^{\prime \prime }(f)=\cup_{j_1,j_2}G_{j_1,j_2}$. Because $\#\mathcal{G}_2^{\prime \prime }(f)>d_2^{\prime \prime }=m(m-1)/2$, according to the pigeonhole principle, there is at least one set $ G_{j_1,j_2} $ containing two distinct elements $g^+$ and $g^-$.
For this set $G_{j_1,j_2}$, assume that $f(\mathfrak{p}_{j_1})=f(\mathfrak{p} _{j_2})=a_{j_0}\in E_q$. Because $G_{j_1,j_2}\subset\mathcal{G}_2^{\prime \prime }(f)$, we know that $a_{j_0}\in\beta^{\circ}$ and thus $1<j_0<q$. Now choose a polygonal Jordan curve $\beta^{\ast}$ on $S$ with an endpoint $ a_{j_0}$, such that $a_{j_0}$ is the only intersection point of $ \beta^{\ast} $ and $\beta$, and $\beta^{\ast}$ separates $\beta$.
Now $g^+$ gives an $f$-lift $\tau^+$ of $\beta^{\ast}$, such that $ (\tau^+)^{\circ}=g^+((\beta^{\ast})^{\circ})$, and its endpoints are $ \mathfrak{p}_{j_1}$ and $\mathfrak{p}_{j_2}$. By adjusting the direction, we can assume that $\tau^+$ has the initial point $\mathfrak{p}_{j_1}$ and the terminal point $\mathfrak{p}_{j_2}$. Obviously $f(\tau^+)=\beta^{\ast}$ and $ f$ restricted to $(\tau^+)^{\circ}$ is a homeomorphism onto $ (\beta^{\ast})^{\circ}$, and $(\tau^+)^{\circ}\subset g^+(S\setminus\beta)$.
Similarly, $g^-$ gives another $f$-lift $\tau^-(\mathfrak{p}_{j_1},\mathfrak{ p}_{j_2})$ of $\beta^{\ast}$. Because $g^+(S\setminus\beta)$ and $ g^-(S\setminus\beta)$ have no common point, $\tau^+-\tau^-$ is a Jordan curve. \end{proof}
Thus, for this case we can also use Lemma \ref{lm:sewdeg}. Summarize our work in this section and we will prove that:
\begin{lemma} \label{lm:main} Let $\Sigma=(f,\overline{\Delta})\in\mathcal{F}_r(L,m)$ be a surface. If $\deg_{\min}(f)>d^{\ast}$, then there exists a surface $ \Sigma_1=(f_1,\overline{\Delta})\in\mathcal{F}_r(L,m)$, such that \begin{align*} \partial\Sigma_1&=\partial\Sigma, \\ H(\Sigma_1)&=H(\Sigma), \end{align*} and moreover, \begin{equation*} \deg_{\min}(f_1)<\deg_{\min}(f), \end{equation*} where $d^{\ast}$ is a constant depending only on $m$ and $q$. \end{lemma}
\begin{proof} We just need to put $d^{\ast }=d_{\infty }+d_{2}+d_{2}^{\prime }+d_{2}^{\prime \prime }$, where $d_{\infty }$, $d_{2}$, $d_{2}^{\prime }$, $ d_{2}^{\prime \prime }$ are constants mentioned in Lemma \ref{lm:GF}, Lemma \ref{lm:GF2}, Lemma \ref{lm:GF2nreg} and Lemma \ref{lm:GF2coin}. Assume $ \deg _{\min }(f)>d^{\ast }$. Then by Lemma \ref{lm:GF}, we have \begin{equation*} \#\mathcal{G}_{\infty }(f)\geq d^{\ast }-d_{\infty }=d_{2}+d_{2}^{\prime }+d_{2}^{\prime \prime }. \end{equation*} By Lemma \ref{lm:GF2}, we have \begin{equation*} \#(\mathcal{G}_{\infty }(f)\setminus \mathcal{G}_{2}(f))\leq d_{2}\mathrm{\ and\ }\#\mathcal{G}_{2}(f)>d_{2}^{\prime }+d_{2}^{\prime \prime }. \end{equation*} By Lemma \ref{lm:GF2nreg}, we have \begin{equation*} \#(\mathcal{G}_{2}(f)\setminus \mathcal{G}_{2}^{\prime }(f))\leq d_{2}^{\prime }\mathrm{\ and\ }\#\mathcal{G}_{2}^{\prime }(f)>d_{2}^{\prime \prime }. \end{equation*}
Assume $\mathcal{G}_2^{\prime }(f)\ne\mathcal{G}_2^{\prime \prime }(f)$. Then there exists a $g\in\mathcal{G}_2^{\prime }(f)\setminus\mathcal{G} _2^{\prime \prime }(f)$. Let $(\tau^{\ast})^+$, $(\tau^{\ast})^-$ be two curves obtained in Lemma \ref{lm:GFbound}. They satisfy the conditions in Lemma \ref{lm:sewdeg}, and the surface $\Sigma_1$ can be constructed by Lemma \ref{lm:sewdeg}. Conversely, if $\mathcal{G}_2^{\prime }(f)=\mathcal{G} _2^{\prime \prime }(f)$, then $\#\mathcal{G}_2^{\prime \prime }(f)>d_2^{\prime \prime }$. And by Lemma \ref{lm:GF2coin}, we also have two curves $\tau^+$, $\tau^-$, which satisfy the conditions in Lemma \ref {lm:sewdeg}. We can construct $\Sigma_1$ as well. So in either case, the required surface $\Sigma_1$ must exist. \end{proof}
Finally, with Lemma \ref{lm:arg}, it is clear that Theorem \ref{thr:main} is a corollary of Lemma \ref{lm:main} inductively.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\begin{abstract} We give a unified interpretation of confluences, contiguity relations and Katz's middle convolutions for linear ordinary differential equations with polynomial coefficients and their generalization to partial differential equations. The integral representations and series expansions of their solutions are also within our interpretation. As an application to Fuchsian differential equations on the Riemann sphere, we construct a universal model of Fuchsian differential equations with a given spectral type, in particular, we construct single ordinary differential equations without apparent singularities corresponding to the rigid local systems, whose existence was an open problem presented by Katz. Furthermore we obtain an explicit solution to the connection problem for the rigid Fuchsian differential equations and the necessary and sufficient condition for their irreducibility. We give many examples calculated by our fractional calculus. \end{abstract}
\maketitle \tableofcontents \section{Introduction} Gauss hypergeometric functions and the functions in their family, such as Bessel functions, Whittaker functions, Hermite functions, Legendre polynomials and Jacobi polynomials etc.\ are the most fundamental and important special functions (cf.~\cite{EMO, Wa, WW}).
Many formulas related to the family have been studied and clarified together with the theory of ordinary differential equations, the theory of holomorphic functions and relations with other fields. They have been extensively used in various fields of mathematics, mathematical physics and engineering.
Euler studied the hypergeometric equation \begin{equation}\label{eq:G}
x(1-x)y''+\bigl(c-(a+b+1)x\bigr)y'-aby=0 \end{equation} with constant complex numbers $a$, $b$ and $c$ and he got the solution \begin{equation}\label{eq:GaussSeries}
F(a,b,c;x) := \sum_{k=0}^\infty\frac{a(a+1)\cdots(a+k-1)\cdot
b(b+1)\cdots(b+k-1)}{c(c+1)\cdots(c+k-1)\cdot k!}x^k. \end{equation} \index{hypergeometric equation/function!Gauss} The series $F(a,b,c;x)$ is now called Gauss hypergeometric series or function and Gauss proved the Gauss summation formula \begin{equation}\label{eq:Gausssum}
F(a,b,c;1) = \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)} \end{equation} when the real part of $c$ is sufficiently large. Then in the study of this function an important concept was introduced by Riemann. That is the Riemann scheme \begin{equation}\label{eq:RSGauss}
\begin{Bmatrix}
x = 0 & 1 & \infty\\
0 & 0 & a&\!\!;\ x\\
1-c&c-a-b & b
\end{Bmatrix} \end{equation} which describes the property of singularities of the function and Riemann proved that this property characterizes the Gauss hypergeometric function.
The equation \eqref{eq:G} is a second order Fuchsian differential equation on the Riemann sphere with the three singular points $\{0,1,\infty\}$. One of the main purpose of this paper is to generalize these results to the general Fuchsian differential equation on the Riemann sphere. In fact, our study will be applied to the following three kinds of generalizations.
One of the generalizations of the Gauss hypergeometric family is the hypergeometric family containing the generalized hypergeometric function ${}_nF_{n-1}(\alpha,\beta;x)$ or the solutions of Jordan-Pochhammer equations. Some of their global structures are concretely described as in the case of the Gauss hypergeometric family.
The second generalization is a class of Fuchsian differential equations such as the Heun equation which is of order 2 and has 4 singular points in the Riemann sphere. In this case, there appear \textsl{accessory parameters}. The global structure of the generic solution is quite transcendental and the Painlev\'e equation which describes the deformations preserving the monodromies of solutions of the equations with an apparent singular point is interesting and has been quite deeply studied and now it becomes an important field of mathematics.
The third generalization is a class of hypergeometric functions of several variables, such as Appell's hypergeometric functions (cf.~\cite{AK}), Gelfand's generalized hypergeometric functions (cf.~\cite{Ge}) and Heckman-Opdam's hypergeometric functions (cf.~\cite{HO}). The author and Shimeno \cite{OS} studied the ordinary differential equations satisfied by the restrictions of Heckman-Opdam's hypergeometric function on singular lines through the origin and we found that some of the equations belong to the even family classified by Simpson \cite{Si}, which is now called a class of \textsl{rigid} differential equations and belongs to the first generalization in the above.
The author's original motivation related to the study in this note is a generalization of Gauss summation formula, namely, to calculate a connection coefficient for a solution of this even family, which is solved in \S\ref{sec:C} as a direct consequence of the general formula \eqref{eq:Icon} of certain connection coefficients described in Theorem~\ref{thm:c}. This paper is the author's first step to a unifying approach for these generalizations and the recent development in general Fuchsian differential equations described below with the aim of getting concrete and computable results. In this paper, we will avoid intrinsic arguments and results if possible and hence the most results can be implemented in computer programs. Moreover the arguments in this paper will be understood without referring to other papers.
Rigid differential equations are the differential equations which are uniquely determined by the data describing the local structure of their solutions at the singular points. From the point of view of the monodromy of the solutions, the rigid systems are the local systems which are uniquely determined by local monodromies around the singular points and Katz \cite{Kz} studied rigid local systems by defining and using the operations called \textsl{middle convolutions} and \textsl{additions}, which enables us to construct and analyze all the rigid local systems. In fact, he proved that any irreducible rigid local system is transformed into a trivial equation $\frac{du}{dz}=0$ by successive application of the operations. In another word, any irreducible rigid local system is obtained by successive applications of the operations to the trivial equation because the operations are invertible.
The arguments there are rather intrinsic by using perverse sheaves. Dettweiler-Reiter \cite{DR, DR2} interprets Katz's operations on monodromy generators and those on the systems of Fuchsian differential equations of Schlesinger canonical form \begin{equation}
\frac{du}{dx} = \sum_{j=1}^p\frac{A_j}{x-c_j}u \end{equation} with constant square matrices $A_1,\dots,A_p$.
Here $A_j$ are called the residue matrices of the system at the singular points $x=c_j$, which describe the local structure of the solutions. For example, the eigenvalues of the monodromy generator at $x=c_j$ are $e^{2\pi\sqrt{-1}\lambda_1},\dots, e^{2\pi\sqrt{-1}\lambda_n}$, where $\lambda_1,\dots,\lambda_n$ are eigenvalues of $A_j$. The residue matrix of the system at $x=\infty$ equals $A_0:=-(A_1+\cdots+A_p)$. These operations are useful also for non-rigid Fuchsian systems.
\index{Deligne-Simpson problem} Related to the Riemann-Hilbert problem, there is a natural problem to determine the condition on matrices $B_0,B_1,\dots,B_p$ of Jordan canonical form such that there exists an irreducible system of Schlesinger canonical form with the residue matrices $A_j$ conjugate to $B_j$ for $j=0,\ldots,p$, respectively. An obvious necessary condition is the equality $\sum_{j=0}^p\Trace B_j=0$. A similar problem for monodromy generators, namely its multiplicative version, is equally formulated. The latter is called a \textsl{mutiplicative} version and the former is called an \textsl{additive} version. Kostov \cite{Ko, Ko2} called them Deligne-Simpson problems and gave an answer under a certain genericity condition. We note that the addition is a kind of a gauge transformation \[u(x)\mapsto (x-c)^\lambda u(x)\] and the middle convolution is essentially an Euler transformation or a transformation by an Riemann-Liouville integral \[u(x)\mapsto \frac1{\Gamma(\mu)}\int_c^x u(t)(x-t)^{\mu-1}dt\] or a fractional derivation.
Crawley-Boevey \cite{CB} found a relation between the Deligne-Simpson problem and representations of certain quivers and gave an explicit answer for the additive Deligne-Simpson problem in terms of a Kac-Moody root system.
Yokoyama \cite{Yo2} defined operations called extensions and restrictions on the systems of Fuchsian ordinary differential equations of Okubo normal form \begin{equation}
\bigl(x - T\bigr)\frac{du}{dx} = Au. \end{equation} Here $A$ and $T$ are constant square matrices such that $T$ are diagonalizable. He proved that the irreducible rigid system of Okubo normal form is transformed into a trivial equation $\frac{du}{dz}=0$ by successive applications of his operations if the characteristic exponents are generic.
The relation between Katz's operations and Yokoyama's operations is clarified by \cite{O4} and it is proved there that their algorithms of reductions of Fuchsian systems are equivalent and so are those of the constructions of the systems.
These operations are quite powerful and in fact if we fix the number of accessory parameters of the systems, they are connected into a finite number of fundamental systems (cf.~\cite[Proposition~8.1 and Theorem~10.2]{O3} and Proposition~\ref{prop:Bineq}), which is a generalization of the fact that the irreducible rigid Fuchsian system is connected to the trivial equation.
Hence it is quite useful to understand how does the property of the solutions transform under these operations. In this point of view, the system of the equations, the integral representation and the monodromy of the solutions are studied by \cite{DR, DR2, HY} in the case of the Schlesinger canonical form. Moreover the equation describing the deformation preserving the monodromy of the solutions doesn't change, which is proved by \cite{HF}. In the case of the Okubo normal form the corresponding transformation of the systems, that of the integral representations of the solutions and that of their connection coefficients are studied by \cite{Yo2}, \cite{Ha} and \cite{Yo3}, respectively. These operation are explicit and hence it will be expected to have explicit results in general Fuchsian systems.
To avoid the specific forms of the differential equations, such as Schlesinger canonical form or Okubo normal form and moreover to make explicit calculations easier under the transformations, we introduce certain operations on differential operators with polynomial coefficients in \S\ref{sec:frac}. The operations in \S\ref{sec:frac} enables us to equally handle equations with irregular singularities or systems of equations with several variables.
The ring of differential operators with polynomial coefficients is called a \textsl{Weyl algebra} and denoted by $W[x]$ in this paper. The endomorphisms of $W[x]$ do not give a wide class of operations and Dixmier \cite{Dix} conjectured that they are the automorphisms of $W[x]$. But when we localize coordinate $x$, namely in the ring $W(x)$ of differential operators with coefficients in rational functions, we have a wider class of operations.
For example, the transformation of the pair $(x, \frac{d}{dx})$ into $(x, \frac{d}{dx} - h(x))$ with any rational function $h(x)$ induces an automorphism of $W(x)$. This operation is called a \textsl{gauge transformation}. The addition in \cite{DR, DR2} corresponds to this operation with $h(x) = \frac\lambda{x-c}$ and $\lambda,\, c\in\mathbb C$, which is denoted by $\Ad\bigl((x-c)^\lambda\bigr)$.
The transformation of the pair $(x,\frac{d}{dx})$ into $(-\frac d{dx},x)$ defines an important automorphism $\Lap$ of $W[x]$, which is called a \textsl{Laplace transformation}. In some cases the Fourier transformation is introduced and it is a similar transformation. Hence we may also localize $\frac {d}{dx}$ and introduce the operators such as $\lambda(\frac{d}{dx} - c)^{-1}$ and then the transformation of the pair $(x,\frac{d}{dx})$ into $(x-\lambda(\frac{d}{dx})^{-1}, \frac{d}{dx})$ defines an endomorphism in this localized ring, which corresponds to the middle convolution or an Euler transformation or a fractional derivation and is denoted by $\Ad(\partial^{-\lambda})$ or $mc_\lambda$. But the simultaneous localizations of $x$ and $\frac{d}{dx}$ produce the operator $(\frac{d}{dx})^{-1}\circ x^{-1}=\sum_{k=0}^\infty k!{x^{-k-1}}(\frac{d}{dx})^{-k-1}$ which is not algebraic in our sense and hence we will not introduce such a microdifferential operator in this paper and we will not allow the simultaneous localizations of the operators.
Since our equation $Pu=0$ studied in this paper is defined on the Riemann sphere, we may replace the operator $P$ in $W(x)$ by a suitable representative $\tilde P\in \mathbb C(x)P\cap W[x]$ with the minimal degree with respect to $x$ and we put $\Red P=\tilde P$. Combining these operations including this replacement gives a wider class of operations on the Weyl algebra $W[x]$. In particular, the operator corresponding to the addition is $\RAd\bigl((x-c)^\lambda\bigr)$ and that corresponding to the middle convolution is $\RAd(\partial^{-\mu})$ in our notation. The operations introduced in \S\ref{sec:frac} correspond to certain transformations of solutions of the differential equations defined by elements of Weyl algebra and we call the calculation using these operations \textsl{fractional calculus of Weyl algebra}.
To understand our operations, we show that, in Example~\ref{ex:midconv}, our operations enables us to construct Gauss hypergeometric equations, the equations satisfied by airy functions and Jordan-Pochhammer equations and to give integral representations of their solutions.
In this paper we mainly study ordinary differential equations and since any ordinary differential equation is \textsl{cyclic}, namely, it is isomorphic to a single differential operator $Pu=0$ (cf.~\S\ref{sec:ODE}), we study a single ordinary differential equation $Pu=0$ with $P\in W[x]$. In many cases, we are interested in a specific function $u(x)$ which is characterized by differential equations and if $u(x)$ is a function with the single variable $x$, the differential operators $P\in W(x)$ satisfying $Pu(x)=0$ are generated by a single operator and hence it is naturally a single differential equation. A relation between our fractional calculus and Katz's middle convolution is briefly explained in \S\ref{sec:DR}.
In \S\ref{sec:reg} we review fundamental results on Fuchsian ordinary differential equations. Our Weyl algebra $W[x]$ is allowed to have some parameters $\xi_1,\ldots$ and in this case the algebra is denoted by $W[x;\xi]$. The position of singular points of the equations and the characteristic exponents there are usually the parameters and the analytic continuation of the parameters naturally leads the confluence of \text{additions} (cf.~\S\ref{sec:VAd}).
Combining this with our construction of equations leads the confluence of the equations. In the case of Jordan-Pochhammer equations, we have versal Jordan-Pochhammer equations. In the case of Gauss hypergeometric equation, we have a unified expression of Gauss hypergeometric equation, Kummer equation and Hermite-Weber equation and get a unified integral representation of their solutions (cf.~Example~\ref{ex:VGHG}). After this section in this paper, we mainly study single Fuchsian differential equations on the Riemann sphere. Equations with irregular singularities will be discussed elsewhere.
In \S\ref{sec:series} and \S\ref{sec:contig} we examine the transformation of series expansions and contiguity relations of the solutions of Fuchsian differential equations under our operations.
The Fuchsian equation satisfied by the generalized hypergeometric series \index{hypergeometric equation/function!generalized} \begin{equation}\label{eq:IGHG}\index{00Fn@${}_nF_{n-1}$} \index{000gammak@$(\gamma)_k,\ (\mu)_\nu$}
\begin{gathered}
{}_nF_{n-1}(\alpha_1,\dots,\alpha_n,\beta_1,\dots,\beta_{n-1};x)
=\sum_{k=0}^\infty
\frac{(\alpha_1)_k\dots(\alpha_n)_k}{
(\beta_1)_k\dots(\beta_{n-1})_{n-1}k!}x^k\\
\text{with\quad}(\gamma)_k :=\gamma(\gamma+1)\cdots(\gamma+k-1)
\end{gathered} \end{equation} is characterized by the fact that it has $(n-1)$-dimensional local holomorphic solutions at $x=1$, which is more precisely as follows. The set of characteristic exponents of the equation at $x=1$ equals $\{0,1,\dots,n-1,-\beta_n\}$ with $\alpha_1+\dots+\alpha_n=\beta_1+\dots+\beta_n$ and those at $0$ and $\infty$ are $\{1-\beta_1,\dots,1-\beta_{n-1},0\}$ and $\{\alpha_1,\dots,\alpha_n\}$, respectively. Then if $\alpha_i$ and $\beta_j$ are generic, the Fuchsian differential equation $Pu=0$ is uniquely characterized by the fact that it has the above set of characteristic exponents at each singular point $0$ or $1$ or $\infty$ and the monodromy generator around the point is \textsl{semisimple}, namely, the local solution around the singular point has no logarithmic term. We express this condition by the (generalized) Riemann scheme \begin{equation}\label{eq:GRSGHG}
\begin{Bmatrix}
x = 0 & 1 & \infty\\
1-\beta_1 & [0]_{(n-1)} & \alpha_1\\
\vdots & & \vdots &;\,x\\
1-\beta_{n-1}& & \alpha_{n-1}\\
0 & -\beta_n & \alpha_n
\end{Bmatrix},\quad
[\lambda]_{(k)}:=
\begin{pmatrix}
\lambda\\
\lambda+1\\
\vdots\\
\lambda+k-1
\end{pmatrix}. \end{equation} In particular, when $n=3$, the (generalized) Riemann scheme is \[
\begin{Bmatrix}
x = 0 & 1 & \infty\\
\begin{matrix}
1 - \beta_1\\
1 - \beta_2
\end{matrix} &
\begin{pmatrix}
0\\
1
\end{pmatrix} &
\begin{matrix}
\alpha_1\\
\alpha_2
\end{matrix} &
\!\!\!\begin{matrix}
\\ &\!\!;\, x
\end{matrix}\\
0 & -\beta_3 & \alpha_3 \end{Bmatrix}. \] The corresponding usual Riemann scheme is obtained from the generalized Riemann scheme by eliminating $\Big($ and $\Big)$. Here $[0]_{(n-1)}$ in the above Riemann scheme means the characteristic exponents $0,1,\dots,n-2$ but it also indicates that the corresponding monodromy generator is semisimple in spite of integer differences of the characteristic exponents. Thus the set of (generalized) characteristic exponents $\{[0]_{(n-1)},-\beta_n\}$ at $x=1$ is defined. Here we remark that the coefficients of the Fuchsian differential operator $P$ which is uniquely determined by the generalized Riemann scheme for generic $\alpha_i$ and $\beta_j$ are polynomial functions of $\alpha_i$ and $\beta_j$ and hence $P$ is naturally defined for any $\alpha_i$ and $\beta_j$ as is given by \eqref{eq:GHP}. Similarly the Riemann scheme of Jordan-Pochhammer equation of order $p$ is \begin{equation} \begin{gathered}
\begin{Bmatrix}
x = c_0 & c_1 &\cdots &c_{p-1} & \infty\\
[0]_{(p-1)} & [0]_{(p-1)} & \cdots&[0]_{(p-1)}& [\lambda'_p]_{(p-1)}
&;\,x\\
\lambda_0 & \lambda_1 & \cdots &\lambda_{p-1}& \lambda_p
\end{Bmatrix},\\
\lambda_0+\cdots+\lambda_{p-1}+\lambda_p+(p-1)\lambda'_p=p-1. \end{gathered} \end{equation} The last equality in the above is called \textsl{Fuchs relation}.
In \S\ref{sec:index} we define the set of generalized characteristic exponents at a regular singular point of a differential equation $Pu=0$. In fact, when the order of $P$ is $n$, it is the set $\{[\lambda_1]_{(m_1)},\dots,[\lambda_k]_{(m_k)}\}$ with a partition $n=m_1+\cdots+m_k$ and complex numbers $\lambda_1,\dots,\lambda_k$. It means that the set of characteristic exponents at the point equals $\{\lambda_j+\nu\,;\,\nu=0,\dots,m_j-1\text{ and }j=1,\dots,k\}$ and the corresponding monodromy generator is semisimple if $\lambda_i-\lambda_j\not\in\mathbb Z$ for $1\le i<j\le k$. In \S\ref{sec:Gexp} we define the set of generalized characteristic exponents without the assumption $\lambda_i-\lambda_j\not\in\mathbb Z$ for $1\le i<j\le k$. Here we only remark that when $\lambda_i=\lambda_1$ for $i=1,\dots,k$, it is also characterized by the fact that the Jordan normal form of the monodromy generator is defined by the dual partition of $n=m_1+\cdots+m_k$ together with the usual characteristic exponents.
Thus for a single Fuchsian differential equation $Pu=0$ on the Riemann sphere which has $p+1$ regular singular points $c_0,\dots,c_p$, we define a (generalized) Riemann scheme \begin{equation}\label{eq:IGRS}
\begin{Bmatrix}
x = c_0 & c_1 & \cdots & c_p\\
[\lambda_{0,1}]_{(m_{0,1})} & [\lambda_{1,1}]_{(m_{1,1})}&\cdots
&[\lambda_{p,1}]_{(m_{p,1})}\\
\vdots & \vdots & \vdots & \vdots&;\,x\\
[\lambda_{0,n_0}]_{(m_{0,n_0})} & [\lambda_{1,n_1}]_{(m_{1,n_1})}&\cdots
&[\lambda_{p,n_p}]_{(m_{p,n_p})}
\end{Bmatrix}. \end{equation} Here $n=m_{j,1}+\cdots+m_{j,n_j}$ for $j=0,\dots,p$ and $n$ is the order of $P$ and $\lambda_{j,\nu}\in\mathbb C$. The $(p+1)$-tuple of partitions of $n$, which is denoted by $\mathbf m=\bigl(m_{j,\nu}\bigr)_{\substack{j=0,\dots,p\\\nu=1,\dots,n_j}}$, is called the \textsl{spectral type} of $P$ and the Riemann scheme \eqref{eq:IGRS}. Here we note that the Riemann scheme \eqref{eq:IGRS} should always satisfy the Fuchs relation \begin{align}\label{eq:IFuchs}
|\{\lambda_{\mathbf m}\}|&:=\sum_{j=0}^p\sum_{\nu=1}^{n_p}
m_{j,\nu}\lambda_{j,\nu}- \ord\mathbf m + \tfrac12\idx\mathbf m
=0,\\
\idx\mathbf m&:=\sum_{j=0}^p\sum_{\nu=1}^{n_p}m_{j,\nu}^2
-(p-1)\ord\mathbf m. \end{align} Here $\idx\mathbf m$ coincides with the \textsl{index of rigidity} introduced by \cite{Kz}.
In \S\ref{sec:index}, after introducing certain representatives of conjugacy classes of matrices and some notation and concepts related to tuples of partitions, we define that the tuple $\mathbf m$ is \textsl{realizable} if there exists a Fuchsian differential operator $P$ with the Riemann scheme \eqref{eq:IGRS} for generic complex numbers $\lambda_{j,\nu}$ under the condition \eqref{eq:IFuchs}. Furthermore, if there exists such an operator $P$ so that $Pu=0$ is irreducible, we define that $\mathbf m$ is \textsl{irreducibly realizable}.
Lastly in \S\ref{sec:index}, we examine the generalized Riemann schemes of the \textsl{product} of Fuchsian differential operators and the \textsl{dual} operators.
In \S\ref{sec:reduction} we examine the transformations of the Riemann scheme under our operations corresponding to the additions and the middle convolutions, which define transformations within Fuchsian differential operators. The operations induce transformations of spectral types of Fuchsian differential operators, which keep the indices of rigidity invariant but change the orders in general. Looking at the spectral types, we see that the combinatorial aspect of the reduction of Fuchsian differential operators is parallel to that of systems of Schlesinger canonical form. In this section, we also examine the combination of these transformation and the \textsl{fractional linear transformations.}
As our interpretation of Deligne-Simpson problem introduced by Kostov, we examine the condition for the existence of a given Riemann scheme in \S\ref{sec:DS}. We determine the conditions on $\mathbf m$ such that $\mathbf m$ is realizable and irreducibly realizable, respectively, in Theorem~\ref{thm:univmodel}. Moreover if $\mathbf m$ is realizable, Theorem~\ref{thm:univmodel} gives an explicit construction of the \textsl{universal Fuchsian differential operator} \begin{equation} \begin{gathered}
P_{\mathbf m}
=\Bigl(\prod_{j=1}^p(x-c_j)^n\Bigr)\frac{d^n}{dx^n}
+\sum_{k=0}^{n-1} a_k(x,\lambda, g)\frac{d^k}{dx^k},\\
\lambda=\bigl(\lambda_{j,\nu}\bigr)
_{\substack{j=0,\dots,p\\ \nu=1,\dots,n_j}},\quad
g=(g_1,\dots,g_N)\in\mathbb C^N \end{gathered} \end{equation} with the Riemann scheme \eqref{eq:IGRS}, which has the following properties.
For fixed complex numbers $\lambda_{j,\nu}$ satisfying \eqref{eq:IFuchs} the operator with the Riemann scheme \eqref{eq:IGRS} satisfying $c_0=\infty$ equals $P_{\mathbf m}$ for a suitable $g\in\mathbb C^N$ up to a left multiplication by an element of $\mathbb C(x)$ if $\lambda_{j,\nu}$ are ``generic", namely, \begin{equation}
(\Lambda(\lambda)|\alpha)\notin
\bigl\{-1,-2,\dots,1-(\alpha|\alpha_{\mathbf m})\bigr\}\text{ for any } \alpha\in\Delta(\mathbf m)
\text{ with } (\alpha|\alpha_{\mathbf m})>1 \end{equation} under the notation used in \eqref{eq:KacIrr}, or $\mathbf m$ is fundamental or \textsl{simply reducible} (cf.~Definition \ref{def:fund} and \S\ref{sec:simpred}), etc. Here $g_1,\dots,g_N$ are called \textsl{accessory parameters} and if $\mathbf m$ is irreducibly realizable, $N=1-\frac12\idx\mathbf m$. In particular, if there is an irreducible and \textsl{locally non-degenerate} (cf.~Definition~\ref{def:locnondeg}) operator $P$ with the Riemann scheme \eqref{eq:IGRS}, then $\lambda_{j,\nu}$ are ``generic".
The coefficients $a_k(x,\lambda, g)$ of the differential operator $P_{\mathbf m}$ are polynomials of the variables $x$, $\lambda$ and $g$. The coefficients satisfy $\frac{\partial^2 a_k}{\partial g_\nu^2}=0$ and furthermore $g_\nu$ can be equal to suitable $a_{i_\nu,j_\nu}$ under the expression $
P_{\mathbf m}=
\sum a_{i,j}x^i\frac{d^j}{dx^j} $ and the pairs $(i_\nu,j_\nu)$ for $\nu=1,\dots,N$ are explicitly given in the theorem.
The universal operator $P_\mathbf m$ is a classically well-known operator in the case of Gauss hypergeometric equation, Jordan-Pochhammer equation or Heun's equation etc.\ and the theorem assures the existence of such a good operator for any realizable tuple $\mathbf m$. We define the tuple $\mathbf m$ is \textsl{rigid} if $\mathbf m $ is irreducibly realizable and moreover $N=0$, namely, $P_{\mathbf m}$ is free from accessory parameters.
In particular, the theorem gives the affirmative answer for the following question. Katz asked a question in the introduction in the book \cite{Kz} whether a rigid local system is realized by a single Fuchsian differential equation $Pu=0$ without apparent singularities (cf.~Corollary~\ref{cor:irred} iii)).
It is a natural problem to examine the Fuchsian differential equation $P_{\mathbf m}u=0$ with an irreducibly realizable spectral type $\mathbf m$ which cannot be reduced to an equation with a lower order by additions and middle convolutions. The tuple $\mathbf m$ with this condition is called \textsl{fundamental}.
The equation $P_{\mathbf m}u=0$ with an irreducibly realizable spectral type $\mathbf m$ can be transformed by the operation $\partial_{max}$ (cf.~Definition~\ref{def:pell}) into a Fuchsian equation $P_{\mathbf m'}v=0$ with a fundamental spectral type $\mathbf m'$. Namely, there exists a non-negative integer $K$ such that $P_{\mathbf m'}=\partial_{\max}^KP_{\mathbf m}$ and we define $f\mathbf m:=\mathbf m'$. Then it turns out that a realizable tuple $\mathbf m$ is rigid if and only if the order of $f\mathbf m$, which is the order of $P_{f\mathbf m}$ by definition, equals 1. Note that the operator $\partial_{\max}$ is essentially a product of suitable operators $\RAd\bigl((x-c_j)^{\lambda_j}\bigr)$ and $\RAd\bigl(\partial^{-\mu}\bigr)$.
In this paper we study the transformations of several properties of the Fuchsian differential equation $P_{\mathbf m}u=0$ under the additions and middle convolutions. If they are understood well, the study of the properties are reduced to those of the equation $P_{f\mathbf m}v=0$, which are of order 1 if $\mathbf m$ is rigid. We note that there are many rigid spectral types $\mathbf m$ and for example there are 187 different rigid spectral types $\mathbf m$ with $\ord\mathbf m\le 8$ as are given in \S\ref{sec:rigidEx}.
As in the case of the systems of Schlesinger canonical form studied by \cite{CB}, the combinatorial aspect of transformations of the spectral type $\mathbf m$ of the Fuchsian differential operator $P$ induced from our fractional operations is described in \S\ref{sec:KacM} by using the terminology of a Kac-Moody root system $(\Pi,W_{\!\infty})$. Here $\Pi$ is the fundamental system of a Kac-Moody root system with the following star-shaped Dynkin diagram and $W_{\!\infty}$ is the Weyl group generated by the \textsl{simple reflections} $s_\alpha$ for $\alpha\in\Pi$. The elements of $\Pi$ are called \textsl{simple roots}.
Associated to a tuple $\mathbf m$ of $(p+1)$ partitions of a positive integer $n$, we define an element $\alpha_{\mathbf m}$ in the positive root lattice (cf.~\S\ref{sec:KM}, \eqref{eq:PIKac}): \begin{equation} \begin{split} \Pi&:=\{\alpha_0,\,\alpha_{j,\nu}\,;\,
j=0,1,\ldots,\ \nu=1,2,\ldots\},\\
W_{\!\infty}&:=\langle s_\alpha\,;\,\alpha\in\Pi\rangle,\\ \alpha_{\mathbf m}&:=n\alpha_0+\sum_{j=0}^p\sum_{\nu=1}^{n_j-1} \Bigl(\sum_{i=\nu+1}^{n_j}m_{j,i}\Bigr)\alpha_{j,\nu},\\
(\alpha_{\mathbf m}&|\alpha_{\mathbf m})=\idx\mathbf m, \end{split}\qquad \begin{xy} \ar@{-} *++!D{\text{$\alpha_0$}} *\cir<4pt>{}="O";
(10,0) *+!L!D{\text{$\alpha_{1,1}$}} *\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *+!L!D{\text{$\alpha_{1,2}$}} *\cir<4pt>{}="B", \ar@{-} "B"; (30,0) *{\cdots}, \ar@{-} "O"; (10,-7) *+!L!D{\text{$\alpha_{2,1}$}} *\cir<4pt>{}="C", \ar@{-} "C"; (20,-7) *+!L!D{\text{$\alpha_{2,2}$}} *\cir<4pt>{}="E", \ar@{-} "E"; (30,-7) *{\cdots} \ar@{-} "O"; (10,8) *+!L!D{\text{$\alpha_{0,1}$}} *\cir<4pt>{}="D", \ar@{-} "D"; (20,8) *+!L!D{\text{$\alpha_{0,2}$}} *\cir<4pt>{}="F", \ar@{-} "F"; (30,8) *{\cdots} \ar@{-} "O"; (10,-13) *+!L!D{\text{$\alpha_{3,1}$}} *\cir<4pt>{}="G", \ar@{-} "G"; (20,-13) *+!L!D{\text{$\alpha_{3,2}$}} *\cir<4pt>{}="H", \ar@{-} "H"; (30,-13) *{\cdots}, \ar@{-} "O"; (7, -13), \ar@{-} "O"; (4, -13), \end{xy} \end{equation} We can define a fractional operation on $P_{\mathbf m}$ which is compatible with the action of $w\in W_{\!\infty}$ on the root lattice (cf.~Theorem~\ref{thm:KatzKac}): \begin{equation}\label{eq:mcKacdg}
\begin{matrix} \bigl\{P_{\mathbf m}:\,\text{Fuchsian differential operators}\bigr\}
&\rightarrow
&\bigl\{(\Lambda(\lambda),\alpha_{\mathbf m})\,;\,\alpha_{\mathbf m}\in\overline\Delta_+\bigr\}
\\
\\%[-10pt]
\downarrow \text{fractional operations}
&\circlearrowright&\quad \downarrow {W_{\!\infty}\text{-action},\
+\tau\Lambda^0_{0,j}}\\
\\%[-10pt] \bigl\{P_{\mathbf m}:\,\text{Fuchsian differential operators}\bigr\}
& \rightarrow
& \bigl\{(\Lambda(\lambda), \alpha_{\mathbf m})\,;\,\alpha_{\mathbf m}
\in\overline\Delta_+\bigr\}.
\end{matrix} \end{equation} Here $\tau\in\mathbb C$ and \begin{equation}
\begin{split}
\Lambda^0&:=\alpha_0+\sum_{\nu=1}^\infty(1+\nu)\alpha_{0,\nu}+
\sum_{j=1}^p\sum_{\nu=1}^\infty(1-\nu)\alpha_{j,\nu},\\
\Lambda^0_{i,j}&:=\sum_{\nu=1}^\infty \nu(\alpha_{i,\nu}-\alpha_{j,\nu}),\\
\Lambda_0&:=\frac12\alpha_0
+\frac12\sum_{j=0}^p\sum_{\nu=1}^\infty(1-\nu)\alpha_{j,\nu},\\
\Lambda(\lambda)&:=-\Lambda_0-\sum_{j=0}^p\sum_{\nu=1}^\infty
\Bigl(\sum_{i=1}^\nu\lambda_{j,i}
\Bigr)\alpha_{j,\nu}
\end{split} \end{equation} and these linear combinations of infinite simple roots are identified with each other if their differences are in $\mathbb C\Lambda^0$. We note that \begin{equation}
|\{\lambda_{\mathbf m}\}|
=(\Lambda(\lambda)+\tfrac12\alpha_{\mathbf m}|\alpha_{\mathbf m}). \end{equation}
The realizable tuples exactly correspond to the elements of the set $\overline\Delta_+$ of positive integer multiples of the positive roots of the Kac-Moody root system whose support contains $\alpha_0$ and the rigid tuples exactly correspond to the positive real roots whose support contain $\alpha_0$. For an element $w\in W_{\!\infty}$ and an element $\alpha\in\overline\Delta_+$ we do not consider $w\alpha$ in the commutative diagram \eqref{eq:mcKacdg} when $w\alpha\notin \overline\Delta_+$.
Hence the fact that any irreducible rigid Fuchsian equation $P_{\mathbf m}u=0$ is transformed into the trivial equation $\frac{dv}{dx}=0$ by our invertible fractional operations corresponds to the fact that there exists $w\in W_{\!\infty}$ such that $w\alpha_{\mathbf m}=\alpha_0$ because $\alpha_{\mathbf m}$ is a positive real root. The \textsl{monotone} fundamental tuples of partitions correspond to $\alpha_0$ or the positive imaginary roots $\alpha$ in the closed
negative Weyl chamber which are indivisible or satisfies $(\alpha|\alpha)<0$. A tuple of partitions $\mathbf m=\bigl(m_{j,\nu}\bigr)_{\substack{j=0,\dots,p\\\nu=1,\dots,n_j}}$ is said to be monotone if $m_{j,1}\ge m_{j,2}\ge \cdots\ge m_{j,n_j}$ for $j=0,\dots,p$. For example, we prove the exact estimate \begin{equation}
\ord\mathbf m\le 3|\idx\mathbf m|+6 \end{equation} for any fundamental tuple $\mathbf m$ in \S\ref{sec:basic}. Since we may assume \begin{equation}
p\le\tfrac12|\idx\mathbf m|+3 \end{equation} for a fundamental tuple $\mathbf m$, there exist only finite number of monotone fundamental tuples with a fixed index of rigidity. We list the fundamental tuples of the index of rigidity $0$ or $-2$ in Remark~\ref{rem:bas0} or Proposition~\ref{prop:bas2}, respectively.
Our results in \S\ref{sec:series}, \S\ref{sec:reduction} and \S\ref{sec:DS} give an integral expression and a power series expression of a local solution of the universal equation $P_{\mathbf m}u=0$ corresponding to the characteristic exponent whose multiplicity is free in the local monodromy. These expressions are in \S\ref{sec:exp}.
In \S\ref{sec:MM} we review the monodromy of solutions of a Fuchsian differential equation from the view point of our operations. The theorems in this section are given by \cite{DR, DR2, Kz, Ko}. In \S\ref{sec:Scott} we review Scott's lemma \cite{Sc} and related results with their proofs, which are elementary but important for the study of the irreducibility of the monodromy.
In \S\ref{sec:reddirect} we examine the condition for the decomposition $P_{\mathbf m}=P_{\mathbf m'}P_{\mathbf m''}$ of universal operators with or without fixing the exponents $\{\lambda_{j,\nu}\}$, which implies the reducibility of the equation $P_{\mathbf m}u=0$. In \S\ref{sec:redred} we study the value of spectral parameters which makes the equation reducible and obtain Theorem~\ref{thm:irred}. In particular we have a necessary and sufficient condition on characteristic exponents so that the monodromy of the solutions of the equation $P_{\mathbf m}u=0$ with a rigid spectral type $\mathbf m$ is irreducible, which is given in Corollary \ref{cor:irred} or Theorem \ref{thm:irrKac}. When $m_{j,1}\ge m_{j,2}\ge \cdots$ for any $j\ge0$, the condition equals \begin{equation}\label{eq:KacIrr}
(\Lambda(\lambda)|\alpha)\notin\mathbb Z\quad(\forall\alpha\in\Delta(\mathbf m)). \end{equation} Here $\Delta(\mathbf m)$ denotes the totality of real positive roots $\alpha$ such that $w_{\mathbf m}\alpha$ are negative and $w_{\mathbf m}$ is the element of $W_{\!\infty}$ with the minimal length so that $\alpha_0=w_{\mathbf m}\alpha_{\mathbf m}$ (cf.~Definition~\ref{def:wm} and Proposition~\ref{prop:wm} v)). The number of elements of $\Delta(\mathbf m)$ equals the length of $w_{\mathbf m}$, which is the minimal length of the expressions of $w_{\mathbf m}$ as products of simple reflections $s_\alpha$ with $\alpha\in\Pi$.
In \S\ref{sec:shift} we construct shift operators between rigid Fuchsian differential equations with the same spectral type such that the differences of the corresponding characteristic exponents are integers. Theorem~\ref{thm:shifm1} gives a recurrence relation of certain solutions of the rigid Fuchsian equations, which is a generalization of the formula \begin{equation}
c\bigl(F(a,b+1,c;x) - F(a,b,c;x)\bigr) = axF(a+1,b+1,c+1;x) \end{equation} and moreover gives relations between the universal operators and the shift operators in Theorem~\ref{thm:shifm1} and Theorem~\ref{thm:sftUniv}. In particular, Thorem~\ref{thm:sftUniv} gives a condition which assures that a universal operator is this shift operator.
The shift operators are useful for the study of Fuchsian differential equations when they are reducible because of special values of the characteristic exponents. Theorem~\ref{thm:isom} give a necessary condition and a sufficient condition so that the shift operator is bijective. In many cases we get a necessary and sufficient condition by this theorem. As an application of a shift operator we examine polynomial solutions of a rigid Fuchsian differential equation of Okubo type in \S\ref{sec:polyn}.
In \S\ref{sec:C1} we study a connection problem of the Fuchsian differential equation $P_{\mathbf m}u=0$. First we give Lemma~\ref{lem:conn} which describes the transformation of a connection coefficient under an addition and a middle convolution. In particular, for the equation $P_{\mathbf m}u=0$ satisfying $m_{0,n_0}=m_{1,n_1}=1$, Theorem~\ref{thm:GC} says that the connection coefficient $c(c_0\!:\!\lambda_{0,n_0}\!\rightsquigarrow\!c_1\!:\!\lambda_{1,n_1})$ from the local solution corresponding to the exponent $\lambda_{0,n_0}$ to that corresponding to $\lambda_{1,n_1}$ in the Riemann scheme \eqref{eq:IGRS} equals the connection coefficient of the reduced equation $P_{f\mathbf m}v=0$ up to the gamma factors which are explicitly calculated.
In particular, if the equation is rigid, Theorem~\ref{thm:c} gives the connection coefficient as a quotient of products of gamma functions and an easier non-zero term. For example, when $p=2$, the easier term doesn't appear and the connection coefficient has the universal formula \begin{equation}\label{eq:Icon}
c(c_0\!:\!\lambda_{0,n_0}\!\rightsquigarrow\!c_1\!:\!\lambda_{1,n_1})
=\frac
{\displaystyle\prod_{\nu=1}^{n_0-1}
\Gamma\bigl(\lambda_{0,n_0}-\lambda_{0,\nu}+1\bigr)
\cdot\prod_{\nu=1}^{n_1-1}
\Gamma\bigl(\lambda_{1,\nu}-\lambda_{1,n_1}\bigr)
}
{\displaystyle\prod_{\substack{\mathbf m'\oplus\mathbf m''=\mathbf m\\
m'_{0,n_0}=m''_{1,n_1}=1}}
\Gamma\bigl(|\{\lambda_{\mathbf m'}\}|\bigr)
}. \end{equation} Here the notation \eqref{eq:IFuchs} is used and $\mathbf m=\mathbf m'\oplus\mathbf m''$ means that $\mathbf m=\mathbf m'+\mathbf m''$ with rigid tuples $\mathbf m'$ and $\mathbf m''$. Moreover the number of gamma factors in the above denominator is equals to that of the numerator. The author conjectured this formula in 2007 and proved it in 2008 (cf.~\cite{O3}). The proof in \S\ref{sec:C1} is different from the original proof, which is explained in \S\ref{sec:C2}.
Suppose $p=2$, $\ord\mathbf m=2$, $m_{j,\nu}=1$ for $0\le j\le 2$ and $1\le\nu\le 2$, Then \eqref{eq:Icon} equals \begin{equation}
\frac{\Gamma(\lambda_{0,2}-\lambda_{0,1}+1)\Gamma(\lambda_{1,2}-\lambda_{1,1})}
{\Gamma(\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,1})
\Gamma(\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,2})}, \end{equation} which implies \eqref{eq:Gausssum} under \eqref{eq:RSGauss}.
The hypergeometric series $F(a,b,c;x)$ satisfies
$\lim_{k\to+\infty}F(a,b,c+k;x)=1$ if $|x|\le 1$, which obviously implies $\lim_{k\to+\infty}F(a,b,c+k;1)=1$. Gauss proves the summation formula \eqref{eq:Gausssum} by this limit formula and the recurrence relation $F(a,b,c;1)=\frac{(c-a)(c-b)}{c(c-a-b)}F(a,b,c+1;1)$. We have $\lim_{k\to+\infty} c(c_0\!:\!\lambda_{0,n_0}+k\!\rightsquigarrow\!c_1\!:\!\lambda_{1,n_1}-k)=1$ in the connection formula \eqref{eq:Icon} (cf.~Corollary~\ref{cor:C}). This suggests a similar limit formula for a local solution of a general Fuchsian differential equation, which is given in \S\ref{sec:estimate}.
In \S\ref{sec:C2} we propose a procedure to calculate the connection coefficient (cf.~Remark~\ref{rem:Cproc}), which is based on the calculation of its zeros and poles. This procedure is different from the proof of Theorem~\ref{thm:c} in \S\ref{sec:C1} and useful to calculate a certain connection coefficient between local solutions with multiplicities in eigenvalues of local monodromies. The coefficient is defined in Definition~\ref{def:GC}.
In \S\ref{sec:ex} we show many examples which explain our fractional calculus in this paper and also give concrete results of the calculus. In \S\ref{sec:basicEx} we list all the fundamental tuples whose indices of rigidity are not smaller than $-6$ and in \S\ref{sec:rigidEx} we list all the rigid tuples whose orders are not larger than 8, most of which are calculated by a computer program \texttt{okubo} explained in \S\ref{sec:okubo}. In \S\ref{sec:PoEx} and \S\ref{sec:GHG} we apply our fractional calculus to Jordan-Pochhammer equations and the hypergeometric family, respectively, which helps us to understand our unifying study of rigid Fuchsian differential equations. In \S\ref{sec:EOEx} we apply our fractional calculus to the even/odd family classified by \cite{Si} and most of the results there have been first obtained by the calculus.
In \S\ref{sec:4Ex}, \S\ref{sec:ord6Ex} and \S\ref{sec:Rob} we study the rigid Fuchsian differential equations of order not larger than 4 and those of order 5 or 6 and the equations belonging to 12 submaximal series classified by \cite{Ro}, respectively. Note that these 12 maximal series contain Yokoyama's list \cite{Yo}. In \S\ref{sec:RobEx}, we explain how we read the condition of irreducibility, connection coefficients, shift operators etc.\ of the corresponding differential equation from the data given in \S\ref{sec:4Ex}--\S\ref{sec:Rob}. In \S\ref{sec:TriEx}, we show some interesting identities of trigonometric functions as a consequence of the concrete value \eqref{eq:Icon} of connection coefficients. We examine Appell's hypergeometric equations in \S\ref{sec:ApEx} by our fractional calculus, which will be further discussed in another paper.
In \S\ref{sec:prob} we give some problems to be studied related to the results in this paper.
In \S\ref{sec:appendix} a theorem on Coxeter groups is given, which was proved by K.~Nuida through a private communication between the author and Nuida. The theorem is useful for the study of the difference of various reductions of Fuchsian differential equations (cf.~Proposition~\ref{prop:wm} v)). The author greatly thanks Nuida for allowing the author to put the theorem with its proof in this paper.
The author express his sincere gratitude to Kazuo Okamoto and Yoshishige Haraoka for the guidance to the subjects developed in this paper and to Kazuki Hiroe for reading the manuscript of this paper.
\section{Fractional operations}\label{sec:frac} \subsection{Weyl algebra} In this section we define several operations on a Weyl algebra. The operations are elementary or well-known but their combinations will be important.
Let $\mathbb C[x_1,\dots,x_n]$ denote the polynomial ring of $n$ independent variables $x_1,\dots,x_n$ over $\mathbb C$ and let $\mathbb C(x_1,\dots,x_n)$ denote the quotient field of $\mathbb C[x_1,\dots,x_n]$. \index{Weyl algebra} The \textsl{Weyl algebra} $W[x_1,\dots,x_n]$ of $n$ variables $x_1,\dots,x_n$ is the algebra over $\mathbb C$ generated by $x_1,\dots,x_n$ and $\frac{\partial}{\partial x_1},\dots,\frac{\partial}{\partial x_n}$ with the fundamental relation \begin{equation}
[x_i,x_j]=[\tfrac{\partial}{\partial x_i},\tfrac{\partial}{\partial x_j}]=0, \quad
[\tfrac{\partial}{\partial x_i},x_j]=\delta_{i,j}\qquad(1\le i,\,j\le n). \end{equation} We introduce a Weyl algebra $W[x_1,\dots,x_n][\xi_1,\dots,\xi_n]$ with parameters $\xi_1,\dots,\xi_N$ by \begin{equation*}
W[x_1,\dots,x_n][\xi_1,\dots,\xi_N]:=
\mathbb C[\xi_1,\dots,\xi_N]\underset{\mathbb C}\otimes W[x_1,\dots,x_n] \end{equation*} and put \begin{align*}
W[x_1,\dots,x_n;\xi_1,\dots,\xi_N]
&:= \mathbb C(\xi_1,\dots,\xi_N)\underset{\mathbb C}\otimes W[x_1,\dots,x_n],\\
W(x_1,\dots,x_n;\xi_1,\dots,\xi_N)
&:= \mathbb C(x_1,\dots,x_n,\xi_1,\dots,\xi_N)
\underset{\mathbb C[x_1,\dots,x_n]}\otimes W[x_1,\dots,x_n]. \end{align*} Here we have \begin{align} [x_i,\xi_\nu]&=[\tfrac{\partial}{\partial{x_i}},\xi_\nu]=0 \quad(1\le i\le n,\ 1\le\nu\le N),\\ \begin{split} \Bigl[\frac{\partial}{\partial{x_i}},\frac gf\Bigr]
&=\frac{\partial}{\partial x_i}\left(\frac gf\right)\\
&=\frac{\frac{\partial g}{\partial x_i}\cdot f
-g\cdot\frac{\partial f}{\partial{x_i}}}{f^2} \quad(f,\ g\in\mathbb C[x_1,\dots,x_n,\xi_1,\dots,\xi_N]) \end{split} \end{align} \index{00Wxxi@$W[x;\xi],\ W(x;\xi)$} and $[\frac{\partial}{\partial x_i},f]=\frac{\partial f}{\partial{x_i}}\in \mathbb C[x_1,\dots,x_n,\xi_1,\dots,\xi_N]$.
For simplicity we put $x=(x_1,\dots,x_n)$ and $\xi=(\xi_1,\dots,\xi_N)$ and the algebras $\mathbb C[x_1,\dots,x_n]$, \ $\mathbb C(x_1,\dots,x_n)$, \ $W[x_1,\dots,x_n][\xi_1,\dots,\xi_N]$, \ $W[x_1,\dots,x_n;\xi_1,\dots,\xi_N]$, $W(x_1,\dots,x_n;\xi_1,\dots,\xi_N)$ etc.\ are also denoted by $\mathbb C[x]$, $\mathbb C(x)$, $W[x][\xi]$, $W[x;\xi]$, $W(x;\xi)$ etc., respectively. Then \begin{equation}
\mathbb C[x,\xi]\subset W[x][\xi]\subset W[x;\xi]\subset W(x;\xi). \end{equation}
The element $P$ of $W(x;\xi)$ is uniquely written by \begin{align}
P = \sum_{\alpha=(\alpha_1,\dots,\alpha_n)\in {\mathbb Z}_{\ge 0}^n}
p_\alpha(x,\xi)
\frac{\partial^{\alpha_1+\cdots+\alpha_n}}{\partial x_1^{\alpha_1}
\cdots \partial x_n^{\alpha_n}}\qquad(p_\alpha(x,\xi)\in\mathbb C(x,\xi)). \end{align} \index{00Z@$\mathbb Z_{\ge 0},\ \mathbb Z_{>0}$} Here $\mathbb Z_{\ge0}=\{0,1,2,\dots\}$. Similar we will denote the set of positive integers by $\mathbb Z_{>0}$. If $P\in W(x;\xi)$ is not zero, the maximal integer $\alpha_1+\dots+\alpha_n$ satisfying $p_\alpha(x,\xi)\ne 0$ is called the \textsl{order} of $P$ and denoted by $\ord P$. \index{differential equation/operator!order}\index{00ord@$\ord$} If $P\in W[x;\xi]$, $p_\alpha(x,\xi)$ are polynomials of $x$ with coefficients in $\mathbb C(\xi)$ and the maximal degree of $p_\alpha(x,\xi)$ as polynomials of $x$ is called the \textsl{degree} of $P$ and denoted by $\deg P$. \index{differential equation/operator!degree}
\subsection{Laplace and gauge transformations and reduced representatives} First we will define some fundamental operations on $W[x;\xi]$. \begin{defn}\label{def:Lap} \textrm{i) }\index{00R@$\Red$} For a non-zero element $P\in W(x;\xi)$ we choose an element $(\mathbb C(x,\xi)\setminus\{0\})P\cap W[x;\xi]$ with the minimal degree and denote it by $\Red P$ and call it a \textsl{reduced representative} of $P$. If $P=0$, we put $\Red P=0$. Note that $\Red P$ is determined up to multiples by non-zero elements of $\mathbb C(\xi)$. \index{reduced representative}
\textrm{ii)\ }\index{00L@$\Lap$} For a subset $I$ of $\{1,\dots,n\}$ we define an automorphism $\Lap_I$ of $W[x;\xi]$: \begin{align}
\Lap_I(\tfrac{\partial}{\partial x_i})=\begin{cases}
x_i&(i\in I)\\
\tfrac{\partial}{\partial x_i}&(i\not\in I)
\end{cases},\ \
\Lap_I(x_i) =\begin{cases}
-\tfrac{\partial}{\partial x_i}&(i\in I)\\
x_i&(i\not\in I)
\end{cases}
\text{ and }\Lap_I(\xi_\nu)=\xi_\nu. \end{align} We put $\Lap=\Lap_{\{1,\dots,n\}}$ and call $\Lap$ the \textsl{Laplace transformation} of $W[x;\xi]$. \index{Laplace transformation}
\textrm{iii)\ } Let $W\!_L(x;\xi)$ be the algebra isomorphic to $W(x;\xi)$ which is defined by the Laplace transformation \begin{equation}\label{eq:WL}
\Lap: W(x;\xi)\ \overset{\sim}\to\ W\!_L(x;\xi)
\ \overset{\sim}\to\ W(x;\xi). \end{equation} For an element $P\in W\!_L(x;\xi)$ we define \begin{equation}
\Red_L(P):= \Lap^{-1}\circ \Red \circ \Lap(P). \end{equation} \end{defn}
Note that the element of $W\!_L(x;\xi)$ is a finite sum of products of elements of $\mathbb C[x]$ and rational functions of $(\frac{\partial}{\partial x_1},\dots,\frac{\partial}{\partial x_n},\xi_1,\dots,\xi_N)$.
We will introduce an automorphism of $W(x;\xi)$. \begin{defn}[gauge transformation]\label{def:gauge}\index{gauge transformation} \index{00Ad@$\Ad$, $\Adei$} Fix an element $(h_1,\dots,h_n)\in\mathbb C(x,\xi)^n$ satisfying \begin{equation}
\frac{\partial h_i}{\partial x_j} = \frac{\partial h_j}{\partial x_i}
\qquad(1\le i,\ j\le n). \end{equation} We define an automorphism $\Adei(h_1,\dots,h_n)$ of $W(x;\xi)$ by \begin{equation} \begin{aligned}
\Adei(h_1,\dots,h_n)(x_i)&=x_i&&(i=1,\dots,n),\\
\Adei(h_1,\dots,h_n)(\tfrac{\partial}{\partial x_i})
&=\tfrac{\partial}{\partial x_i}-h_i&&(i=1,\dots,n),\\
\Adei(h_1,\dots,h_n)(\xi_\nu)&=\xi_\nu
&&(\nu=1,\dots,N). \end{aligned} \end{equation} Choose functions $f$ and $g$ satisfying $\frac{\partial g}{\partial x_i}=h_i$ for $i=1,\dots,n$ and put $f=e^g$ and \begin{align}
\Ad(f)&=\Ade(g)=\Adei(h_1,\dots,h_n). \end{align} \end{defn}
We will define a homomorphism of $W(x;\xi)$. \index{coordinate transformation} \begin{defn}[coordinate transformation]\label{def:coord} \index{coordinate transformation} \index{00T@$T_{\phi}^*$} Let $\phi=(\phi_1,\dots,\phi_n)$ be an element of $\mathbb C(x_1,\dots,x_m,\xi)^n$ such that the rank of the matrix \begin{equation}
\Phi:=\Bigl(\frac{\partial \phi_j}{\partial x_i}
\Bigr)_{\substack{1\le i\le m \\ 1\le j\le n}} \end{equation} equals $n$ for a generic point $(x,\xi)\in\mathbb C^{m+N}$. Let $\Psi=\bigl(\psi_{i,j}(x,\xi)\bigr)_{\substack{1\le i\le n\\1\le j\le m}}$ be an left inverse of $\Phi$, namely, $\Psi\Phi$ is an identity matrix of size $n$ and $m\ge n$. Then a homomorphism $T^*_{\phi}$ from $W(x_1,\dots,x_n;\xi)$ to $W(x_1,\dots,x_m;\xi)$ is defined by \begin{equation}
\begin{aligned}
T^*_\phi(x_i)&=\phi_i(x)&&(1\le i\le n),\\
T^*_\phi(\tfrac{\partial}{\partial x_i})&=\sum_{j=1}^m
\psi_{i,j}(x,\xi)\tfrac{\partial}{\partial x_j}
&&(1\le i\le n).
\end{aligned} \end{equation} If $m>n$, we choose linearly independent elements $h_\nu=(h_{\nu,1},\dots,h_{\nu,m})$ of $\mathbb C(x,\xi)^m$ for $\nu=1,\dots,m-n$ such that $\psi_{i,1}h_{\nu,1}+\cdots+\psi_{i,m}h_{\nu,m}=0$ for $i=1,\dots,n$ and $\nu=1,\dots,m-n$ and put \begin{equation}
\mathcal K^*(\phi):=\sum_{\nu=1}^{m-n}\mathbb C(x,\xi)\sum_{j=1}^m
h_{\nu,j}\tfrac{\partial}{\partial x_j}\in W(x;\xi). \end{equation} \end{defn} The meaning of these operations are clear as follows. \begin{rem} Let $P$ be an element of $W(x;\xi)$ and let $u(x)$ be an analytic solution of the equation $Pu=0$ with a parameter $\xi$. Then under the notation in Definitions~\ref{def:Lap}--\ref{def:gauge}, we have $(\Red P)u(x)=\bigl(\Ad(f)(P)\bigr)(f(x)u(x))=0$. Note that $\Red P$ is defined up to the multiplications of non-zero elements of $\mathbb C(\xi)$.
If a Laplace transform\index{Laplace transformation} \begin{equation}
(\mathcal R_k u)(x)
=\int_C e^{-x_1t_1-\cdots-x_kt_k}u(t_1,\dots,t_k,x_{k+1},\dots,x_n)
dt_1\cdots dt_k \end{equation} of $u(x)$ is suitably defined, then $\bigl(\Lap_{\{1,\dots,k\}}(\Red P)\bigr)(\mathcal R_ku)=0$, which follows from the equalities $\frac{\partial \mathcal R_ku}{\partial x_i}=\mathcal R_k(-x_iu)$ and $0=\int_C\frac{\partial}{\partial t_i}\bigl(e^{-x_1t_1-\cdots-x_kt_k} u(t,x_{k+1},\ldots)\bigr)dt =-x_i\mathcal R_ku+\mathcal R_k(\frac{\partial u}{\partial t_i})$ for $i=1,\dots,k$. Moreover we have \[
f(x)\mathcal R_k \Red Pu=f(x)\bigl(L_{\{1,\dots,k\}}(\Red P)\bigr)(\mathcal R_k u)
=\bigl(\Ad(f)L_{\{1,\dots,k\}}(\Red P)\bigr)\bigl(f(x)\mathcal R_k u\bigr). \] Under the notation of Definition~\ref{def:coord}, we have $T_\phi^*(P) u(\phi_1(x),\dots,\phi_n(x))=0$ and $Q u\bigl(\phi_1(x),\dots,\phi_n(x)\bigr)=0$ for $Q\in\mathcal K^*(\phi)$.
Another transformation of $W[x;\xi]$ based on an integral transformation frequently used will be given in Proposition~\ref{prop:1-tx}. \end{rem}
We introduce some notation for combinations of operators we have defined. \begin{defn} \index{00Rad@$\RAd$, $\RAdL$} Retain the notation in Definition \ref{def:Lap}--\ref{def:coord} and recall that $f=e^g$ and $h_i=\frac{\partial g}{\partial x_i}$. \begin{align}
\RAd(f)
&\phantom{:}=
\RAde(g)=\RAdei(h_1,\dots,h_n):=\Red\circ\Adei(h_1,\dots,h_n),\\
\begin{split}
\AdL(f)&\phantom{:}=\AdeL(h)=\AdeiL(h_1,\dots,h_n)\\
&:=\Lap^{-1}\circ\Adei(h_1,\dots,h_n)\circ\Lap, \end{split} \\
\begin{split}
\RAdL(f)&\phantom{:}=\RAdeL(h)=\RAdeiL(h_1,\dots,h_n)\\
&:=\Lap^{-1}\circ\RAdei(h_1,\dots,h_n)\circ\Lap, \end{split} \\
\Ad(\partial_{x_i}^{\mu})&:=\Lap^{-1}\circ\Ad(x_i^\mu)\circ \Lap, \\
\RAd(\partial_{x_i}^{\mu})&:=\Lap^{-1}\circ\RAd(x_i^\mu)\circ \Lap. \end{align} Here $\mu$ is a complex number or an element of $\mathbb C(\xi)$ and $\Ad(\partial_{x_i}^{\mu})$ defines an endomorphism of $W_L(x;\xi)$. \end{defn}
We will sometimes denote $\frac{\partial}{\partial x_i}$ by $\partial_{x_i}$ or $\partial_i$ for simplicity. If $n=1$, we usually denote $x_1$ by $x$ and $\frac{\partial}{\partial x_1}$ by $\frac d{dx}$ or $\partial_x$ or $\partial$. We will give some examples.
Since the calculation $\Ad\bigl(x^{-\mu})\partial=x^{-\mu}\circ \partial\circ x^{\mu} =x^{-\mu}(x^{\mu}\partial + \mu x^{\mu-1})=\partial + \mu x^{-1}$ is allowed, the following calculation is justified by the isomorphism \eqref{eq:WL}: \begin{align*}
\Ad(\partial^{-\mu})x^m&=\partial^{-\mu}\circ x^m\circ \partial^\mu\\
&=(x^m\partial^{-\mu}+\tfrac{(-\mu)m}{1!}x^{m-1}\partial^{-\mu-1}+ \tfrac{(-\mu)(-\mu-1)m(m-1)}{2!}x^{m-2}\partial^{-\mu-2}\\
&\qquad{}
+\cdots+\tfrac{(-\mu)(-\mu-1)\cdots(-\mu-m+1)m!}{m!}\partial^{-\mu-m})\partial^\mu\\
&=\sum_{\nu=0}^m(-1)^\nu(\mu)_\nu\binom{m}{\nu}x^{m-\nu}\partial^{-\nu}. \end{align*} This calculation is in a ring of certain pseudo-differential operators according to Leibniz's rule. In general, we may put $\Ad(\partial^{-\mu})P=\partial^{-\mu}\circ P\circ \partial^\mu$ for $P\in W[x;\xi]$ under Leibniz's rule. Here $m$ is a positive integer and we use the notation \index{000gammak@$(\gamma)_k,\ (\mu)_\nu$} \begin{equation}
(\mu)_\nu:=\prod_{i=0}^{\nu-1}(\mu+i),\quad
\binom{m}{\nu}:=\frac{\Gamma(m+1)}{\Gamma(m-\nu+1)\Gamma(\nu+1)} =\frac{m!}{(m-\nu)!\nu!}. \end{equation} \subsection{Examples of ordinary differential operators} In this paper we mainly study ordinary differential operators. We give examples of the operations we have defined, which are related to classical differential equations. \begin{exmp} [$n=1$] For a rational function $h(x,\xi)$ of $x$ with a parameter $\xi$ we denote by $\int h(x,\xi)dx$ the function $g(x,\xi)$ satisfying $\frac{d}{dx}g(x,\xi)=h(x,\xi)$. Put $f(x,\xi)=e^{g(x,\xi)}$ and define \index{000thetaz@$\vartheta$} \begin{equation}
\vartheta := x\frac{d}{dx}. \end{equation} Then we have the following identities. \begin{align} \Adei(h)\partial &= \partial - h = \Ad(e^{\int h(x)dx})\partial
= e^{\int h(x)dx}\circ \partial \circ e^{-\int h(x)dx}, \allowdisplaybreaks\\
\Ad(f)x &=x,\quad \AdL(f)\partial = \partial,\\ \Ad(\lambda f)&=\Ad(f)\quad \AdL(\lambda f)=\AdL(f),\\ \Ad(f)\partial&=\partial - h(x,\xi)\ \Rightarrow\ \AdL(f)x=x + h(\partial,\xi), \allowdisplaybreaks\\
\Ad\bigl((x-c)^\lambda\bigr)
&=\Ade\bigl(\lambda\log(x-c)\bigr)
=\Adei\bigl(\tfrac{\lambda}{x-c}\bigr), \allowdisplaybreaks\\
\Ad\bigl((x-c)^\lambda\bigr)x&=x,\quad
\Ad\bigl((x-c)^\lambda\bigr)\partial=\partial - \tfrac{\lambda}{x-c}, \allowdisplaybreaks\\
\RAd\bigl((x-c)^\lambda\bigr)\partial
&=\Ad\bigl((x-c)^\lambda\bigr)\bigl((x-c)\partial\bigr)
=(x-c)\partial - \lambda, \allowdisplaybreaks\\ \begin{split}
\RAdL\bigl((x-c)^\lambda\bigr)x&=
L^{-1}\circ \RAd\bigl((x-c)^\lambda\bigr)(-\partial)\\
&= L^{-1}\bigl((x-c)(-\partial)+\lambda\bigr)\\
&=
(\partial-c)x+\lambda=x\partial - cx +1+\lambda, \end{split} \allowdisplaybreaks\\
\RAdL\bigl((x-c)^\lambda\bigr)\partial&=\partial,\quad
\RAdL\bigl((x-c)^\lambda\bigr)\bigl((\partial-c)x\bigr)
=(\partial-c)x+\lambda, \allowdisplaybreaks\\ \Ad(\partial^{\lambda})\vartheta &= \AdL(x^{\lambda})\vartheta=\vartheta+\lambda, \allowdisplaybreaks\index{000deltas@$\partial^{\lambda}$}\\ \Ad\bigl(e^{\tfrac {\lambda(x-c)^m}{m}}\bigr)x &= x,\quad \Ad\bigl(e^{\tfrac {\lambda(x-c)^m}{m}}\bigr)\partial = \partial - \lambda(x-c)^{m-1}, \allowdisplaybreaks\\
\RAdL\bigl(e^{\frac {\lambda(x-c)^m}{m}}\bigr)x&= \begin{cases}
x+\lambda(\partial-c)^{m-1}&(m\ge1),\\
(\partial-c)^{1-m}x + \lambda&(m\le -1), \end{cases}\allowdisplaybreaks\\
T_{(x-c)^m}^*(x)&=(x-c)^m,\quad T_{(x-c)^m}^*(\partial)=\tfrac1m (x-c)^{1-m}\partial. \end{align} Here $m$ is a non-zero integer and $\lambda$ is a non-zero complex number. \end{exmp}
Some operations are related to Katz's operations defined by \cite{Kz}. The operation $\RAd\bigl((x-c)^\mu\bigr)$ corresponds to the \textsl{addition} given in \cite{DR} and the operator \begin{equation}\index{00mc@$mc_\mu$}
mc_\mu:=\RAd(\partial^{-\mu})=\RAdL(x^{-\mu}) \end{equation} corresponds to Katz's \textsl{middle convolution} and the Euler transformation or the Riemann-Liouville integral \index{Euler transformation}\index{Riemann-Liouville integral} \index{fractional derivation} (cf.~\cite[\S5.1]{Kh}) or the fractional derivation \index{00Ic@$I_c^\mu,\ \tilde I_c^\mu$} \begin{equation}\label{eq:fracdif}
(I^\mu_c(u))(x) = \frac1{\Gamma(\mu)}\int_c^x u(t)(x-t)^{\mu-1}dt. \end{equation} \index{addition}\index{middle convolution} Here $c$ is suitably chosen. In most cases, $c$ is a singular point of the multi-valued holomorphic function $u(x)$. The integration may be understood through an analytic continuation with respect to a parameter or in the sense of generalized functions. When $u(x)$ is a multi-valued holomorphic function on the punctured disk around $c$, we can define the complex integral \begin{equation}\label{eq:EuPh}
(\tilde I_c^\mu (u))(x)
:= \int^{(x+,c+,x-,c-)}\!\!\!\!u(z)(x-z)^{\mu-1}dz\quad \ \ \raisebox{5pt}{\begin{xy} (-2,-7) *{c}="c", (32,-7) *{x}="x", (0,0) *{\times} *\cir<3.0mm>{l_d} *\cir<4.5mm>{l_dr}, (30,0) *{\times} *\cir<3.0mm>{r_u} *\cir<4.5mm>{ur_lu}, (12,-7) *{\text{starting point}} \ar@{<-{*}} (2.8,0) ; (14,0); \ar@{-} (14,0); (27.2,0); \ar@{-} (0,-2.85); (13.5,-2,85); \ar@{->} (13.5,-2.85); (27,-2.85); \ar@{<-} (30,2.85) ; (3,2.85); \ar@{->} (0,-4.3) ; (24,-4.3): \ar@{-} (24,-4.3) ; (27,3); \ar@{.>} "c";(0,0), \ar@{.>} "x";(30,0), \ar@{<.} (2.8,0);(1,-7) \end{xy}} \end{equation} through Pochhammer contour\index{Pochhammer contour} $(x+,c+,x-,c-)$ along a double loop circuit (cf.\ \cite[12.43]{WW}). If $(z-c)^{-\lambda}u(z)$ is a meromorphic function in a neighborhood of the point $c$, we have \begin{equation}
(\tilde I_c^\mu(u))(x)=\bigl(1-e^{2\pi\lambda\sqrt{-1}}\bigr)
\bigl(1-e^{2\pi\mu\sqrt{-1}}\bigr)
\int_c^xu(t)(x-t)^{\mu-1}dt. \end{equation} For example, we have \begin{align}
\begin{split}
I_c^\mu\bigl((x-c)^\lambda\bigr)
&=\frac1{\Gamma(\mu)}\int_c^x (t-c)^\lambda(x-t)^{\mu-1}dt\\
&=\frac{(x-c)^{\lambda+\mu}}{\Gamma(\mu)}\int_0^1 s^\lambda(1-s)^{\mu-1}ds
\quad(x-t=(1-s)(x-c))\\
&=\frac{\Gamma(\lambda+1)}{\Gamma(\lambda+\mu+1)}(x-c)^{\lambda+\mu},
\end{split}\\
\tilde I_c^\mu\bigl((x-c)^\lambda\bigr)
&=\frac{4\pi^2e^{\pi(\lambda+\mu)\sqrt{-1}}}
{\Gamma(-\lambda)\Gamma(1-\mu)\Gamma(\lambda+\mu+1)}(x-c)^{\lambda+\mu+1}. \end{align} For $k\in\mathbb Z_{\ge0}$ we have \begin{align}
\tilde I_c^\mu\bigl((x-c)^k\log(x-c)\bigr)=
\frac{-4\pi^2k!e^{\pi\lambda\sqrt{-1}}}{\Gamma(1-\mu)\Gamma(\mu+k+1)} (x-c)^{\mu+k+1}. \end{align}
We note that since \begin{align*}
\tfrac{d}{dt}\big(u(t)(x-t)^{\mu-1}\bigr)
&=u'(t)(x-t)^{\mu-1}
-\tfrac{d}{dx}\bigl(u(t)(x-t)^{\mu-1}\bigr)\\ \intertext{and}
\tfrac{d}{dt}\big(u(t)(x-t)^{\mu}\bigr)
&=u'(t)(x-t)^{\mu}-u(t)\tfrac{d}{dx}(x-t)^{\mu}\\
&=xu'(t)(x-t)^{\mu-1} - tu'(t)(x-t)^{\mu-1}
-\mu u(t)(x-t)^{\mu-1}, \end{align*} we have \begin{equation}\label{eq:Icp}
\begin{split}
I_c^\mu(\partial u)&=\partial I_c^\mu(u),\\
I_c^\mu(\vartheta u)&=(\vartheta - \mu)I_c^\mu(u).
\end{split} \end{equation}
\begin{rem}\label{rem:defIc} {\rm i)} \ The integral \eqref{eq:fracdif} is naturally well-defined and the equalities \eqref{eq:Icp} are valid if $\RE\lambda > 1$ and $\lim_{x\to c}x^{-1}u(x)=0$. Depending on the definition of $I_c^\lambda$, they are also valid in many cases, which can be usually proved in this paper by analytic continuations with respect to certain parameters (for example, cf.~\eqref{eq:IcP}). Note that \eqref{eq:Icp} is valid if $I_c^\mu$ is replaced by $\tilde I_c^\mu$ defined by \eqref{eq:EuPh}.
{\rm ii)} Let $\epsilon$ be a positive number and let $u(x)$ be a holomorphic function on \[
U^+_{\epsilon,\theta}:=\{x\in\mathbb C\,;\,|x-c|<\epsilon\text{ and }
e^{-i\theta}(x-c)\notin(-\infty,0]\}. \] Suppose that there exists a positive number $\delta$
such that $|u(x)(x-c)^{-k}|$ is bounded on
$\{x\in U_{\epsilon.\theta}^+\,;\,|\Arg(x-c)-\theta|<\delta\}$ for any $k>0$. Note that the function $Pu(x)$ also satisfies this estimate for $P\in W[x]$. Then the integration \eqref{eq:fracdif} is defined along a suitable path $C:\gamma(t)$ $(0\le t\le 1$) such that $\gamma(0)=c$, $\gamma(1)=x$ and
$|\Arg\bigl(\gamma(t)-c\bigr)-\theta|<\delta$ for $0<t<\frac12$ and the equalities \eqref{eq:Icp} are valid. \end{rem}
\begin{exmp}\label{ex:midconv} We apply additions, middle convolutions and Laplace transformations to the trivial ordinary differential equation \begin{equation}
\frac{du}{dx} = 0, \end{equation} which has the solution $u(x)\equiv 1$.
\textrm{i)\ }(Gauss hypergeometric equation). \index{hypergeometric equation/function!Gauss} Put \begin{equation}
\begin{split}
P_{\lambda_1,\lambda_2,\mu}:\!&=
\RAd\bigl(\partial^{-\mu}\bigr)\circ\RAd\bigl(
x^{\lambda_1}(1-x)^{\lambda_2}\bigr)\partial\\
&= \RAd(\partial^{-\mu}\bigr)\circ\Red
(\partial-\tfrac{\lambda_1}{x}+\tfrac{\lambda_2}{1-x})\\
&=\RAd(\partial^{-\mu}\bigr)\bigl(x(1-x)\partial
-\lambda_1(1-x)+\lambda_2 x\bigr)\\
&=\RAd(\partial^{-\mu}\bigr)
\bigl((\vartheta-\lambda_1)-x(\vartheta-\lambda_1-\lambda_2)\bigr)\\
&=\Ad(\partial^{-\mu}\bigr)
\bigl((\vartheta+1-\lambda_1)\partial-(\vartheta+1)(\vartheta-\lambda_1-\lambda_2)\bigr)\\
&=(\vartheta+1-\lambda_1-\mu)\partial-(\vartheta+1-\mu)(\vartheta-\lambda_1-\lambda_2-\mu)\\
&=(\vartheta+\gamma)\partial-(\vartheta+\beta)(\vartheta+\alpha)\\
&=x(1-x)\partial^2+\bigl(\gamma-(\alpha+\beta+1)x\bigr)\partial-\alpha\beta
\end{split} \end{equation} with \begin{equation} \begin{cases}
\alpha=-\lambda_1-\lambda_2-\mu, \\
\beta=1-\mu,\\
\gamma=1-\lambda_1-\mu. \end{cases} \end{equation} We have a solution \begin{equation}\label{eq:Gmid}
\begin{split}
u(x)&=I_0^\mu(x^{\lambda_1}(1-x)^{\lambda_2})\\
&=\frac1{\Gamma(\mu)}
\int_0^x t^{\lambda_1}(1-t)^{\lambda_2}(x-t)^{\mu-1}dt\\
&=\frac{x^{\lambda_1+\mu}}{\Gamma(\mu)}
\int_0^1 s^{\lambda_1}(1-s)^{\mu-1}(1-xs)^{\lambda_2}ds
\quad(t=xs)\\
&=\frac{\Gamma(\lambda_1+1)x^{\lambda_1+\mu}}{\Gamma(\lambda_1+\mu+1)}
F(-\lambda_2,\lambda_1+1,\lambda_1+\mu+1;x)\\
&=\frac{\Gamma(\lambda_1+1)x^{\lambda_1+\mu}
(1-x)^{\lambda_2+\mu}}{\Gamma(\lambda_1+\mu+1)}
F(\mu,\lambda_1+\lambda_2+\mu,\lambda_1+\mu+1;x)\\
&=\frac{\Gamma(\lambda_1+1)x^{\lambda_1+\mu}(1-x)^{-\lambda_2}}
{\Gamma(\lambda_1+\mu+1)}
F(\mu,-\lambda_2,\lambda_1+\mu+1;\frac x{x-1})
\end{split} \end{equation} of the Gauss hypergeometric equation $P_{\lambda_1,\lambda_2,\mu}u=0$ with the Riemann scheme \begin{equation}
\begin{Bmatrix}
x=0 & 1 & \infty\\
0 &0&1-\mu&\!\!;\ x\\
\lambda_1+\mu &\lambda_2+\mu&-\lambda_1-\lambda_2-\mu
\end{Bmatrix}, \end{equation} which is transformed by the middle convolution $mc_\mu$ from the Riemann scheme\index{Riemann scheme} \[
\begin{Bmatrix}
x= 0 & 1 & \infty\\
\lambda_1 & \lambda_2 & -\lambda_1-\lambda_2&\!\!;x
\end{Bmatrix} \] of $x^{\lambda_1}(1-x)^{\lambda_2}$. Here using Riemann's $P$ symbol, we note that \begin{align*}
&P\begin{Bmatrix}
x=0 & 1 & \infty\\
0 &0&1-\mu&\!\!;\ x\\
\lambda_1+\mu &\lambda_2+\mu&-\lambda_1-\lambda_2-\mu
\end{Bmatrix}\allowdisplaybreaks\\
&=x^{\lambda_1+\mu}
P\begin{Bmatrix}
x=0 & 1 & \infty\\
-\lambda_1-\mu &0&\lambda_1+1&\!\!;\ x\\
0 &\lambda_2+\mu&-\lambda_2
\end{Bmatrix}\allowdisplaybreaks\\
&=x^{\lambda_1+\mu}(1-x)^{\lambda_2+\mu}
P\begin{Bmatrix}
x=0 & 1 & \infty\\
-\lambda_1-\mu &-\lambda_2-\mu&\lambda_1+\lambda_2+\mu+1&\!\!;\ x\\
0 &0&\mu
\end{Bmatrix}\allowdisplaybreaks\\
&=x^{\lambda_1+\mu}
P\begin{Bmatrix}
x=0 & 1 & \infty\\
-\lambda_1-\mu &\lambda_1+1&0&\!\!;\ \dfrac x{x-1}\\
0 &-\lambda_2 &\lambda_2+\mu
\end{Bmatrix}\allowdisplaybreaks\\
&=x^{\lambda_1+\mu}(1-x)^{-\lambda_2}
P\begin{Bmatrix}
x=0 & 1 & \infty\\
-\lambda_1-\mu &\lambda_1+\lambda_2+1&
-\lambda_2&\!\!;\ \dfrac x{x-1}\\
0 & 0 &\mu
\end{Bmatrix}. \end{align*} In general, the Riemann scheme and its relation to $mc_\mu$ will be studied in \S\ref{sec:index} and the symbol `$P$' will be omitted for simplicity.
The function $u(x)$ defined by \eqref{eq:Gmid} corresponds to the characteristic exponent $\lambda_1+\mu$ at the origin and depends meromorphically on the parameters $\lambda_1,\lambda_2$ and $\mu$. The local solutions corresponding to the characteristic exponents $\lambda_2+\mu$ at $1$ and $-\lambda_1-\lambda_2-\mu$ at $\infty$ are obtained by replacing $I_0^\mu$ by $I_1^\mu$ and $I_\infty^\mu$, respectively.
When we apply $\Ad(x^{\lambda_1'}(x-1)^{\lambda_2'})$ to $P_{\lambda_1,\lambda_2,\mu}$, the resulting Riemann scheme is \begin{equation}
\begin{Bmatrix}
x=0 & 1 & \infty\\
\lambda_1' &\lambda_2'&1-\lambda_1'-\lambda_2'-\mu&\!\!;\ x\\
\lambda_1+\lambda_1'+\mu &\lambda_2+\lambda_2'+\mu&-\lambda_1-\lambda_2-\lambda_1'-\lambda_2'-\mu,
\end{Bmatrix}. \end{equation} Putting $\lambda_{1,1}=\lambda_1'$, $\lambda_{1,2}=\lambda_1+\lambda_1'+\mu$, $\lambda_{2,1}=\lambda_2'$, $\lambda_{2,2}=\lambda_2+\lambda_2'+\mu$, $\lambda_{0,1}=1-\lambda_1'-\lambda_2'-\mu$ and $\lambda_{0,2}=-\lambda_1-\lambda_2-\lambda_1'-\lambda_2'-\mu$, we have the Fuchs relation \begin{equation}
\lambda_{0,1}+\lambda_{0,2}+\lambda_{1,1}+\lambda_{1,2}+\lambda_{2,1}
+\lambda_{2,2}=1 \end{equation} and the corresponding operator \begin{equation}\label{eq:GH2} \begin{split}
P_{\lambda}&=x^2(x-1)^2\partial^2+x(x-1)
\bigl((\lambda_{0,1}+\lambda_{0,2}+1)x+\lambda_{1,1}+\lambda_{1,2}-1\bigr)
\partial\\ &{}\quad+\lambda_{0,1}\lambda_{0,2}x^2+
(\lambda_{2,1}\lambda_{2,2}-\lambda_{0,1}\lambda_{0,2}-\lambda_{1,1}\lambda_{1,2} )x+\lambda_{1,1}\lambda_{1,2} \end{split}\end{equation} has the Riemann scheme \begin{equation}\label{eq:GRSGG}
\begin{Bmatrix}
x=0 & 1 & \infty\\
\lambda_{0,1} &\lambda_{1,1}&\lambda_{2,1}&\!\!;\ x\\
\lambda_{0,2} &\lambda_{1,2}&\lambda_{2,2}
\end{Bmatrix}. \end{equation} By the symmetry of the transposition $\lambda_{j,1}$ and $\lambda_{j,2}$ for each $j$, we have integral representations of other local solutions.
\textrm{ii)\ }(Airy equations).\index{Airy equation} For a positive integer $m$ we put \begin{equation} \begin{split}
P_m &:= \Lap\circ\Ad(e^\frac{x^{m+1}}{m+1})\partial\\
&\phantom{:}= \Lap(\partial - x^m) = x - (-\partial)^m. \end{split} \end{equation} Thus the equation \begin{equation}
\frac{d^mu}{dx^m}-(-1)^mxu=0 \end{equation} has a solution \begin{equation}
u_j(x) = \int_{C_j}\exp\left(\frac{z^{m+1}}{m+1}-xz\right)
dz\qquad(0\le j\le m), \end{equation} where the path $C_j$ of the integration is \begin{equation*}
C_j: z(t)= e^{\frac{(2j-1)\pi\sqrt{-1}}{m+1}-t}
+ e^{\frac{(2j+1)\pi\sqrt{-1}}{m+1}+t} \quad(-\infty<t<\infty). \end{equation*} Here we note that $u_0(x)+\cdots+u_m(x)=0$. The equation has the symmetry under the rotation $x\mapsto e^{\frac{2\pi\sqrt{-1}}{m+1}}x$.
\textrm{iii)\ }(Jordan-Pochhammer equation).\index{Jordan-Pochhammer} For $\{c_1,\dots,c_p\}\in\mathbb C\setminus\{0\}$ put \begin{align*}
P_{\lambda_1,\dots,\lambda_p,\mu} :\!&=
\RAd(\partial^{-\mu})\circ\RAd\Bigl(\prod_{j=1}^p (1-c_jx)^{\lambda_j}\Bigr)\partial\\
&=\RAd(\partial^{-\mu})\circ
\Red\Bigl(\partial+\sum_{j=1}^p\frac{c_j\lambda_j}{1-c_jx}\Bigr)\\
&=\RAd(\partial^{-\mu})\Bigl(p_0(x)\partial+q(x)\Bigr)\\
&=\partial^{-\mu+p-1}\Bigl(p_0(x)\partial+q(x)\Bigr)\partial^{\mu}
=\sum_{k=0}^pp_k(x)\partial^{p-k} \intertext{with}
p_0(x)&=\prod_{j=1}^p(1-c_jx),\quad
q(x)=p_0(x)\sum_{j=1}^p\frac{c_j\lambda_j}{1-c_jx},\\
p_k(x)&=\binom{-\mu+p-1}{k}
p_0^{(k)}(x) +
\binom{-\mu+p-1}{k-1}
q^{(k-1)}(x),\\
\binom{\alpha}{\beta}
:\!&=
\frac{\Gamma(\alpha+1)}{\Gamma(\beta+1)\Gamma(\alpha-\beta+1)}
\quad(\alpha,\beta\in\mathbb C). \end{align*} We have solutions \[
u_j(x)=\frac{1}{\Gamma(\mu)}\int_{\frac1{c_j}}^x\prod_{\nu=1}^p
(1-c_\nu t)^{\lambda_\nu}(x-t)^{\mu-1}dt
\quad(j=0,1,\dots,p,\ c_0=0) \] of the Jordan-Pochhammer equation $P_{\lambda_1,\dots,\lambda_p,\mu}u=0$ with the Riemann scheme \index{000lambda@$[\lambda]_{(k)}$} \begin{equation}
\begin{Bmatrix}
x=\frac1{c_1} & \cdots & \frac1{c_p} & \infty\\
[0]_{(p-1)} & \cdots & [0]_{(p-1)}& [1-\mu]_{(p-1)}&\!\!;\,x\\
\lambda_1+\mu&\cdots & \lambda_p+\mu & -\lambda_1-\dots-\lambda_p-\mu
\end{Bmatrix}. \end{equation}
Here and hereafter we use the notation \begin{equation}\label{eq:mult}
[\lambda]_{(k)}:=
\begin{pmatrix}
\lambda \\ \lambda+1 \\ \vdots \\ \lambda+k-1
\end{pmatrix} \end{equation} for a complex number $\lambda$ and a non-negative integer $k$. If the component $[\lambda]_{(k)}$ is appeared in a Riemann scheme, it means the corresponding local solutions with the exponents $\lambda+\nu$ for $\nu=0,\dots,k-1$ have a semisimple local monodromy when $\lambda$ is generic. \end{exmp} \subsection{Ordinary differential equations}\label{sec:ODE} We will study the ordinary differential equation \begin{equation}\label{eq:M}
\mathcal M : Pu = 0 \end{equation} with an element $P\in W(x;\xi)$ in this paper. The solution $u(x,\xi)$ of $\mathcal M$ is at least locally defined for $x$ and $\xi$ and holomorphically or meromorphically depends on $x$ and $\xi$. Hence we may replace $P$ by $\Red P$ and we similarly choose $P$ in $W[x;\xi]$.
We will identify $\mathcal M$ with the left $W(x;\xi)$-module $W(x;\xi)/W(x;\xi)P$. Then we may consider \eqref{eq:M} as the fundamental relation of the generator $u$ of the module $\mathcal M$.
The results in this subsection are standard and well-known but for our convenience we briefly review them. First note that $W(x;\xi)$ is a (left) Euclidean ring:
Let $P$, $Q\in W(x;\xi)$ with $P\ne0$. Then there uniquely exists $R$, $S\in W(x;\xi)$ such that \begin{equation}\label{eq:Div}
Q = SP + R\qquad(\ord R<\ord P). \end{equation} Hence we note that $\dim_{\mathbb C(x,\xi)}\bigl(W(x;\xi)/W(x;\xi)P\bigr)=\ord P$. We get $R$ and $S$ in \eqref{eq:Div} by a simple algorithm as follows. Put \begin{equation}\label{eq:PQ}
P=a_n\partial^n+\cdots+a_1\partial+a_0 \text{ \ and \ } Q=b_m\partial^m+\cdots+b_1\partial+b_0 \end{equation} with $a_n\ne 0$, $b_m\ne 0$. Here $a_n$, $b_m\in\mathbb C(x,\xi)$. The division \eqref{eq:Div} is obtained by the induction on $\ord Q$. If $\ord P>\ord Q$, \eqref{eq:Div} is trivial with $S=0$. If $\ord P \le \ord Q$, \eqref{eq:Div} is reduced to the equality $Q'=S'P+R$ with $Q'=Q - a_n^{-1}b_m\partial^{m-n}P$ and $S'=S-a_n^{-1}b_m\partial^{m-n}$ and then we have $S'$ and $R$ satisfying $Q'=S'P+R$ by the induction because $\ord Q'<\ord Q$. The uniqueness of \eqref{eq:Div} is clear by comparing the highest order terms of \eqref{eq:Div} in the case when $Q=0$.
By the standard Euclid algorithm using the division \eqref{eq:Div} we have $M$, $N\in W(x;\xi)$ such that \begin{equation}\label{eq:Euc}
MP+NQ=U,\ P\in W(x;\xi)U \text{ \ and \ } Q\in W(x;\xi)U. \end{equation} Hence in particular any left ideal of $ W(x;\xi)$ is generated by a single element of $W[x;\xi]$, namely, $W(x;\xi)$ is a principal ideal domain. \begin{defn} The operators $P$ and $Q$ in $W(x;\xi)$ are defined to be \textsl{mutually prime} if one of the following equivalent conditions is valid. \index{differential equation/operator!mutually prime} \begin{align}
&W(x;\xi)P + W(x;\xi)Q = W(x;\xi),\\
&\text{there exists }R\in W(x;\xi)\text{ satisfying }
RQu=u\text{ for the equation }Pu=0,\\
&\begin{cases}
\text{the simultaneous equation }Pu=Qu=0\text{ has not a non-zero solution}\\
\text{for a generic value of }\xi.
\end{cases} \end{align} \end{defn}
Moreover we have the following. \index{differential equation/operator!cyclic} \begin{equation}\label{eq:Wsing} \text{Any left $W(x;\xi)$-module $\mathcal R$ with
$\dim_{\mathbb C(x,\xi)}\mathcal R<\infty$ is \textsl{cyclic},} \end{equation} namely, it is generated by a single element. Hence any system of ordinary differential equations is isomorphic to a single differential equation under the algebra $W(x;\xi)$.
To prove \eqref{eq:Wsing} it is sufficient to show that the direct sum $\mathcal M\oplus\mathcal N$ of $\mathcal M: Pu=0$ and $\mathcal N: Qv=0$ is cyclic. In fact $M\oplus\mathcal N=W(x;\xi)w$ with $w=u+(x-c)^nv\in\mathcal M\oplus\mathcal N$ and $n=\ord P$ if $c\in\mathbb C$ is generic. For the proof we have only to show $\dim_{\mathbb C(x,\xi)}W(x;\xi)w\ge m+n$ and we may assume that $P$ and $Q$ are in $W[x;\xi]$ and they are of the form \eqref{eq:PQ}. Fix $\xi$ generically and we choose $c\in\mathbb C$ such that $a_n(c)b_m(c)\ne 0$. Since the function space $V=\{\phi(x)+(x-c)^n\varphi(x)\,;\,P\phi(x)=Q\varphi(x)=0\}$ is of dimension $m+n$ in a neighborhood of $x=c$, $\dim_{W(x;\xi)}W(x;\xi)w\ge m+n$ because the relation $Rw=0$ for an operator $R\in W(x;\xi)$ implies $R\psi(x)=0$ for $\psi\in V$.
Thus we have the following standard definition. \begin{defn}\label{def:irred} Fix $P\in W(x;\xi)$ with $\ord P>0$. The equation \eqref{eq:M} is \textsl{irreducible} if and only if one of the following equivalent conditions is valid. \index{differential equation/operator!irreducible} \begin{align} &\text{The left $W(x;\xi)$-module $\mathcal M$ is simple.} \label{eq:irrsimp}\allowdisplaybreaks\\ &\text{The left $W(x;\xi)$-ideal $W(x;\xi)P$ is maximal.}\allowdisplaybreaks\\ &P=QR\text{ with }Q,\,R\in W(x;\xi)\text{ implies }
\ord Q\cdot\ord R=0.\label{eq:QR}\allowdisplaybreaks\\ &\forall Q\not\in W(x;\xi)P,\ \exists M,\ N\in W(x;\xi)\text{ satisfying }MP+NQ=1. \label{eq:irrEuc}\allowdisplaybreaks\\
&\begin{cases}
ST\in W(x;\xi)P\text{ with }S, T\in W(x;\xi)
\text{ and }\ord S<\ord P \\
\Rightarrow S=0\text{ or }T\in W(x;\xi)P.
\end{cases}\label{eq:ST} \end{align} The equivalence of the above conditions is standard and easily proved. The last condition may be a little non-trivial.
Suppose \eqref{eq:ST} and $P=QR$ and $\ord Q\cdot\ord R\ne 0$. Then $R\notin W(x;\xi)P$ and therefore $Q=0$, which contradicts to $P=QR$. Hence \eqref{eq:ST} implies \eqref{eq:QR}.
Suppose \eqref{eq:irrsimp}, \eqref{eq:irrEuc}, $ST\in W(x;\xi)P$ and $T\notin W(x;\xi)P$. Then there exists $P'$ such that $\{J\in W(x;\xi)\,;\,JT\in W(x;\xi)P\}= W(x;\xi)P'$, $\ord P'=\ord P$ and moreover $P'v=0$ is also simple. Since $Sv=0$ with $\ord S<\ord P'$, we have $S$=0.
In general, a system of ordinary differential equations is defined to be irreducible if it is simple as a left $W(x;\xi)$-module. \end{defn}
\begin{rem} Suppose the equation $\mathcal M$ given in \eqref{eq:M} is irreducible.
{\rm i) } Let $u(x,\xi)$ be a non-zero solution of $\mathcal M$, which is locally defined for the variables $x$ and $\xi$ and meromorphically depends on $(x,\xi)$. If $S\in W[x;\xi]$ satisfies $Su(x,\xi)=0$, then $S\in W(x;\xi)P$. Therefore $u(x,\xi)$ determines $\mathcal M$.
{\rm ii) } Suppose $\ord P>1$. Fix $R\in W(x;\xi)$ such that $\ord R<\ord P$ and $R\ne 0$. For $Q\in W(x;\xi)$ and a positive integer $m$, the condition $R^mQu=0$ is equivalent to $Qu=0$. Hence for example, if $Q_1u+\partial^mQ_2u=0$ with certain $Q_j\in W(x;\xi)$, we will allow the expression $\partial^{-m}Q_1u+Q_2u=0$ and $\partial^{-m}Q_1u(x,\xi)+Q_2u(x,\xi)=0$.
{\rm iii) } For $T\not\in W(x;\xi)P$ we construct a differential equation $Qv=0$ satisfied by $v=Tu$ as follows. Put $n=\ord P$. We have $R_j\in W(x;\xi)$ such that $\partial^jTu=R_ju$ with $\ord R_j<\ord P$. Then there exist $b_0,\dots,b_n\in\mathbb C(x,\xi)$ such that $b_nR_n+\cdots+b_1R_1+b_0R_0=0$. Then $Q=b_n\partial^n+\cdots+b_1\partial+b_0$. \end{rem}
\subsection{Okubo normal form and Schlesinger canonical form}\label{sec:DR} \index{Schlesinger canonical form}\index{Okubo normal form}
In this subsection we briefly explain the interpretation of Katz's middle convolution (cf.~\cite{Kz}) by \cite{DR} and its relation to our fractional operations.
For constant square matrices $T$ and $A$ of size $n'$, the ordinary differential equation \begin{equation}\label{eq:ONF}
(xI_{n'}-T)\frac{du}{dx}=Au \end{equation} is called \textsl{Okubo normal form} of Fuchsian system when $T$ is a diagonal matrix. \index{Okubo normal form} Then \begin{equation}
mc_\mu\bigl((xI_{n'}-T)\partial -A\bigr)=(xI_{n'}-T)\partial -(A+\mu I_{n'}) \end{equation} for generic $\mu\in\mathbb C$, namely, the system is transformed into \begin{equation}\label{eq:mcONF}
(xI_{n'}-T)\frac{du_\mu}{dx}=\bigl(A+\mu I_{n'}\bigr)u_\mu \end{equation} by the operation $mc_\mu$. Hence for a solution $u(x)$ of \eqref{eq:ONF}, the Euler transformation $u_\mu(x)=I_c^\mu(u)$ of $u(x)$ satisfies \eqref{eq:mcONF}.
For constant square matrices $A_j$ of size $m$ and the \textsl{Schlesinger canonical form} \begin{equation}\label{eq:SCF}
\frac{d v}{dx} = \sum_{j=1}^p\frac{A_j}{x-c_j}v \end{equation} of a Fuchsian system of the Riemann sphere, we have\\[-20pt] \begin{equation}\label{eq:S2O}
\frac{du}{dx} = \sum_{j=1}^p
\frac{\tilde A_j}{x-c_j}u,\quad
\tilde A_j:= \bordermatrix{
& & \cr
& & \cr
j)&A_1&\cdots&A_p\cr
& & \cr}
\text{ \ and \ }
u:=\begin{pmatrix}
\frac{v}{x-c_1}\\\vdots\\\frac{v}{x-c_p}
\end{pmatrix}. \end{equation} Here $\tilde A_j$ are square matrices of size $pm$. The addition $\Ad\bigl((x-c_k)^{\mu_k}\bigr)$ transforms $A_j$ into $A_j+\mu_k\delta_{j,k}I_m$ for $j=1,\dots,p$ in the system \eqref{eq:SCF}. Putting \[A=\tilde A_1+\cdots+\tilde A_p\text{ \ and \ }
T=\left(\begin{smallmatrix}
c_1I_n\\&\ddots\\&&c_pI_n
\end{smallmatrix}\right), \] the equation \eqref{eq:S2O} is equivalent to \eqref{eq:ONF} with $n'=pm$. Define square matrices of size $n'$ by \begin{align}
\tilde A&:=
\begin{pmatrix}
A_1\\
& \ddots\\
&& A_p
\end{pmatrix},\\[-5pt]
\tilde A_j(\mu)
&:= \bordermatrix{
& & & & \underset{\smile}{j} \cr
& & \cr
j)&A_1&\cdots&A_{j-1}&A_j+\mu&A_{j+1}&\cdots& A_p\cr
& & \cr}.\label{eq:conSch} \end{align} Then $\ker\tilde A$ and $\ker(A+\mu)$ are invariant under $\tilde A_j(\mu)$ for $j=1,\dots,p$ and therefore $\tilde A_j(\mu)$ induce endomorphisms of $V:=\mathbb C^{pm}/\bigl(\ker\tilde A+\ker (A+\mu)\bigr)$, which correspond to square matrices of size $N:=\dim V$, which we put $\bar A_j(\mu)$, respectively, under a fixed basis of $V$. Then the middle convolution $mc_\mu$ of \eqref{eq:SCF} is the system \begin{equation}\label{eq:mcS}
\frac{dw}{dx} = \sum_{j=1}^p \frac{\bar A_j(\mu)}{x-c_j}w \end{equation} of rank $N$, which is defined and studied by \cite{DR, DR2}. Here $\ker\tilde A \cap\ker(A+\mu)=\{0\}$ if $\mu\ne0$.
We define another realization of the middle convolution as in \cite[\S2]{O2}. Suppose $\mu\ne0$. The square matrices of size $n'$ \begin{align}
A_j^\vee(\mu)
&:= \bordermatrix{
& && \underset{\smile}{j} && \cr
& && A_1\cr
& && \vdots\cr
j\,{\text{\tiny$)$}} &&& A_j+\mu && \cr
& && \vdots\cr
& && A_p
}\text{ \ and \ }
A^\vee(\mu) := A^\vee_1(\mu)+\cdots+A^\vee_p(\mu) \end{align} satisfy \begin{align}
\tilde A(A+\mu I_{n'}) &= A^\vee(\mu)\tilde A
=
\Bigl(A_iA_j+\mu\delta_{i,j}A_i\Bigr) _{\substack{1\le i\le p \\ 1\le j\le p}}\in M(n',\mathbb C),\\ \tilde A (A+\mu I_{n'})\tilde A_j(\mu) &= A_j^\vee(\mu)\tilde A(A+\mu I_{n'}). \end{align} Hence $w^\vee :=\tilde A (A+\mu I_{n'}) u$ satisfies \begin{align}
\frac{d w^\vee}{dx}
&= \sum_{j=1}^p\frac{A_j^\vee(\mu)}{x-c_j}w^\vee,\\
\sum_{j=1}^p\frac{A_j^\vee(\mu)}{x-c_j}
&=\biggl(\frac{A_i+\mu\delta_{i,j}I_m}{x-c_j}\biggr)
_{\substack{ 1\le i\le p,\\1\le j\le p}}\notag \end{align} and $\tilde A(A+\mu I_{n'})$ induces the isomorphism \begin{equation} \tilde A(A+\mu I_{n'}):
V=\mathbb C^{n'}/(\mathcal K+\mathcal L_\mu)
\ \overset{\sim}\to \ V^\vee:=\IM \tilde A(A+\mu I_{n'})
\subset\mathbb C^{n'}. \end{equation}
Hence putting $\bar A_j^\vee(\mu):=A_j^\vee(\mu)|_{V^\vee}$, the system \eqref{eq:mcS} is isomorphic to the system \begin{equation}\label{eq:SCFmcv}
\frac{d w^\vee}{dx} = \sum_{j=1}^p\frac{\bar A_j^\vee(\mu)}{x-c_j}
w^\vee \end{equation} of rank $N$, which can be regarded as a middle convolution $mc_\mu$ of \eqref{eq:SCF}. Here \begin{equation}
w^\vee=\begin{pmatrix}w^\vee_1\\ \vdots\\ w_p^\vee\end{pmatrix},\quad
w^\vee_j = \sum_{\nu=1}^p(A_jA_\nu + \mu\delta_{j,\nu})
(u_\mu)_\nu
\quad(j=1,\dots,p) \end{equation} and if $v(x)$ is a solution of \eqref{eq:SCF}, then \begin{equation}
w^\vee(x)=\biggl(\sum_{\nu=1}^p(A_jA_\nu+\mu \delta_{j,\nu})
I_c^\mu\Bigl(\frac{v(x)}{x-c_\nu}\Bigr)\biggr)_{j=1,\dots,p} \end{equation} satisfies \eqref{eq:SCFmcv}.
Since any non-zero homomorphism between irreducible $W(x)$-modules is an isomorphism, we have the following remark (cf.~\S\ref{sec:ODE} and \S\ref{sec:contig}). \begin{rem}\label{rem:SCFmc} Suppose that the systems \eqref{eq:SCF} and \eqref{eq:SCFmcv} are irreducible. Moreover suppose the system \eqref{eq:SCF} is isomorphic to a single Fuchsian differential equation $P\tilde u=0$ as left $W(x)$-modules and the equation $mc_\mu(P)\tilde w=0$ is also irreducible. Then the system \eqref{eq:SCFmcv} is isomorphic to the single equation $mc_\mu(P)\tilde w=0$ because the differential equation satisfied by $I_c^\mu(\tilde u(x))$ is isomorphic to that of $I_c^\mu(Q\tilde u(x))$ for a non-zero solution $v(x)$ of $P\tilde u=0$ and an operator $Q\in W(x)$ with $Q\tilde u(x)\ne0$ (cf.~\S\ref{sec:contig}, Remark~\ref{rem:midisom} iii) and Proposition~\ref{prop:irred}).
In particular if the systems are rigid and their spectral parameters are generic, all the assumptions here are satisfied (cf.~Remark~\ref{rem:generic} ii) and Corollary~\ref{cor:irred}). \end{rem}
Yokoyama \cite{Yo2} defines extension and restriction operations among the systems of differential equations of Okubo normal form. The relation of Yokoyama's operations to Katz's operations is clarified by \cite{O4}, which shows that they are equivalent from the view point of the construction and the reduction of systems of Fuchsian differential equations.
\section{Confluences} \subsection{Regular singularities}\label{sec:reg} \index{regular singularity} In this subsection we review fundamental facts related to the regular singularities of the ordinary differential equations.
\subsubsection{Characteristic exponents}\index{characteristic exponent} The ordinary differential equation \begin{equation}
a_n(x)\tfrac{d^n u}{dx^n}+a_{n-1}(x)\tfrac{d^{n-1} u}{dx^{n-1}}+
\cdots + a_1(x)\tfrac{d u}{dx}+a_0(x)u=0\label{eq:ode} \end{equation} of order $n$ with meromorphic functions $a_j(x)$ defined in a neighborhood of $c\in\mathbb C$ has a singularity at $x=c$ if the function $\frac{a_j(x)}{a_n(x)}$ has a pole at $x=c$ for a certain $j$. The singular point $x=c$ of the equation is a \textsl{regular singularity} if it is a removable singularity of the functions $b_j(x):=(x-c)^{n-j}a_j(x)a_n(x)^{-1}$ for $j=0,\dots,n$. In this case $b_j(c)$ are complex numbers and the $n$ roots of the \textsl{indicial equation}\index{indicial equation} \begin{equation}
\sum_{j=0}^n b_j(c)s(s-1)\cdots(s-j+1)=0 \end{equation} are called the \textsl{charactersitic exponents} of \eqref{eq:ode} at $c$.
Let $\{\lambda_1,\dots,\lambda_n\}$ be the set of these characteristic exponents at $c$.
If $\lambda_j-\lambda_1\notin\mathbb Z_{>0}$ for $1<j\le n$, then \eqref{eq:ode} has a unique solution $(x-c)^{\lambda_1}\phi_1(x)$ with a holomorphic function $\phi_1(x)$ in a neighborhood of $c$ satisfying $\phi_1(c)=1$.
\begin{defn}\label{def:exp} The regular singularity and the characteristic exponents for the differential operator \begin{equation}\label{eq:ODP}
P=a_n(x)\tfrac{d^n}{dx^n}+a_{n-1}(x)\tfrac{d^{n-1}}{dx^{n-1}}+
\cdots+a_1(x)\tfrac d{dx}+a_0(x) \end{equation} are defined by those of the equation \eqref{eq:ode}, respectively. Suppose $P$ has a regular singularity at $c$. We say $P$ is \textsl{normalized at $c$} if $a_n(x)$ is holomorphic at $c$ and \index{regular singularity!normalized} \begin{equation}
a_n(c)=a_n^{(1)}(c)=\cdots=a_n^{(n-1)}(c)=0\text{ \ and \ }
a_n^{(n)}(c)\ne 0. \end{equation} In this case $a_j(x)$ are analytic and have zeros of order at least $j$ at $x=c$ for $j=0,\dots,n-1$.
\subsubsection{Local solutions} \index{regular singularity!local solution} \index{00O@$\mathcal O,\ \hat{\mathcal O},\ \mathcal O_c,\ \mathcal O_c(\mu,m)$} The ring of convergent power series at $x=c$ is denoted by $\mathcal O_c$ and for a complex number $\mu$ and a non-negative integer $m$ we put \begin{equation}\label{def:Ocm}
\mathcal O_c(\mu,m) :=\bigoplus_{\nu=0}^m
\mathcal (x-c)^\mu\log^\nu(x-c)\mathcal O_c. \end{equation} \end{defn} Let $P$ be a differential operator of order $n$ which has a regular singularity at $x=c$ and let $\{\lambda_1,\cdots,\lambda_n\}$ be the corresponding characteristic exponents. Suppose $P$ is normalized at $c$. If a complex number $\mu$ satisfies $\lambda_j-\mu\notin\{0,1,2,\dots\}$ for $j=1,\dots,n$, then $P$ defines a linear bijective map \begin{equation}\label{eq:Pbij}
P: \mathcal O_c(\mu,m)\ \overset{\sim}{\to}\ \mathcal O_c(\mu,m) \end{equation} for any non-negative integer $m$.
Let $\hat {\mathcal O}_c$ be the ring of formal power series $\sum_{j=0}^\infty a_j(x-c)^j$ $(a_j\in\mathbb C)$ of $x$ at $c$. For a domain $U$ of $\mathbb C$ we denote by $\mathcal O(U)$ the ring of holomorphic functions on $U$. Put \index{00Brc@$B_r(c)$} \begin{equation}\label{eq:Brc}
B_r(c):=\{x\in\mathbb C\,;\,|x-c|<r\} \end{equation} for $r>0$ and \begin{align}
\hat{\mathcal O}_c(\mu,m) &:=\bigoplus_{\nu=0}^m
\mathcal (x-c)^\mu\log^\nu(x-c)\hat{\mathcal O}_c,\\
\mathcal O_{B_r(c)}(\mu,m) &:= \bigoplus_{\nu=0}^m
\mathcal (x-c)^\mu\log^\nu(x-c)\mathcal O_{B_r(c)} . \end{align} Then $\mathcal O_{B_r(c)}(\mu,m)\subset\mathcal O_c(\mu,m) \subset \hat {\mathcal O}_c(\mu,m)$.
Suppose $a_j(x)\in\mathcal O\bigl(B_r(c)\bigr)$ and $a_n(x)\ne 0$ for $x\in B_r(c)\setminus\{c\}$ and moreover $\lambda_j-\mu\notin \{0,1,2,\ldots\}$, we have \begin{align}
P&: \mathcal O_{B_r(c)}(\mu,m) \ \overset{\sim}{\to} \ \
\mathcal O_{B_r(c)}(\mu,m),\label{eq:PBr}\\
P&: \hat{\mathcal O}_c(\mu,m) \ \overset{\sim}{\to} \ \
\hat{\mathcal O}_c(\mu,m).\label{eq:Phat} \end{align} The proof of these results are reduced to the case when $\mu=m=c=0$ by the translation $x\mapsto x-c$, the operation $\Ad\bigl(x^{-\mu}\bigr)$, and the fact $P(\sum_{j=0}^m f_j(x)\log^jx) = (Pf_m(x))\log^jx+ \sum_{j=0}^{m-1}\phi_j(x)\log^jx$ with suitable $\phi_j(x)$ and moreover we may assume \begin{align*}
P &= \prod_{j=0}^n(\vartheta -\lambda_j) - xR(x,\vartheta),\\
xR(x,\vartheta) &= x\sum_{j=0}^{n-1}r_j(x)\vartheta^j
\quad(r_j(x)\in\mathcal O\bigl(B_r(c)\bigr)). \end{align*} When $\mu=m=0$, \eqref{eq:Phat} is easy and \eqref{eq:PBr} and hence \eqref{eq:Pbij} are also easily proved by the method of majorant series (for example, cf.~\cite{O0}).
For the differential operator \[
Q = \tfrac{d^n}{dx^n}+b_{n-1}(x)\tfrac{d^{n-1}}{dx^{n-1}}
+ \cdots +b_1(x)\tfrac{d}{dx} + b_0(x) \] with $b_j(x)\in\mathcal O\bigl(B_r(c)\bigr)$, we have a bijection \begin{equation}
\begin{matrix}
Q:&\mathcal O\bigl(B_r(c)\bigr) & \overset{\sim}{\to} &
\mathcal O\bigl(B_r(c)\bigr)\oplus\mathbb C^n\phantom{ABCDEFG}\\
& \rotatebox{90}{$\in$} & &\rotatebox{90}{$\in$}\phantom{ABCD}\\
&u(x)&\mapsto &Pu(x)\oplus \bigl(u^{(j)}(c)\bigr)_{0\le j\le n-1}
\end{matrix} \end{equation} because $Q(x-c)^n$ has a regular singularity at $x=c$ and the characteristic exponents are $-1,-2,\dots,-n$ and hence \eqref{eq:PBr} assures that for any $g(x)\in\mathbb C[x]$ and $f(x)\in\mathcal O\bigl(B_r(c)\bigr)$ there uniquely exists $v(x)\in\mathcal O\bigl(B_r(c)\bigr)$ such that $Q(x-c)^nv(x)= f(x)-Qg(x)$.
If $\lambda_\nu-\lambda_1\notin\mathbb Z_{>0}$, the characteristic exponents of $R:=\Ad\bigl((x-c)^{-\lambda_1-1}\bigr)P$ at $x=c$ are $\lambda_\nu-\lambda_1-1$ for $\nu=1,\dots,n$ and therefore $R=S(x-c)$ with a differential operator $R$ whose coefficients are in $\mathcal O\bigl(B_r(c)\bigr)$. Then there exists $v_1(x)\in\mathcal O\bigl(B_r(c)\bigr)$ such that $-S1=S(x-c) v_1(x)$, which means $P\bigl((x-c)^{\lambda_1}(1+(x-c)v_1(x))\bigr)=0$. Hence if $\lambda_i-\lambda_j\notin\mathbb Z$ for $1\le i<j\le n$, we have solutions $u_\nu(x)$ of $Pu=0$ such that \begin{equation}
u_\nu(x)=(x-c)^{\lambda_\nu}\phi_\nu(x) \end{equation} with suitable $\phi_\nu\in\mathcal O\bigl(B_r(c)\bigr)$ satisfying $\phi_\nu(c)=1$ for $\nu=1,\dots,n$.
Put $k=\#\{\nu\,;\,\lambda_\nu=\lambda_1\}$ and $m=\#\{\nu\,;\,\lambda_\nu-\lambda_1\in\mathbb Z_{\ge0}\}$. Then we have solutions $u_\nu(x)$ of $Pu=0$ for $\nu=1,\dots,k$ such that \begin{equation}
u_\nu(x) - (x-c)^{\lambda_1}\log^{\nu-1}(x-c)
\in\mathcal O_{B_r(c)}(\lambda_1+1,m-1). \end{equation} If $\mathcal O_{B_r(c)}$ is replaced by $\hat{\mathcal O}_c$, the solution \[u_{\nu}(x) = (x-c)^{\lambda_1}\log^{\nu-1}(x-c)+
\sum_{i=1}^\infty\sum_{j=0}^{m-1}c_{\nu,i,j}(x-c)^{\lambda_1+i}\log^j(x-c)
\in \hat{\mathcal O}_c(\lambda_1,m-1) \] is constructed by inductively defining $c_{\nu,i,j}\in\mathbb C$. Since \[ \begin{split}
&P\Bigl(\sum_{i=N+1}^\infty\sum_{j=0}^{m-1}c_{\nu,i,j}(x-c)^{\lambda_1+i}\log^j(x-c) \Bigr)=
-P\Bigl((x-c)^{\lambda_1}\log^{\nu-1}(x-c)\\
&\qquad +\sum_{i=1}^N c_{\nu,i,j}(x-c)^{\lambda_1+i}\log^j(x-c) \Bigr)
\in\mathcal O_{B_r(c)}(\lambda_1+N,m-1) \end{split} \] for an integer $N$ satisfying $\RE(\lambda_\ell-\lambda_1)<N$ for $\ell=1,\dots,n$, we have \[
\sum_{i=N+1}^\infty\sum_{j=0}^{m-1}c_{\nu,i,j}(x-c)^{\lambda_1+i}\log^j(x-c)\in
\mathcal O_{B_r(c)}(\lambda_1+N,m-1) \] because of \eqref{eq:PBr} and \eqref{eq:Phat}, which means $u_{\nu}(x)\in\mathcal O_{B_r(c)}(\lambda_1,m)$.
\subsubsection{Fuchsian differential equations} \index{Fuchsian differential equation/operator} The regular singularity at $\infty$ is similarly defined by that at the origin under the coordinate transformation $x\mapsto\frac1x$. When $P\in W(x)$ and the singular points of $P$ in $\overline{\mathbb C}:=\mathbb C\cup\{\infty\}$ are all regular singularities, the operator $P$ and the equation $Pu=0$ are called \textsl{Fuchsian}. \index{Fuchsian differential equation/operator} Let $\overline{\mathbb C}'$ be the subset of $\overline{\mathbb C}$ deleting singular points $c_0,\ldots,c_p$ from $\overline{\mathbb C}$. Then the solutions of the equation $Pu=0$ defines a map \begin{equation}\label{eq:Fmap}
\mathcal F:\ \overline{\mathbb C}'\supset U:\text{(simply connected domain)}
\mapsto \mathcal F(U)\subset\mathcal O(U) \end{equation} by putting $\mathcal F(U):=\{u(x)\in\mathcal O(U)\,;\,Pu(x)=0\}$. Put \[
U_{j,\epsilon,R}=
\begin{cases}
\{x=c_j+re^{\sqrt{-1}\theta}\,;\,0<r<\epsilon,\ R<\theta<R+2\pi\}
&(c_j\ne\infty)\\
\{x = re^{\sqrt{-1}\theta}\,;\, r>\epsilon^{-1},\ R<\theta<R+2\pi\}
&(c_j=\infty).
\end{cases} \] For simply connected domains $U$, $V\subset\overline{\mathbb C}'$, the map $\mathcal F$ satisfies \begin{align}
&\mathcal F(U) \subset\mathcal O(U)\text{ \ and \ } \dim\mathcal F(U)= n,
\label{eq:F0}\\
&V\subset U\ \Rightarrow\ \mathcal F(V)=\mathcal F(U)|_V,\label{eq:F1}\\
&\begin{cases}
\exists \epsilon > 0,\ \forall \phi\in\mathcal F(U_{j,\epsilon,R}),\
\exists C>0, \exists m>0\text{\ such that\ }\\
|\phi(x)|<
\begin{cases}
C|x-c_j|^{-m} & (c_j\ne\infty,\ x\in U_{j,\epsilon,R}),\\
C|x|^m & (c_j=\infty,\ x\in U_{j,\epsilon,R})\\
& \text{\quad for }j=0,\dots,p,\ \forall R\in\mathbb R.
\end{cases}
\end{cases}\label{eq:F2} \end{align} Then we have the bijection \begin{equation}\label{eq:FisP}
\begin{matrix}
\bigl\{\partial^n+\displaystyle\sum_{j=0}^{n-1}a_j(x)\partial^j\in W(x)\,:\,\text{Fuchsian}\bigr\}
&\overset{\sim}\to&
\bigl\{\mathcal F\text{ satisfying \eqref{eq:F0}--\eqref{eq:F2}}\bigr\}\\[-8pt]
\rotatebox{90}{$\in$}&&\rotatebox{90}{$\in$}\\
P&\mapsto&\bigl\{U\mapsto\{u\in\mathcal O(U)\,;\,Pu=0\}\bigr\}.
\end{matrix} \end{equation} Here if $\mathcal F(U)=\sum_{j=1}^n\mathbb C\phi_j(x)$, \begin{equation}\label{eq:F2E}
a_j(x)=(-1)^{n-j}\frac{\det\Phi_j}{\det\Phi_n}\text{ \ with \ }
\Phi_j=\begin{pmatrix}
\phi_1^{(0)}(x) & \cdots & \phi_n^{(0)}(x)\\
\vdots & \vdots & \vdots\\
\phi_1^{(j-1)}(x) & \cdots & \phi_n^{(j-1)}(x)\\
\phi_1^{(j+1)}(x) & \cdots & \phi_n^{(j+1)}(x)\\
\vdots & \vdots & \vdots\\
\phi_1^{(n)}(x) & \cdots & \phi_n^{(n)}(x)
\end{pmatrix}. \end{equation} The elements $\mathcal F_1$ and $\mathcal F_2$ of the right hand side of \eqref{eq:FisP} are naturally identified if there exists a simply connected domain $U$ such that $\mathcal F_1(U)=\mathcal F_2(U)$.
Let \[
P = \partial^n+a_{n-1}(x)\partial^{n-1}+\cdots+a_0(x) \] be a Fuchsian differential operator with $p+1$ regular singular points $c_0=\infty$,$c_1,\dots,c_p$ and let $\lambda_{j,1},\dots,\lambda_{j,n}$ be the characteristic exponents of $P$ at $c_j$, respectively. Since $a_{n-1}(x)$ is holomorphic at $x=\infty$ and $a_{n-1}(\infty)=0$, there exists $a_{n-1,j}\in\mathbb C$ such that $a_{n-1}(x)=
-\sum_{j=1}^p\frac{a_{n-1,j}}{x-c_j}$. For $c\in\mathbb C$ we have $x^n(\partial^n - cx^{-1}\partial^{n-1}\bigr)=
\vartheta^n-\bigl(c+\frac{n(n-1)}2\bigr)\vartheta^{n-1}+
c_{n-2}\vartheta^{n-2}+\cdots+c_0$ with $c_j\in\mathbb C$. Hence we have \[
\lambda_{j,1}+\cdots+\lambda_{j,n}
=\begin{cases}
-\sum_{j=1}^pa_{n-1,j}-\frac{n(n-1)}{2}&(j=0),\\
a_{n-1,j}+\frac{n(n-1)}{2}&(j=1,\dots,p),
\end{cases} \] and the \textsl{Fuchs relation}\index{Fuchs relation} \begin{equation}\label{eq:FC0}
\sum_{j=0}^p\sum_{\nu=1}^n\lambda_{j,\nu} = \frac{(p-1)n(n-1)}{2}. \end{equation}
Suppose $Pu=0$ is reducible. Then $P=SR$ with $S,\, R\in W(x)$ so that $n'=\ord R<n$. Since the solution $v(x)$ of $Rv=0$ satisfies $Pv(x)=0$, $R$ is also Fuchsian. Note that the set of $m$ characteristic exponents $\{\lambda'_{j,\nu}\,;\,\nu=1,\dots,n'\}$ of $Rv=0$ at $c_j$ is a subset of $\{\lambda_{j,\nu}\,;\,\nu=1,\dots,n\}$. The operator $R$ may have other singular points $c_1',\ldots,c_q'$ called \textsl{apparent singular points}\index{apparent singularity} where any local solutions at the points is analytic. Hence the set characteristic exponents at $x=c'_j$ are $\{\lambda'_{j,\nu}\,\:\,\nu=1,\ldots,n'\}$ such that $0\le \mu_{j,1}<\mu_{j,2}<\cdots<\mu_{j,n'}$ and $\mu_{j,\nu}\in\mathbb Z$ for $\nu=1,\dots,n'$ and $j=1,\dots,q$. Since $\mu_{j,1}+\cdots+\mu_{j,n'}\ge\frac{n'(n'-1)}2$, the Fuchs relation for $R$ implies \begin{equation}\label{eq:FC1}
\mathbb Z\ni \sum_{j=0}^p\sum_{\nu=1}^{n'} \lambda'_{j,\nu}\le
\frac{(p-1)n'(n'-1)}2. \end{equation}
\index{differential equation/operator!irreducible} Fixing a generic point $q$ and pathes $\gamma_j$ around $c_j$ as in \eqref{fig:mon} and moreover a base $\{u_1,\dots,u_n\}$ of local solutions of the equation $Pu=0$ at $q$, we can define monodromy generators $M_j\in GL(n,\mathbb C)$. We call the tuple $\mathbf M=(M_0,\dots,M_p)$ the \textsl{monodromy} of the equation $Pu=0$. The monodromy $\mathbf M$ is defined to be \textsl{irreducible} if there exists no subspace $V$ of $\mathbb C^n$ such that $M_j V\subset V_j$ for $j=0,\dots,p$ and $0<\dim V<n$, which is equivalent to the condition that $P$ is irreducible. \index{monodromy}
Suppose $Qv=0$ is another Fuchsian differential equation of order $n$ with the same singular points. The monodromy $\mathbf N=(N_0,\dots,N_p)$ is similarly defined by fixing a base $\{v_1,\dots,v_n\}$ of local solutions of $Qv=0$ at $q$. Then \begin{equation}\label{eq:isoWM} \begin{split} \mathbf M\sim\mathbf N&\ \overset{\text{def}}\Leftrightarrow \
\exists g\in GL(n,\mathbb C)\text{ such that }
N_j=gM_jg^{-1}\ (j=0,\dots,p)\\
& \ \Leftrightarrow\ Qv=0
\text{ is $W(x)$-isomorphic to }
Pu=0. \end{split} \end{equation}
If $Qv=0$ is $W(x)$-isomorphic to $Pu=0$, the isomorphism defines an isomorphism between their solutions and then $N_j=M_j$ under the bases corresponding to the isomorphism.
Suppose there exists $g\in GL(n,\mathbb C)$ such that $N_j=gM_jg^{-1}$ for $j=0,\dots,p$. The equations $Pu=0$ and $Qu=0$ are $W(x)$-isomorphic to certain first order systems $U'=A(x)U$ and $V'=B(x)V$ of rank $n$, respectively. We can choose bases $\{U_1,\dots,U_n\}$ and $\{V_1,\dots,V_n\}$ of local solutions of $PU=0$ and $QV=0$ at $q$, respectively, such that their monodromy generators corresponding $\gamma_j$ are same for each $j$. Put $\tilde U=(U_1,\dots,U_n)$ and $\tilde V=(V_1,\dots,V_n)$. Then the element of the matrix $\tilde V\tilde U^{-1}$ is holomorphic at $q$ and can be extended to a rational function of $x$ and then $\tilde V\tilde U^{-1}$ defines a $W(x)$-isomorphism between the equations $U'=A(x)U$ and $V'=B(x)V$.
\begin{exmp}[apparent singularity]
The differential equation \begin{equation}\label{eq:ApSing}
x(x-1)(x-c)\tfrac{dy^2}{dx} + (x^2-2cx+c)\tfrac{dy}{dx}=0 \end{equation} is a special case of Heun's equation \eqref{eq:Heun} with $\alpha=\beta=\lambda=0$ and $\gamma=\delta=1$. It has regular singularities at $0$, $1$, $c$ and $\infty$ and its Riemann scheme equals \index{Heun's equation} \begin{equation}
\begin{Bmatrix}
x= \infty & 0 & 1 & c\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 2
\end{Bmatrix}. \end{equation}
\index{Wronskian} The local solution at $x=c$ corresponding to the characteristic exponent 0 is holomorphic at the point and therefore $x=c$ is an apparent singularity, which corresponds to the zero of the Wronskian $\det\Phi_n$ in \eqref{eq:F2E}. Note that the equation \eqref{eq:ApSing} has the solutions $1$ and $c\log x+(1-c)\log(x-1)$. \index{apparent singularity}
The equation \eqref{eq:ApSing} is not $W(x)$-isomorphic to Gauss hypergeometric equation if $c\ne 0$ and $c\ne 1$, which follows from the fact that $c$ is a modulus of the isomorphic classes of the monodromy. It is easy to show that any tuple of matrices $\mathbf M=(M_0,M_1,M_2)\in GL(2,\mathbb C)$ satisfying $M_2M_1M_0=I_2$ is realized as the monodromy of the equation obtained by applying a suitable addition $\RAd\bigl(x^{\lambda_0}(1-x)^{\lambda_1}\bigr)$ to a certain Gauss hypergeometric equation or the above equation. \end{exmp}
\subsection{A confluence} \index{confluence} The non-trivial equation $(x-a)\tfrac{du}{dx} = \mu u$ obtained by the addition $\RAd\bigl((x-a)^\mu\bigr)\partial$ has a solution $(x-a)^\mu$ and regular singularities at $x=c$ and $\infty$. To consider the confluence of the point $x=a$ to $\infty$ we put $a=\frac 1c$. Then the equation is \[\bigl((1-cx)\partial+c\mu\bigr)u=0\] and it has a solution $u(x)=(1-cx)^\mu$.
The substitution $c=0$ for the operator $(1-cx)\partial+c\mu\in W[x;c,\mu]$ gives the trivial equation $\tfrac{du}{dx}=0$ with the trivial solution $u(x)\equiv1$. To obtain a nontrivial equation we introduce the parameter $\lambda=c\mu$ and we have the equation \[\bigl((1-cx)\partial+\lambda\bigr)u=0\] with the solution $(1-cx)^{\frac \lambda c}$. The function $(1-cx)^{\frac \lambda c}$ has the holomorphic parameters $c$ and $\lambda$ and the substitution $c=0$ gives the equation $(\partial+\lambda)u=0$ with the solution $e^{-\lambda x}$. Here $(1-cx)\partial+\lambda=\RAdei\bigl(\frac{\lambda}{1-cx}\bigr)\partial =\RAd\bigl((1-cx)^{\frac{\lambda}{c}}\bigr)\partial$.
This is the simplest example of the confluence and we define a confluence of simultaneous additions in this subsection.
\subsection{Versal additions}\label{sec:VAd}\index{versal!addition} \index{confluence!addition} For a function $h(c,x)$ with a holomorphic parameter $c\in\mathbb C$ we put \begin{equation}
\begin{split}
h_n(c_1,\dots,c_n,x)
&:=\frac1{2\pi\sqrt{-1}}\int_{|z|=R}
\frac{h(z,x)dz}{\prod_{j=1}^n(z-c_j)}\\
&=\sum_{k=1}^n\frac{h(c_k,x)}{\prod_{1\le i\le n,\, i\ne k}
(c_k-c_i)}
\end{split} \end{equation} with a sufficiently large $R>0$. Put \begin{equation}
h(c,x):=c^{-1}\log(1-c x)= -x - \frac c2x^2-\frac{c^2}3x^3 -\frac{c^3}4x^4-\cdots. \end{equation} Then \begin{equation}
(1-c x)h'(c,x)=-1 \end{equation} and \begin{equation}\label{eq:hnid} \begin{split}
h'_n(c_1,\dots,c_n,x)\prod_{1\le i\le n}(1-c_ix)
&=-\sum_{k=1}^n\frac{\prod_{1\le i\le n,\,i\ne k}(1-c_ix)}
{\prod_{1\le i\le n,\,i\ne k}(c_k-c_i)}\\
&=-x^{n-1}. \end{split} \end{equation} The last equality in the above is obtained as follows. Since the left hand side of \eqref{eq:hnid} is a holomorphic function of $(c_1,\dots,c_n)\in\mathbb C^n$ and the coefficient of $x^m$ is homogeneous of degree $m-n+1$, it is zero if $m<n-1$. The coefficient of $x^{n-1}$ proved to be $-1$ by putting $c_1=0$. Thus we have \begin{align}
h_n(c_1,\dots,c_n,x)&=-\int_0^x\frac{t^{n-1}dt}{\prod_{1\le i\le n}(1-c_it)},\\ \begin{split}
e^{\lambda_n h_n(c_1,\dots,c_n,x)}
&\circ\Bigl(\prod_{1\le i\le n}\bigl(1-c_ix\bigr)\Bigr)\partial\circ
e^{-\lambda_n h_n(c_1,\dots,c_n,x)}\\&=
\Bigl(\prod_{1\le i\le n}\bigl(1-c_ix\bigr)\Bigr)\partial
+\lambda_n x^{n-1}, \end{split}\\
e^{\lambda_n h_n(c_1,\dots,c_n,x)}&=
\prod_{k=1}^n\Bigl(1-c_kx\Bigr)^{\frac{\lambda_n}{c_k\prod_{\substack{1\le i\le n\\i\ne k}}(c_k-c_i)}}. \end{align} \begin{defn}[versal addition] We put \index{addition!confluence}\index{00AdV@$\AdV$, $\AdV^0$} \begin{align} \begin{split}
\AdV_{(\frac1{c_1},\dots,\frac1{c_p})}(\lambda_1,\dots,\lambda_p)&:=
\Ad\left(
\prod_{k=1}^p
\Bigl(1-c_kx\Bigr)^{
\sum_{n=k}^p{\frac{\lambda_n}{c_k\prod_{\substack{1\le i\le n\\i\ne k}}
(c_k-c_i)}}}\right)\\
&\phantom{:}=\Adei\left(-\sum_{n=1}^p\frac{\lambda_n x^{n-1}}
{\prod_{i=1}^n(1-c_ix)}\right), \end{split} \\ \RAdV_{(\frac1{c_1},\dots,\frac1{c_p})}(\lambda_1,\dots,\lambda_p)&=
\Red\circ \AdV_{(\frac1{c_1},\dots,\frac1{c_p})}(\lambda_1,\dots,\lambda_p). \end{align} We call $\RAdV_{(\frac1{c_1},\dots,\frac1{c_p})}(\lambda_1,\dots,\lambda_p)$ a \textsl{versal addition} at the $p$ points $\frac1{c_1},\dots,\frac1{c_p}$. \end{defn} \index{00RAdV@$\RAdV$} Putting \begin{align*}
h(c,x) := \log(x-c), \end{align*} we have \begin{align*}
h'_n(c_1,\dots,c_n,x)\prod_{1\le i\le n}(x-c_i)
= \sum _{k=1}^n \frac{\prod_{1\le i\le n,\ i\ne k}(x-c_i)}{\prod_{1\le i\le n,\ i\ne k}(c_k-c_i)}=1 \end{align*} and the \textsl{conflunence of additions around the origin} is defined by \index{addition!confluence!around the origin} \begin{align} \begin{split}
\AdV_{(a_1,\dots,a_p)}^0(\lambda_1,\dots,\lambda_p):\!&=
\Ad\left(
\prod_{k=1}^p
(x-a_k)^{\sum_{n=k}^p
{\frac{\lambda_n}{\prod_{\substack{1\le i\le n\\i\ne k}}
(a_k-a_i)}}}\right)\\
&=\Adei
\left(\sum_{n=1}^p\frac{\lambda_n}{\prod_{1\le i\le n}(x-a_i)}\right), \end{split} \\ \RAdV_{(a_1,\dots,a_p)}^0(\lambda_1,\dots,\lambda_p)&=
\Red\circ \AdV_{(a_1,\dots,a_p)}^0(\lambda_1,\dots,\lambda_p). \end{align} \begin{rem} Let $g_k(c,x)$ be meromorphic functions of $x$ with the holomorphic parameter $c=(c_1,\dots,c_p)\in\mathbb C^p$ for $k=1,\dots,p$ such that \[
g_k(c,x) \in
\sum_{i=1}^p\mathbb C\frac{1}{1-c_ix}
\text{ \ if \ }0\ne c_i\ne c_j\ne 0\quad(1\le i< j\le p,\ 1\le k\le p). \] Suppose $g_1(c,x),\dots,g_p(c,x)$ are linearly independent for any fixed $c\in\mathbb C^p$. Then there exist entire functions $a_{i,j}(c)$ of $c\in\mathbb C^p$ such that \[
g_k(x,c) = \sum_{n=1}^p \frac{a_{k,n}(c)x^{n-1}}{\prod_{i=1}^n(1-c_ix)} \] and $\bigl(a_{i,j}(c)\bigr)\in GL(p,\mathbb C)$ for any $c\in\mathbb C^p$ (cf.~\cite[Lemma~6.3]{O1}). Hence the versal addition is essentially unique. \end{rem}
\subsection{Versal operators}\label{sec:Versal}\index{versal!operator} If we apply a middle convolution to a versal addition of the trivial operator $\partial$, we have a \textsl{versal Jordan-Pochhammer} operator. \index{Jordan-Pochhammer!versal} \begin{align}
P:\!&=\RAd(\partial^{-\mu})\circ
\RAdV_{(\frac1{c_1},\dots,\frac1{c_p})}(\lambda_1,\dots,\lambda_p)\partial
\\
&=\RAd(\partial^{-\mu})\circ\Red\Bigl(
\partial+\sum_{k=1}^p\frac{\lambda_kx^{k-1}}{\prod_{\nu=1}^k(1-c_\nu x)}
\Bigr)
\notag\\
&=\partial^{-\mu+p-1}\Bigl(p_0(x)\partial+q(x)\Bigr)\partial^{\mu}
=\sum_{k=0}^p p_k(x)\partial^{p-k}\notag \intertext{with}
p_0(x) &=\prod_{j=1}^p(1-c_jx),\quad
q(x) =\sum_{k=1}^p\lambda_kx^{k-1}\prod_{j=k+1}^p(1-c_jx), \notag\\
p_k(x)&=\binom{-\mu+p-1}{k}
p_0^{(k)}(x) +
\binom{-\mu+p-1}{k-1}
q^{(k-1)}(x) .\notag \end{align} We naturally obtain the integral representation of solutions of the versal Jordan-Pochhammer equation $Pu=0$, which we show in the case $p=2$ as follows. \begin{exmp}\label{ex:VGHG} We have the \textsl{versal Gauss hypergeometric} operator \index{hypergeometric equation/function!Gauss!versal} \index{hypergeometric equation/function!Gauss!confluence} \begin{align*}
P_{c_1,c_2;\lambda_1,\lambda_2,\mu}:\!&=\RAd(\partial^{-\mu})\circ
\RAdV_{(\frac1{c_1},\frac1{c_2})}(\lambda_1,\lambda_2)\partial\allowdisplaybreaks\\
&=\RAd(\partial^{-\mu})\circ
\RAd\left((1-c_1x)^{\frac{\lambda_1}{c_1}+\frac{\lambda_2}{c_1(c_1-c_2)}} (1-c_2x)^{\frac{\lambda_2}{c_2(c_2-c_1)}}\right)\\
&=\RAd(\partial^{-\mu})\circ
\RAdei\left(-\tfrac{\lambda_1}{1-c_1x}
-\tfrac{\lambda_2 x}{(1-c_1x)(1-c_2x)} \right)
\partial\allowdisplaybreaks\\
&=\RAd(\partial^{-\mu})\circ
\Red\left(\partial+\tfrac{\lambda_1}{1-c_1x}+\tfrac{\lambda_2x}{(1-c_1x)(1-c_2x)}
\right)\allowdisplaybreaks\\
&=\Ad(\partial^{-\mu})\left(
\partial(1-c_1x)(1-c_2x)\partial+\partial(\lambda_1(1-c_2x)+\lambda_2 x)
\right)\allowdisplaybreaks\\
&=\bigl((1-c_1x)\partial+c_1(\mu-1)\bigr)\bigl((1-c_2x)\partial + c_2\mu\bigr)
\\
&\quad{}
+\lambda_1\partial + (\lambda_2-\lambda_1c_2)(x\partial+1-\mu)\allowdisplaybreaks\\
&=(1-c_1x)(1-c_2x)\partial^2\\
&\quad{}+\bigl((c_1+c_2)(\mu-1)+\lambda_1
+(2c_1c_2(1-\mu)+\lambda_2-\lambda_1c_2)x\bigr)\partial\\
&\quad{}+
(\mu-1)(c_1c_2\mu+\lambda_1c_2-\lambda_2), \end{align*} whose solution is obtained by applying $I_c^\mu$ to \[ K_{c_1,c_2;\lambda_1,\lambda_2}(x)=
(1-c_1x)^{\frac{\lambda_1}{c_1}+\frac{\lambda_2}{c_1(c_1-c_2)}}(1-c_2x)^{\frac{\lambda_2}{c_2(c_2-c_1)}} \] The equation $Pu=0$ has the Riemann scheme \begin{equation}
\begin{Bmatrix}
x = \frac1{c_1} & \frac 1{c_2} & \infty\\
0 & 0 &1-\mu&\!\!;\,x\\
\frac{\lambda_1}{c_1}+\frac{\lambda_2}{c_1(c_1-c_2)} + \mu
& \frac{\lambda_2}{c_2(c_2-c_1)} + \mu
&-\frac{\lambda_1}{c_1}+\frac{\lambda_2}{c_1c_2}-\mu
\end{Bmatrix}. \end{equation} Thus we have the following well-known confluent equations \index{Kummer's equation}
\begin{align*} P_{c_1,0;\lambda_1,\lambda_2,\mu}&=
(1-c_1x)\partial^2+\bigl(c_1(\mu-1)+\lambda_1+\lambda_2x\bigr)\partial
-\lambda_2(\mu-1),&&\text{(Kummer)}\\ K_{c_1,0;\lambda_1,\lambda_2}&=
(1-c_1x)^{\frac{\lambda_1}{c_1}+\frac{\lambda_2}{c_1^2}}
\exp({\tfrac{\lambda_2 x}{c_1}}), \allowdisplaybreaks\\ P_{0,0;0,-1,\mu} &=\partial^2-x\partial+(\mu-1),&&\text{(Hermite)}\\ \Ad(e^{\frac{1}4x^2})&P_{0,0;0,1,\mu}= (\partial-\tfrac{1}2x)^2+x(\partial-\tfrac{1}2x)
-(\mu-1)\allowdisplaybreaks\\ &=\partial^2+(\tfrac12-\mu-\tfrac{x^2}4), &&\text{(Weber)}\\ K_{0,0;0,\mp1}&=
\exp\Bigl(\int_0^x \!\pm tdt\Bigr)=\exp(\pm \tfrac{x^2}{2}). \end{align*} The solution \begin{align*}
D_{-\mu}(x)&:=(-1)^{-\mu}e^{\frac{x^2}4}I_\infty^\mu(e^{-\frac{x^2}2})
= \frac{e^{\frac{x^2}4}}{\Gamma(\mu)}
\int_x^\infty e^{-\frac{t^2}2}(t-x)^{\mu-1}dt\\
&= \frac{e^{\frac{x^2}4}}{\Gamma(\mu)}
\int_0^\infty e^{-\frac{(s+x)^2}2}s^{\mu-1}ds
= \frac{e^{-\frac{x^2}4}}{\Gamma(\mu)}
\int_0^\infty e^{-xs-\frac{t^2}2}s^{\mu-1}ds\\
&\sim x^{-\mu}e^{-\frac{x^2}4}
{}_2F_0(\tfrac\mu2,\tfrac\mu2+\tfrac12;-\tfrac2{x^2})
=\sum_{k=0}^\infty x^{-\mu}e^{-\frac{x^2}4}
\frac{(\tfrac\mu2)_k(\tfrac\mu2+\tfrac12)_k}{k!}\bigl(-\tfrac2{x^2}\bigr)^k \end{align*} of Weber's equation $\frac{d^2u}{dx^2}=(\frac{x^2}4+\mu-\frac12)u$ is called a parabolic cylinder function (cf.~\cite[\S16.5]{WW}). Here the above last line is an asymptotic expansion when $x\to+\infty$. \index{Weber's equation}\index{parabolic cylinder function}
The normal form of Kummer equation is obtained by the coordinate transformation $y=x-\frac1{c_1}$ but we also obtain it as follows: \begin{align*}
P_{c_1;\lambda_1,\lambda_2,\mu}
:\!&=\RAd(\partial^{-\mu})\circ\Red\circ \Ad(x^{\lambda_2})\circ \AdV_{\frac1{c_1}} (\lambda_1)\partial \allowdisplaybreaks\\
&=\RAd(\partial^{-\mu})\circ\Red\bigl(
\partial - \tfrac{\lambda_2}{x}+\tfrac{\lambda_1}{1-c_1x}\bigr) \allowdisplaybreaks\\
&=\Ad(\partial^{-\mu})\bigl(\partial x(1-c_1x)\partial
-\partial(\lambda_2-(\lambda_1+c_1\lambda_2)x)\bigr) \allowdisplaybreaks\\
&=(x\partial +1 -\mu)\bigl((1-c_1x)\partial+c_1\mu)-\lambda_2\partial
+(\lambda_1+c_1\lambda_2)(x\partial +1 -\mu) \allowdisplaybreaks\\
&=x(1-c_1x)\partial^2+\bigl(1-\lambda_2-\mu
+(\lambda_1+c_1(\lambda_2+2\mu-2))x\bigr)\partial\\
&\quad{}
+(\mu-1)\bigl(\lambda_1+c_1(\lambda_2+\mu)\bigr), \allowdisplaybreaks\\
P_{0;\lambda_1,\lambda_2,\mu}
&=x\partial^2+(1-\lambda_2-\mu +\lambda_1x)\partial
+\lambda_1(\mu-1),\\
P_{0;-1,\lambda_2,\mu}
&=x\partial^2+(1-\lambda_2-\mu - x)\partial
+1-\mu\qquad\text{(Kummer)}, \allowdisplaybreaks\\
K_{c_1;\lambda_1,\lambda_2}(x)&:=x^{\lambda_2}
(1-c_1x)^{\frac{\lambda_1}{c_1}},\quad
K_{0;\lambda_1,\lambda_2}(x)=x^{\lambda_2}\exp(-\lambda_1x). \end{align*} The Riemann scheme of the equation $P_{c_1;\lambda_1,\lambda_2,\mu}u=0$ is \begin{align}
\begin{Bmatrix}
x=0 & \frac1{c_1} & \infty \\
0 & 0 &1-\mu&\!\!;\,x\\
\lambda_2+\mu & \frac{\lambda_1}{c_1}+\mu&-\frac{\lambda_1}{c_1}-\lambda_2-\mu
\end{Bmatrix} \end{align} and the local solution at the origin corresponding to the characteristic exponent $\lambda_2+\mu$ is given by \begin{align*} I_0^\mu(K_{c_1;\lambda_1,\lambda_2})(x)
= \frac1{\Gamma(\mu)}\int_0^x
t^{\lambda_2}
(1-c_1t)^{\frac{\lambda_1}{c_1}}(x-t)^{\mu-1}dt. \end{align*} In particular, we have a solution \begin{align*}
u(x)&=I_0^\mu(K_{0;-1,\lambda_2})(x)=
\frac1{\Gamma(\mu)}\int_0^x t^{\lambda_2}e^{t}(x-t)^{\mu-1}dt\\
&=\frac{x^{\lambda_2+\mu}}{\Gamma(\mu)}\int_0^1
s^{\lambda_2}(1-s)^{\mu-1}e^{xs}ds\qquad(t=xs)\\
&=\frac{\Gamma(\lambda_2+1)x^{\lambda_2+\mu}}{\Gamma(\lambda_2+\mu+1)}
{}_1F_1(\lambda_2+1,\mu+\lambda_2+1;x) \end{align*} of the Kummer equation $P_{0;-1,\lambda_2,\mu}u=0$ corresponding to the exponent $\lambda_2+\mu$ at the origin. If $c_1\notin(-\infty,0]$ and $x\notin[0,\infty]$ and $\lambda_2\notin\mathbb Z_{\ge0}$, the local solution at $-\infty$ corresponding to the exponent $-\lambda_2-\frac{\lambda_1}{c_1}-\mu$ is given by \begin{align*}
&\frac1{\Gamma(\mu)}
\int_{-\infty}^x
(-t)^{\lambda_2}
(1-c_1t)^{\frac{\lambda_1}{c_1}}(x-t)^{\mu-1}dt\\
&=\frac{(-x)^{\lambda_2}}{\Gamma(\mu)}\int_0^\infty
\left(1-\frac{s}{x}\right)^{\lambda_2}
\bigl(1+c_1(s-x)\bigr)^{\frac{\lambda_1}{c_1}}
s^{\mu-1}ds\qquad(s=x-t)
\\
&\hspace{-20pt}\xrightarrow[c_1\to+0]{\lambda_1=-1}\quad\\
&
\frac{(-x)^{\lambda_2}}{\Gamma(\mu)}\int_{0}^\infty
\left(1-\frac{s}{x}\right)^{\lambda_2}e^{x-s}
s^{\mu-1}ds\\ &=
\frac{(-x)^{\lambda_2}e^x}{\Gamma(\mu)}\int_{0}^\infty
s^{\mu-1}e^{-s}\left(1-\frac{s}{x}\right)^{\lambda_2}ds\\ &\sim \sum_{n=0}^\infty \frac{\Gamma(\mu+n)\Gamma(-\lambda_2+n)}{\Gamma(\mu)\Gamma(-\lambda_2)n!x^n}(-x)^{\lambda_2}
e^x =(-x)^{\lambda_2}e^{x}{}_2F_0(-\lambda_2,\mu;\tfrac1x). \end{align*} Here the above last line is an asymptotic expansion of a rapidly decreasing solution of the Kummer equation when $\mathbb R\ni-x\to+\infty$. The Riemann scheme of the equation $P_{0;-1,\lambda_2,\mu}u=0$ can be expressed by \begin{equation}
\begin{Bmatrix}
x = 0 & \infty & (1)\\
0 & 1-\mu & 0 \\
\lambda_2+\mu & -\lambda_2 & 1
\end{Bmatrix}. \end{equation}\index{Riemann scheme} In general, the expression $
\begin{Bmatrix}
\infty & (r_1) & \cdots & (r_k)\\
\lambda & \alpha_1 & \cdots &\alpha_k
\end{Bmatrix} $ with $0<r_1<\cdots<r_k$ means the existence of a solution $u(x)$ satisfying \begin{equation}
u(x)\sim x^{-\lambda}
\exp\Bigl(\sum_{\nu=1}^k\alpha_\nu\frac{x^{r_\nu}}{r_\nu}\Bigr)
\text{ \ for \ }|x|\to\infty \end{equation} under a suitable restriction of $\Arg x$. Here $k\in\mathbb Z_{\ge 0}$ and $\lambda$, $\alpha_\nu\in\mathbb C$. \end{exmp}
\section{Series expansion}\label{sec:series} In this section we review the Euler transformation and remark on its relation to middle convolutions.
First we note the following which will be frequently used: \begin{gather}
\int_0^1 t^{\alpha-1}(1-t)^{\beta-1} dt =
\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)},\label{eq:beta}
\allowdisplaybreaks\\
\begin{split}
(1-t)^{-\gamma}&=\sum_{\nu=0}^\infty
\frac{(-\gamma)(-\gamma-1)\cdots(-\gamma-\nu+1)}{\nu!}(-t)^\nu\\
&= \sum_{\nu=0}^\infty\frac{\Gamma(\gamma+\nu)}{\Gamma(\gamma)\nu!}t^\nu
= \sum_{\nu=0}^\infty\frac{(\gamma)_\nu}{\nu!} t^\nu.\noindent
\end{split}\label{eq:seriesP} \end{gather} The integral \eqref{eq:beta} converges if $\RE \alpha>0$ and $\RE \beta>0$ and the right hand side is meromorphically continued to $\alpha\in\mathbb C$ and $\beta\in\mathbb C$. If the integral in \eqref{eq:beta} is interpreted in the sense of generalized functions, \eqref{eq:beta} is valid if $\alpha\notin\{0,-1,-2,\ldots\}$ and $\beta\notin\{0,-1,-2,\ldots\}$.
Euler transformation $I_c^\mu$ is sometimes expressed by $\partial^{-\mu}$ and as is shown in (\cite[\S5.1]{Kh}), we have \begin{align} \begin{split}
I_c^\mu u(x):\!&=\frac{1}{\Gamma(\mu)}\int_c^x (x-t)^{\mu-1} u(t) dt
\label{eq:Icdef}\\
&=\frac{(x-c)^{\mu}}{\Gamma(\mu)}\int_0^1
(1-s)^{\mu-1}u((x-c)s+c)ds, \end{split}\allowdisplaybreaks\\
I_c^\mu\circ I_c^{\mu'} &= I_c^{\mu+\mu'},\allowdisplaybreaks\label{eq:Icprod}\\
I_c^{-n}u(x)&=\frac{d^n}{dx^n}u(x),\allowdisplaybreaks\label{eq:Icdif}\\ \begin{split}
I_c^{\mu}\sum_{n=0}^\infty c_n(x-c)^{\lambda+n}&=
\sum_{n=0}^\infty
\frac{\Gamma(\lambda+n+1)}{\Gamma(\lambda+\mu+n+1)}
c_n(x-c)^{\lambda+\mu+n}\\
&=\frac{\Gamma(\lambda+1)}{\Gamma(\lambda+\mu+1)}
\sum_{n=0}^\infty\frac{(\lambda+1)_nc_n}{(\lambda+\mu+1)_n}(x-c)^{\lambda+\mu+n}, \end{split}\label{eq:IcP}\allowdisplaybreaks\\
I_\infty^{\mu}\sum_{n=0}^\infty c_n x^{\lambda-n}&=
e^{\pi\sqrt{-1}\mu}\sum_{n=0}^\infty
\frac{\Gamma(-\lambda-\mu+n)}{\Gamma(-\lambda+n)}
c_n x^{\lambda+\mu-n}.\label{eq:IinfP} \end{align} Moreover the following equalities which follow from \eqref{eq:Gmid} are also useful. \begin{equation} \begin{split}
&I_0^{\mu}\sum_{n=0}^\infty c_nx^{\lambda+n}(1-x)^\beta\\
&\ =\frac{\Gamma(\lambda+1)}{\Gamma(\lambda+\mu+1)}
\sum_{m,\,n=0}^\infty\frac{(\lambda+1)_{m+n}(-\beta)_mc_n}
{(\lambda+\mu+1)_{m+n}m!}x^{\lambda+\mu+m+n}\\
&\ =\frac{\Gamma(\lambda+1)}{\Gamma(\lambda+\mu+1)}(1-x)^{-\beta}
\sum_{m,\,n=0}^\infty\frac{(\lambda+1)_n(\mu)_m(-\beta)_mc_n}
{(\lambda+\mu+1)_{m+n}m!}x^{\lambda+\mu+n}\Bigl(\frac x{x-1}\Bigr)^m. \end{split}\label{eq:I0H} \end{equation}
If $\lambda\notin\mathbb Z_{<0}$ (resp.\ $\lambda+\mu\notin\mathbb Z_{\ge0})$ and moreover the power series $\sum_{n=0}^\infty c_nt^n$ has a positive radius of convergence, the equalities \eqref{eq:IcP} (resp,\ \eqref{eq:IinfP}) is valid since $I_c^\mu$ (resp.\ $I_\infty^\mu$) can be defined through analytic continuations with respect to the parameters $\lambda$ and $\mu$. Note that $I_c^\mu$ is an invertible map of $\mathcal O_c(x-c)^\lambda$ onto $\mathcal O_c(x-c)^{\lambda+\mu}$ if $\lambda\notin\{-1,-2,-3,\ldots\}$ and $\lambda+\mu \notin\{-1,-2,-3,\ldots\}$.
\begin{prop}\label{prop:RAdIc} Let\/ $\lambda$ and\/ $\mu$ be complex numbers satisfying\/ $\lambda\notin\mathbb Z_{<0}$. Differentiating the equality \eqref{eq:IcP} with respect to\/ $\lambda$, we have the linear map \begin{equation}\label{eq:Iclog}
I_c^\mu: \mathcal O_c(\lambda,m) \to \mathcal O_c(\lambda+\mu,m) \end{equation} under the notation \eqref{def:Ocm}, which is also defined by \eqref{eq:Icdef} if\/ $\RE\lambda > -1$ and\/ $\RE\mu > 0$. Here\/ $m$ is a non-negative integer. Then we have \begin{equation}\label{eq:Iclogtop}
I_c^\mu\bigl(\sum_{j=0}^m \phi_j\log^j(x-c)\bigr)
- I_c^\mu(\phi_m)\log^m(x-c)\in\mathcal O(\lambda+\mu,m-1) \end{equation} for\/ $\phi_j\in\mathcal O_c$ and\/ $I_c^\mu$ satisfies \eqref{eq:Icp}. The map \eqref{eq:Iclog} is bijective if\/ $\lambda+\mu\notin\mathbb Z_{<0}$. In particular for\/ $k\in\mathbb Z_{\ge0}$ we have\/ $I_c^\mu\partial^k=\partial^kI_c^\mu=I_c^{\mu-k}$ on\/ $\mathcal O_c(\lambda,m)$ if\/ $\lambda-k\notin\{-1,-2,-3,\dots\}$.
Suppose that\/ $P\in W[x]$ and\/ $\phi\in\mathcal O_c(\lambda,m)$ satisfy\/ $P\phi=0$, $P\ne0$ and\/ $\phi\ne0$. Let\/ $k$ and\/ $N$ be non-negative integers such that \begin{equation}
\partial^k P = \sum_{i=0}^N\sum_{j\ge 0}a_{i,j}\partial^i\bigl((x-c)\partial\bigr)^j \end{equation} with suitable\/ $a_{j,j}\in\mathbb C$ and put\/ $Q=\sum_{i=0}^N\sum_{j\ge0}c_{i,j}\partial^i\bigl((x-c)\partial - \mu\bigr)^j$. Then if\/ $\lambda\notin\{N-1,N-2,\ldots,0,-1,\ldots\}$, we have \begin{equation}
I_c^\mu\partial^kPu=QI_c^\mu(u) \text{ \ for \ }u\in\mathcal O_c(\lambda,m) \end{equation} and in particular\/ $QI_c^\mu(\phi)=0$.
Fix\/ $\ell\in\mathbb Z$. For\/ $u(x)=\sum_{i=\ell}^\infty\sum_{j=0}^mc_{i,j}(x-c)^i\log^j(x-c) \in\mathcal O_c(\ell,m)$ we put\/ $(\Gamma_Nu)(x)=\sum_{\nu=\max\{\ell,N-1\}}^\infty \sum_{j=0}^mc_{i,j}(x-c)^i\log^j(x-c)$. Then \[
\Bigl(\prod_{\ell-N\le\nu\le N-1}\bigl((x-c)\partial-\nu\bigr)^{m+1}\Bigr)
\partial^kP\bigl(u(x) - (\Gamma_Nu)(x)\bigr)=0 \] ane therefore \begin{equation}
\begin{split}
&\Bigl(\prod_{\ell-N\le\nu\le N-1}\bigl((x-c)\partial-\mu-\nu\bigr)^{m+1}\Bigr)
QI_c^\mu(\Gamma_Nu)\\
&\quad=
I_c^\mu\Bigl(\prod_{\ell-N\le\nu\le N-1}\bigl((x-c)\partial-\nu\bigr)^{m+1}\Bigr)
\partial^kPu.
\end{split} \end{equation} In particular, $\prod_{\ell-N\le\nu\le N-1}\bigl((x-c)\partial-\mu-\nu\bigr)^{m+1} \cdot QI_c^\mu\bigl(\Gamma_N(u)\bigr)=0$ if\/ $Pu=0$.
Suppose moreover\/ $\lambda\notin\mathbb Z$ and $\lambda+\mu\notin\mathbb Z$ and\/ $Q=ST$ with\/ $S$, $T\in W[x]$ such that\/ $x=c$ is not a singular point of the operator\/ $S$. Then\/ $TI_c^\mu(\phi)=0$. In particular, \begin{equation} \bigl(\RAd(\partial^{-\mu})P\bigr)I_c^\mu(\phi)=0. \end{equation} Hence if the differential equation\/ $\bigl(\RAd(\partial^{-\mu})P\bigr)v=0$ is irreducible, we have \begin{equation}
W(x)\bigl(\RAd(\partial^{-\mu})P\bigr)=\{T\in W(x)\,;\,
TI_c^\mu(\phi)=0\}. \end{equation} The statements above are also valid even if we replace\/ $x-c$, $I_c^\mu$ by \/$\frac1x$, $I_\infty^\mu$, respectively. \end{prop} \begin{proof} It is clear that \eqref{eq:Iclog} is well-defined and \eqref{eq:Iclogtop} is valid. Then \eqref{eq:Iclog} is bijective because of \eqref{eq:IcP} and \eqref{eq:Iclogtop}. Since \eqref{eq:Icp} is valid when $m=0$, it is also valid when $m=1,2,\ldots$ by the definition of \eqref{eq:Iclog}.
The equalities \eqref{eq:IcP} and \eqref{eq:IinfP} assure that $QI_c^\mu(\phi)=0$. Note that $TI_c^\mu(\phi)\in\mathcal O(\lambda+\mu-N,m)$ with a suitable positive integer $N$. Since $\lambda+\mu-N\notin\mathbb Z$ and any solution of the equation $Sv=0$ is holomorphic at $x=c$, the equality $S\bigl(TI_c^\mu(\phi)\bigr)=0$ implies $TI_c^\mu(\phi)=0$.
The remaining claims in the theorem are similarly clear. \end{proof} \begin{rem} {\rm i)} Let $\gamma: [0,1]\to \mathbb C$ be a path such that $\gamma(0)=c$ and $\gamma(1)=x$. Suppose $u(x)$ is holomorphic along the path $\gamma(t)$ for $0<t\le 1$ and $u(\gamma(t))=\phi(\gamma(t))$ for $0<t\ll 1$ with a suitable function $\phi\in\mathcal O_c(\lambda,m)$. Then $I_c^\mu(u)$ is defined by the integration along the path $\gamma$. In fact, if the path $\gamma(t)$ with $t\in[0,1]$ splits into the three paths corresponding to the decomposition $[0,1]=[0,\epsilon]\cup [\epsilon,1-\epsilon]\cup[1-\epsilon,1]$ with $0<\epsilon\ll1$. Let $c_1=c,\dots,c_p$ be points in $\mathbb C^n$ and suppose moreover $u(x)$ is extended to a multi-valued holomorphic function on $\mathbb C\setminus\{c_1,\dots,c_p\}$. Then $I_c^x(u)$ also defines a multi-valued holomorphic function on $\mathbb C\setminus\{c_1,\dots,c_p\}$.
{\rm ii)} Proposition~\ref{prop:RAdIc} is also valid if we replace $\mathcal O_c(\lambda,m)$ by the space of functions given in Remark~\ref{rem:defIc} ii). In fact the above proof also works in this case. \end{rem}
\section{Contiguity relation}\label{sec:contig} The following proposition is clear from Proposition~\ref{prop:RAdIc}. \begin{prop}\label{prop:config0} Let $\phi(x)$ be a non-zero solution of an ordinary differential equation $Pu=0$ with an operator $P\in W[x]$. Let $P_j$ and $S_j\in W[x]$ for $j=1,\dots,N$ so that $\sum_{j=1}^N P_jS_j\in W[x]P$. Then for a suitable $\ell\in\mathbb Z$ we have \begin{equation}
\sum Q_j\bigl(I_c^\mu(\phi_j)\bigr)=0 \end{equation} by putting \begin{equation}
\begin{split}
\phi_j&=S_j\phi,\\
Q_j &= \partial^{\ell-\mu}\circ P_j\circ\partial^{\mu}\in W[x],
\end{split}
\quad(j=1,\dots,N), \end{equation} if $\phi(x)\in\mathcal O(\lambda,m)$ with a non-negative integer $m$ and a complex number $\lambda$ satisfying $\lambda\notin\mathbb Z$ and $\lambda+\mu\notin\mathbb Z$ or $\phi(x)$ is a function given in Remark~\ref{rem:defIc} ii). If $P_j=\sum_{k\ge 0,\ \ell\ge 0}c_{j,k,\ell}\partial^k\vartheta^\ell$ with $c_{j,k,\ell}\in\mathbb C$, then we can assume $\ell\le 0$ in the above. Moreover we have \begin{equation}
\partial\bigl(I_c^{\mu+1}(\phi_1)\bigr) = I_c^\mu(\phi_1). \end{equation} \end{prop}
\begin{proof} Fix an integer $k$ such that $\partial^k P_j=\tilde P_j(\partial,\vartheta)=\sum_{i_1,i_2} c_{i_1,i_2}\partial^{i_1}\vartheta^{i_2}$ with $c_{i_1,i_2}\in\mathbb C$. Since $0=\sum_{j=1}^N \partial^k P_jS_j\phi$, Proposition~\ref{prop:RAdIc} proves $0=\sum_{j=1}^N I_c^\mu(\tilde P_j(\partial,\vartheta)S_j\phi)= \sum_{j=1}^N \tilde P_j(\partial,\vartheta-\mu)I_c^\mu(S_j\phi)$, which implies the first claim of the proposition.
The last claim is clear from \eqref{eq:Icprod} and \eqref{eq:Icdif}. \end{proof} \begin{cor} Let $P(\xi)$ and $K(\xi)$ be non-zero elements of $W[x;\xi]$. If we substitute $\xi$ and $\mu$ by generic complex numbers, we assume that there exists a solution $\phi_\xi(x)$ satisfying the assumption in the preceding proposition and that $I_c^\mu(\phi_\xi)$ and $I_c^\mu(K(\xi)\phi_\xi)$ satisfy irreducible differential equations $T_1(\xi,\mu)v_1=0$ and $T_2(\xi,\mu)v_2=0$ with $T_1(\xi,\mu)$ and $T_2(\xi,\mu)\in W(x;\xi,\mu)$, respectively. Then the differential equation $T_1(\xi,\mu)v_1=0$ is isomorphic to $T_2(\xi,\mu)v_2=0$ as $W(x;\xi,\mu)$-modules. \end{cor} \begin{proof} Since $K(\xi)\cdot 1 - 1\cdot K(\xi)=0$, we have $Q(\xi,\mu)I_c^\mu(\phi_\xi)=\partial^\ell I_c^\mu(K(\xi)\phi_\xi)$ with $Q(\xi,\mu)=\partial^{\ell-\mu}\circ K(\xi)\circ\partial^{\mu}$. Since $\partial^{\ell}I_c^\mu(\phi_\xi)\ne0$ and the equations $T_j(\xi,\mu)v_j=0$ are irreducible for $j=1$ and $2$, there exist $R_1(\xi,\mu)$ and $R_2(\xi,\mu)\in W(x;\xi,\mu)$ such that $I_c^\mu(\phi_\xi)=R_1(\xi,\mu)Q(\xi,\mu)I_c^\mu(\phi_\xi) =R_1(\xi,\mu)\partial^\ell I_c^\mu(K(\xi)\phi_\xi)$ and $I_c^\mu(K(\xi)\phi_\xi)=R_2(\xi,\mu)\partial^\ell I_c^\mu(K(\xi)\phi_\xi) =R_2(\xi,\mu)Q(\xi,\mu)I_c^\mu(\phi_\xi)$. Hence we have the corollary. \end{proof} Using the proposition, we get the contiguity relations with respect to the parameters corresponding to powers of linear functions defining additions and the middle convolutions.
For example, in the case of Gauss hypergeometric functions, we have \begin{align*} u_{\lambda_1,\lambda_2,\mu}(x):\!&=I_0^\mu(x^{\lambda_1}(1-x)^{\lambda_2}),\\ u_{\lambda_1,\lambda_2,\mu-1}(x)&=\partial u_{\lambda_1,\lambda_2,\mu}(x),\\ \partial u_{\lambda_1+1,\lambda_2,\mu}(x)&=(x\partial+1-\mu)u_{\lambda_1,\lambda_2,\mu}(x),\\ \partial u_{\lambda_1,\lambda_2+1,\mu}(x)&=((1-x)\partial+\mu-1)u_{\lambda_1,\lambda_2,\mu}(x). \end{align*} Here Proposition~\ref{prop:config0} with $\phi=x^{\lambda_1}(1-x)^{\lambda_2}$, $(P_1,S_1,P_2,S_2)=(1,x,-x,1)$ and $\ell=1$ gives the above third identity.
Since $P_{\lambda_1,\lambda_2,\mu}u_{\lambda_1,\lambda_2,\mu}(x)=0$ with \begin{align*} P_{\lambda_1,\lambda_2,\mu}&
=\bigl(x(1-x)\partial+(1-\lambda_1-\mu-(2-\lambda_1-\lambda_2-2\mu)x\bigr)\partial\\ &\quad{}
-(\mu-1)(\lambda_1+\lambda_2+\mu) \intertext{as is given in Example~\ref{ex:midconv}, the inverse of the relation $u_{\lambda_1,\lambda_2,\mu-1}(x)=\partial u_{\lambda_1,\lambda_2,\mu}(x)$ is} u_{\lambda_1,\lambda_2,\mu}(x)&= -\frac{x(1-x)\partial+(1-\lambda_1-\mu-(2-\lambda_1-\lambda_2-2\mu)x\bigr)} {(\mu-1)(\lambda_1+\lambda_2+\mu)} u_{\lambda_1,\lambda_2,\mu-1}(x). \end{align*} The equalities $u_{\lambda_1,\lambda_2,\mu-1}(x)=\partial u_{\lambda_1,\lambda_2,\mu}(x)$ and \eqref{eq:Gmid} mean \begin{align*} &\frac{\Gamma(\lambda_1+1)x^{\lambda_1+\mu-1}}{\Gamma(\lambda_1+\mu)} F(-\lambda_2,\lambda_1+1,\lambda_1+\mu;x)\\ &=\frac{\Gamma(\lambda_1+1)x^{\lambda_1+\mu-1}}{\Gamma(\lambda_1+\mu)} F(-\lambda_2,\lambda_1+1,\lambda_1+\mu+1;x)\\ &\quad{}+\frac{\Gamma(\lambda_1+1)x^{\lambda_1+\mu}}{\Gamma(\lambda_1+\mu+1)}
\frac{d}{dx}F(-\lambda_2,\lambda_1+1,\lambda_1+\mu+1;x) \end{align*} and therefore $u_{\lambda_1,\lambda_2,\mu-1}(x)=\partial u_{\lambda_1,\lambda_2,\mu}(x)$ is equivalent to \[
(\gamma-1)F(\alpha,\beta,\gamma-1;x)=(\vartheta+\gamma-1)F(\alpha,\beta,\gamma;x). \] The contiguity relations are very important for the study of differential equations. For example the author's original proof of the connection formula \eqref{eq:Icon} announced in \cite{O3} is based on the relations (cf.~\S\ref{sec:C2}).
Some results related to contiguity relations will be given in \S\ref{sec:shift} but we will not go further in this subject and it will be discussed in another paper.
\section{Fuchsian differential equation and generalized Riemann scheme} \label{sec:index} \subsection{Generalized characteristic exponents}\label{sec:Gexp} We examine the Fuchsian differential equations \begin{equation}
P = a_n(x)\tfrac{d^n}{dx^n}
+a_{n-1}(x)\tfrac{d^{n-1}}{dx^{n-1}}+\cdots+a_0(x)
\label{eq:P} \end{equation} with given local monodromies at regular singular points. For this purpose we first study the condition so that monodromy generators of the solutions of a Fuchsian differential equation is semisimple even when its exponents are not free of multiplicity. \begin{lem}\label{lem:block} Suppose that the operator \eqref{eq:P} defined in a neighborhood of the origin has a regular singularity at the origin. We may assume $a_\nu(x)$ are holomorphic at $0$ and $a_n(0) = a_n'(0)=\cdots=a_n^{(n-1)}(0)=0$ and $a_n^{(n)}(0)\ne0$. Then the following conditions are equivalent for a positive integer $k$. \begin{align}
P &= x^kR&&\text{with a suitable holomorphic differential operator $R$}
\label{C:divblock}\\[-2pt]
& && \text{at the origin,}\notag \allowdisplaybreaks\\
Px^\nu &= o(x^{k-1})&&\text{for \ \ }\nu=0,\ldots,k-1,
\label{C:applyblock} \allowdisplaybreaks\\
Pu &=0&&\text{has a solution $x^\nu+o(x^{k-1})$ for }
\nu=0,\dots,k-1,\label{C:solblock} \allowdisplaybreaks\\
P &= \sum_{j\ge0}x^jp_j(\vartheta)&&\text{with polynomials }p_j
\text{\ satisfying\ }p_j(\nu)=0\label{C:formblock}\\[-10pt]
& &&\text{for \ }0\le\nu< k-j\text{ \ and \ }
j=0,\dots,k-1.\notag \end{align} \end{lem} \begin{proof} \eqref{C:divblock} $\Rightarrow$ \eqref{C:applyblock} $\Leftrightarrow$ \eqref{C:solblock} is clear.
Assume \eqref{C:applyblock}. Then $Px^\nu=o(x^{k-1})$ for $\nu=0,\dots,k-1$ implies $a_j(x)=x^kb_j(x)$ for $j=0,\dots,k-1$. Since $P$ has a regular singularity at the origin, $a_j(x)=x^jc_j(x)$ for $j=0,\dots,n$. Hence we have \eqref{C:divblock}.
Since $Px^\nu = \sum_{j=0}^\infty x^{\nu+j}p_j(\nu)$, the equivalence \eqref{C:applyblock} $\Leftrightarrow$ \eqref{C:formblock} is clear. \end{proof} \begin{defn} Suppose $P$ in \eqref{eq:P} has a regular singularity at $x=0$. Under the notation \eqref{eq:mult} we define that $P$ has a (\textsl{generalized}) \textsl{characteristic exponent} \index{characteristic exponent!generalized} $[\lambda]_{(k)}$ at $x=0$ if $x^{n-k}\Ad(x^{-\lambda})(a_n(x)^{-1}P)\in W[x]$. \end{defn}
Note that Lemma~\ref{lem:block} shows that $P$ has a characteristic exponent $[\lambda]_{(k)}$ at $x=0$ if and only if \begin{equation}
x^na_n(x)^{-1}P=\sum_{j\ge 0}x^jq_j(\vartheta)\prod_{0\le i< k-j} (\vartheta - \lambda- i) \end{equation} with polynomials $q_j(t)$. By a coordinate transformation we can define generalized characteristic exponents for any regular singular point as follows.
\begin{defn}[generalized characteristic exponents] \label{def:RScheme} Suppose $P$ in \eqref{eq:P} has regular singularity at $x=c$. Let $n=m_1+\cdots+m_q$ be a partition of the positive integer $n$ and let $\lambda_1,\dots,\lambda_q$ be complex numbers. We define that $P$ has the (set of \textsl{generalized}) \textsl{characteristic exponents} $\{[\lambda_1]_{(m_1)},\dots,[\lambda_q]_{(m_q)}\}$ and the \textsl{spectral type} $\{m_1,\dots,m_q\}$ at $x=c\in\mathbb C\cup\{\infty\}$ if there exist polynomials $q_\ell(s)$ such that \begin{equation}\label{eq:GEXP}
(x-c)^na_n(x)^{-1}P=\sum_{\ell\ge 0}(x-c)^\ell
q_\ell\bigl((x-c)\partial\bigr)
\prod_{\nu=1}^q\,\prod_{0\le i< m_\nu-\ell}
\bigl((x-c)\partial-\lambda_\nu-i\bigr) \end{equation} in the case when $c\ne\infty$ and \begin{equation}\label{eq:gcexp}
x^{-n}a_n(x)^{-1}P=\sum_{\ell\ge 0}x^{-\ell}
q_\ell\bigl(\vartheta)
\prod_{\nu=1}^q\,\prod_{0\le i< m_\nu-\ell}
\bigl(\vartheta+\lambda_\nu+i\bigr) \end{equation} in the case when $c=\infty$. Here if $m_j=1$, $[\lambda_j]_{(m_j)}$ may be simply written as $\lambda_j$. \end{defn}
\begin{rem}\label{rem:GCexp} {\rm i) } In Definition~\ref{def:RScheme} we may replace the left hand side of \eqref{eq:GEXP} by $\phi(x)a_n(x)^{-1}P$ where $\phi$ is analytic function in a neighborhood of $x=c$ such that $\phi(c)=\cdots=\phi^{(n-1)}(c)=0$ and $\phi^{(n)}(c)\ne 0$. In particular when $a_n(c)=\cdots=a_n^{(n)}(c)=0$ and $a_n(c)\ne 0$, $P$ is said to be \textsl{normalized} at the singular point $x=c$ and the left hand side of \eqref{eq:GEXP} can be replaced by $P$. \index{Fuchsian differential equation/operator!normalized}
In particular when $c=0$ and $P$ is normalized at the regular singular point $x=0$, the condition \eqref{eq:GEXP} is equivalent to \begin{equation}
\prod_{\nu=1}^k\prod_{0\le i<m_\nu-\ell}(s-\lambda_\nu-i)\bigm|p_j(s)
\qquad(\forall\ell=0,1,\dots,\max\{m_1,\dots,m_k\}-1) \end{equation} under the expression $
P = \sum_{j=0}^\infty x^jp_j(\vartheta) $.
{\rm ii) } In Definition~\ref{def:RScheme} the condition that the operator $P$ has a set of generalized characteristic exponents $\{\lambda_1,\dots,\lambda_n\}$ is equivalent to the condition that it is the set of the usual characteristic exponents.
{\rm iii) } Any one of $\{\lambda,\lambda+1,\lambda+2\}$, $\{[\lambda]_{(2)},\lambda+2\}$ and $\{\lambda,[\lambda+1]_{(2)}\}$ is the set of characteristic exponents of \[
P= (\vartheta-\lambda)(\vartheta-\lambda-1)(\vartheta - \lambda - 2+x)
+x^2(\vartheta-\lambda+1) \] at $x=0$ but $\{[\lambda]_{(3)}\}$ is not.
{\rm iv) } Suppose $P$ has a holomorphic parameter $t\in B_1(0)$ (cf.~\eqref{eq:Brc}) and $P$ has regular singularity at $x=c$. Suppose the set of the corresponding characteristic exponents is $\{[\lambda_1(t)]_{(m_1)},\ldots,[\lambda_q(t)]_{(m_q)}\}$ for $t\in B_1(0)\setminus\{0\}$ with $\lambda_\nu(t)\in\mathcal O\bigl(B_1(0)\bigr)$. Then this is also valid in the case $t=0$, which clearly follows from the definition.
When \[
P=\sum_{\ell\ge 0}x^{-\ell}
q_\ell\bigl((x-c)\partial\bigr)
\prod_{\nu=1}^q\,\prod_{0\le i< m_\nu-\ell}
\bigl((x-c)\partial-\lambda_\nu-i\bigr), \] we put \[
P_t=\sum_{\ell\ge 0}x^{-\ell}
q_\ell\bigl((x-c)\partial\bigr)
\prod_{\nu=1}^q\,\prod_{0\le i< m_\nu-\ell}
\bigl((x-c)\partial-\lambda_\nu-\nu t-i\bigr). \] Here $\lambda_\nu\in\mathbb C$, $q_0\ne 0$ and $\ord P = m_1+\dots+m_q$. Then the set of the characteristic exponents of $P_t$ is $\{[\tilde\lambda_1(t)]_{(m_1)},\ldots, [\tilde\lambda_q(t)]_{(m_q)}\}$ with $\tilde\lambda_j(t)=\lambda_j+jt$. Since $\tilde \lambda_i(t)-\tilde\lambda_j(t)\notin\mathbb Z$
for $0<|t|\ll 1$, we can reduce certain claims to the case when the values of characteristic exponents are generic. Note that we can construct local independent solutions which holomorphically depend on $t$ (cf.~\cite{Or}). \end{rem} \begin{lem}\label{lem:GRS} {\rm i) } Let $\lambda$ be a complex number and let $p(t)$ be a polynomial such that $p(\lambda)\ne 0$. Then for non-negative integers $k$ and $m$ we have the exact sequence \begin{equation*}
0\longrightarrow
\mathcal O_0(\lambda,k-1)
\longrightarrow
\mathcal O_0(\lambda,m+k-1)
\xrightarrow{p(\vartheta)(\vartheta-\lambda)^k}{}
\mathcal O_0(\lambda,m-1)
\longrightarrow 0 \end{equation*} under the notation \eqref{def:Ocm}.
{\rm ii) } Let $m_1,\dots,m_q$ be non-negative integers. Let $P$ be a differential operator of order $n$ whose coefficients are in $\mathcal O_0$ such that \begin{equation}\label{eq:GRS0}
P = \sum_{\ell=0}^\infty x^\ell r_\ell(\vartheta)
\prod_{\nu=1}^q\,\prod_{0\le k< m_\nu-\ell}\bigl(\vartheta-k\bigr) \end{equation} with polynomials $r_\ell$. Put $m_{max}=\max\{m_1,\dots,m_q\}$ and suppose $r_0(\nu)\ne 0$ for $\nu=0,\dots,m_{max}-1$.
Let\/ ${\mathbf m}^\vee=(m^\vee_1,\dots,m^\vee_{m_{max}})$ be the dual partition of $\mathbf m:=(m_1,\dots,m_q)$, namely,\index{dual partition} \begin{equation}\label{eq:dualP}
m^\vee_\nu=\#\{j\,;\,m_j\ge \nu\}. \end{equation} Then for $i=0,\ldots,m_{max}-1$ and $j=0,\ldots,m^\vee_{i+1}-1$ we have the functions \begin{equation}
u_{i,j}(x) = x^i\log^j x+ \sum_{\mu=i+1}^{m_{max}-1}
\sum_{\nu=0}^{j}
c_{i,j}^{\mu,\nu}x^\mu \log^\nu x \end{equation} such that $c_{i,j}^{\mu,\nu}\in\mathbb C$ and $Pu_{i,j}\in\mathcal O_0(m_{max},j)$.
{\rm iii) } Let\/ $m'_1,\dots,m'_{q'}$ be non-negative integers and let $P'$ be a differential operator of order $n'$ whose coefficients are in $\mathcal O_0$ such that \begin{equation}
P'= \sum_{\ell=0}^\infty x^\ell r'_\ell(\vartheta)
\prod_{\nu=1}^q\,\prod_{0\le k< m'_\nu-\ell}
\bigl(\vartheta-m_\nu - k\bigr) \end{equation} with polynomials $q'_\ell$. Then for a differential operator $P$ of the form \eqref{eq:GRS0} we have \begin{equation}
P'P= \sum_{\ell=0}^\infty x^\ell\Bigl(\sum_{\nu=0}^\ell
r'_{\ell-\nu}(\vartheta+\nu) r_\nu(\vartheta)\Bigr)
\prod_{\nu=1}^q\,\prod_{0\le k< m_\nu+m'_\nu-\ell}
\bigl(\vartheta- k\bigr). \end{equation} \end{lem} \begin{proof} {\rm i) } The claim is easy if $(p,k) = (1,1)$ or $(\vartheta-\mu,0)$ with $\mu\ne\lambda$. Then the general case follows from induction on $\deg p(t)+k$.
{\rm ii) } Put $P=\sum_{\ell\ge 0}x^\ell p_\ell(\vartheta)$ and $m^\vee_\nu=0$ if $\nu>m_{max}$. Then for a non-negative integer $\nu$, the multiplicity of the root $\nu$ of the equation $p_\ell(t)=0$ is equal or larger than $m^\vee_{\nu+\ell+1}$ for $\ell=1,2,\dots$. If $0\le \nu\le m_{max}-1$, the multiplicity of the root $\nu$ of the equation $p_0(t)=0$ equals $m^\vee_{\nu+1}$.
For non-negative integers $i$ and $j$, we have \[
x^\ell p_\ell(\vartheta)x^i\log^j x=x^{i+\ell}\sum_{0\le \nu\le j - m^\vee_{i+\ell+1}}
c_{i,j,\ell,\nu}\log^\nu x \] with suitable $c_{i,j,\ell,\nu}\in\mathbb C$. In particular, $p_0(\vartheta)x^i\log^j x=0$ if $j<m^\vee_i$. If $\ell>0$ and $i+\ell<m_{\max}$, there exist functions \[
v_{i,j,\ell} = x^{i+\ell} \sum_{\nu=0}^j a_{i,j,\ell,\nu}
\log^\nu x \] with suitable $a_{i,j,\ell,\nu}\in\mathbb C$ such that $p_0(\vartheta)v_{i,j,\ell} = x^\ell p_\ell(\vartheta)x^i\log^j x$ and we define a $\mathbb C$-linear map $Q$ by \[
Qx^i\log^j x= -\sum_{\ell=1}^{m_{max}-i-1}v_{i,j,\ell}
= -\sum_{\ell=1}^{m_{max}-i-1}\sum_{\nu=0}^j
a_{i,j,\ell,\nu}x^{i+\ell}\log^\nu x, \] which implies $p_0(\vartheta)Qx^i\log^jx =-\sum_{\ell=1}^{m_{max}-i-1}x^\ell p_\ell(\vartheta)x^i\log^j$ and $Q^{m_{max}}=0$. Putting $Tu:=\sum_{\nu=0}^{m_{max}-1}Q^\nu u$ for $u\in\sum_{i=0}^{m_{max}-1}\sum_{j=0}^{q-1}\mathbb Cx^i\log^jx$, we have \begin{align*}
P Tu &\equiv
p_0(\vartheta)Tu+\sum_{\ell=1}^{m_{max}-1}x^\ell p_\ell(\vartheta)Tu
&&\mod \mathcal O_0(m_{max},j)\\
&\equiv p_0(\vartheta)(1-Q)Tu &&\mod \mathcal O_0(m_{max},j)\\
&\equiv p_0(\vartheta)(1-Q)(1+Q+\cdots+Q^{m_{max}-1})u
&&\mod \mathcal O_0(m_{max},j)\\
&= p_0(\vartheta)u. \end{align*} Hence if $j<m^\vee_i$, $PTx^i\log^j x\equiv0\mod\mathcal O_0(m_{max},j)$ and $u_{i,j}(x):=Tx^i\log^j x$ are required functions.
{\rm iii) } Since \begin{align*} &x^{\ell'} r'_{\ell'}(\vartheta)
\prod_{\nu=1}^q\,\prod_{0\le k'< m'_{\nu}-\ell'}
(\vartheta - m_\nu - k')\cdot x^{\ell} r_{\ell}(\vartheta)
\prod_{\nu=1}^q\,\prod_{0\le k< m_\nu-\ell}
(\vartheta - k)\\ &=x^{\ell+\ell'}r'_{\ell'}(\vartheta+\ell)r_\ell(\vartheta)
\prod_{\nu=1}^q
\prod_{0\le k'< m'_{\nu}-\ell'}
(\vartheta - m_\nu - k'+\ell)
\prod_{0\le k< m_\nu-\ell}
(\vartheta - k)\\ &=x^{\ell+\ell'}r'_{\ell'}(\vartheta+\ell)r_\ell(\vartheta)
\prod_{\nu=1}^q
\prod_{0\le k< m_\nu+m_{\nu'}-\ell-\ell'}
(\vartheta - k), \end{align*} we have the claim. \end{proof}
\begin{defn}[generalized Riemann scheme]\label{def:GRS} \index{Riemann scheme!generalized} Let $P\in W[x]$. Then we call $P$ is \textsl{Fuchsian} in this paper when $P$ has at most regular singularities in $\mathbb C\cup\{\infty\}$. Suppose $P$ is Fuchsian with regular singularities at $x=c_0=\infty$, $c_1$,\dots, $c_p$ and the functions $\frac{a_j(x)}{a_n(x)}$ are holomorphic on $\mathbb C\setminus\{c_1,\dots,c_p\}$ for $j=0,\dots,n$. Moreover suppose $P$ has the set of characteristic exponents $\{[\lambda_{j,1}]_{(m_{j,1})},\dots,[\lambda_{j,n_j}]_{(m_{j,n_j})}\}$ at $x=c_j$. Then we define the Riemann scheme of $P$ or the equation $Pu=0$ by \begin{equation}\label{eq:GRS}
\begin{Bmatrix}
x = c_0=\infty & c_1 & \cdots & c_p\\
[\lambda_{0,1}]_{(m_{0,1})} & [\lambda_{1,1}]_{(m_{1,1})}&\cdots
&[\lambda_{p,1}]_{(m_{p,1})}\\
\vdots & \vdots & \vdots & \vdots\\
[\lambda_{0,n_0}]_{(m_{0,n_0})} & [\lambda_{1,n_1}]_{(m_{1,n_1})}&\cdots
&[\lambda_{p,n_p}]_{(m_{p,n_p})}
\end{Bmatrix}. \end{equation} \end{defn} \begin{rem} The Riemann scheme \eqref{eq:GRS} always satisfies the Fuchs relation\index{Fuchs relation} (cf.~\eqref{eq:FC0}): \begin{equation}\label{eq:Fuchs}
\sum_{j=0}^p\sum_{\nu=1}^{n_j}\sum_{i=0}^{m_{j,\nu}-1}
\bigl(\lambda_{j,\nu}+i\bigr)
=\frac{(p-1)n(n-1)}{2}. \end{equation} \end{rem}
\begin{defn}[spectral type]\index{spectral type|see{tuple of partitions}} In Definition~\ref{def:GRS} we put \[
\mathbf m=(m_{0,1},\dots,m_{0,n_0};
m_{1,1},\dots;m_{p,1},\dots,m_{p,n_p}), \] which will be also written as $m_{0,1}m_{0,2}\cdots m_{0,n_0},m_{1,1}\cdots,m_{p,1}\cdots m_{p,n_p}$ for simplicity. Then $\mathbf m$ is a $(p+1)$-tuple of partitions of $n$ and we define that $\mathbf m$ is the \textsl{spectral type} of $P$. \index{spectral type}
If the set of (usual) characteristic exponents \begin{equation}\label{eq:Lambdaj}
\Lambda_j:=\{\lambda_{j,\nu}+i\,;\,0\le i\le m_{j,\nu}-1\text{ and } \nu=1,\dots,n_\nu\} \end{equation} of the Fuchsian differential operator $P$ at every regular singular point $x=c_j$ are $n$ different complex numbers, $P$ is said to have \textsl{distinct exponents}. \index{characteristic exponent!distinct} \end{defn} \begin{rem} We remark that the Fuchsian differential equation $\mathcal M: Pu=0$ is irreducible (cf.~Definition~\ref{def:irred}) if and only if the monodromy of the equation is irreducible.
If $P=QR$ with $Q$ and $R\in W(x;\xi)$, the solution space of the equation $Qv=0$ is a subspace of that of $\mathcal M$ and closed under the monodromy and therefore the monodromy is reducible. Suppose the space spanned by certain linearly independent solutions $u_1,\dots,u_m$ is invariant under the monodromy. We have a non-trivial simultaneous solution of the linear relations $b_mu^{(m)}_j+\cdots+b_1u^{(1)}_j+b_0u_j=0$ for $j=1,\dots,m$. Then $\frac{b_j}{b_m}$ are single-valued holomorphic functions on $\mathbb C\cup\{\infty\}$ excluding finite number of singular points. In view of the local behavior of solutions, the singularities of $\frac{b_j}{b_m}$ are at most poles and hence they are rational functions. Then we may assume $R=b_m\partial^m+\cdots+b_0\in W(x;\xi)$ and $P\in W(x;\xi)R$.
Here we note that $R$ is Fuchsian but $R$ may have a singularity which is not a singularity of $P$ and is an \textsl{apparent singularity}. \index{apparent singularity} For example, we have \begin{equation} x(1-x)\partial^2+(\gamma-\alpha x)\partial+\alpha =\Bigl(\frac\gamma\alpha -x\Bigr)^{-1} \Bigl(x(1-x)\partial +(\gamma - \alpha x)\Bigr) \biggl(\Bigl(\frac\gamma\alpha-x\Bigr)\partial+1\biggr). \end{equation} We also note that the equation $\partial^2u=xu$ is irreducible and the monodromy of its solutions is reducible. \end{rem} \subsection{Tuples of partitions}\index{tuple of partitions} For our purpose it will be better to allow some $m_{j,\nu}$ equal 0 and we generalize the notation of tuples of partitions as in \cite{O3}. \begin{defn}\label{def:tuples} Let $\mathbf m =\bigl(m_{j,\nu}\bigr)_{\substack{j=0,1,\ldots\\ \nu=1,2,\ldots}}$ be an ordered set of infinite number of non-negative integers indexed by non-negative integers $j$ and positive integers $\nu$. Then $\mathbf m$ is called a \textsl{$(p+1)$-tuple of partitions of $n$} if the following two conditions are satisfied. \begin{align}
\sum_{\nu=1}^\infty m_{j,\nu}&=n\qquad(j=0,1,\ldots),\\
m_{j,1} &= n\qquad(\forall j > p). \end{align} A $(p+1)$-tuple of partition $\mathbf m$ is called \textsl{monotone}\index{tuple of partitions!monotone} if\begin{equation}
m_{j,\nu} \ge m_{j,\nu+1}\quad(j=0,1,\ldots,\ \nu=1,2,\ldots) \end{equation} \index{tuple of partitions!trivial} and called \textsl{trivial} if $m_{j,\nu}=0$ for $j=0,1,\ldots$ and $\nu=2,3,\ldots$. \index{tuple of partitions!standard} Moreover $\mathbf m$ is called \textsl{standard} if $\mathbf m$ is monotone and $m_{j,2}>0$ for $j=0,\dots,p$. \index{tuple of partitions!indivisible} \index{tuple of partitions!divisible} \index{00gcd@$\gcd$} The greatest common divisor of $\{m_{j,\nu};j=0,1,\ldots,\ \nu=1,2,\ldots\}$ is denoted by $\gcd\mathbf m$ and $\mathbf m$ is called \textsl{divisible} (resp.~\textsl{indivisible}) if $\gcd\mathbf m\ge2$ (resp.~$\gcd\mathbf m=1$). The totality of $(p+1)$-tuples of partitions of $n$ are denoted by ${\mathcal P}_{p+1}^{(n)}$ and we put \index{00P@$\mathcal P,\ \mathcal P_{p+1},\ \mathcal P^{(n)},\ \mathcal P_{p+1}^{(n)}$} \index{00ord@$\ord$} \index{00Pidx@$\Pidx$} \begin{align}
{{\mathcal P}}_{p+1} &:=
\bigcup_{n=0}^\infty {{\mathcal P}}_{p+1}^{(n)},\quad
{{\mathcal P}}^{(n)} :=
\bigcup_{p=0}^\infty {{\mathcal P}}_{p+1}^{(n)},\quad
{{\mathcal P}} :=
\bigcup_{p=0}^\infty {{\mathcal P}}_{p+1},\\
\ord\mathbf m &:= n\quad\text{if \ }
\mathbf m\in{{\mathcal P}}^{(n)}, \allowdisplaybreaks\\
\mathbf 1&:=(1,1,\ldots)=
\bigl(m_{j,\nu}=\delta_{\nu,1}\bigr)_{\substack{j=0,1,\ldots\\\nu=1,2,\ldots}}\in\mathcal P^{(1)}, \allowdisplaybreaks\\
\idx(\mathbf m,\mathbf m')&:=
\sum_{j=0}^p\sum_{\nu=1}^\infty m_{j,\nu}m'_{j,\nu}
-(p-1)\ord\mathbf m\cdot\ord\mathbf m', \allowdisplaybreaks\index{00idx@$\idx$}\\ \idx \mathbf m&:=\idx(\mathbf m,\mathbf m) =
\sum_{j=0}^p\sum_{\nu=1}^\infty m_{j,\nu}^2-(p-1)\ord\mathbf m^2, \allowdisplaybreaks\\ \Pidx\mathbf m&:=1-\frac{\idx\mathbf m}2. \end{align} \end{defn} Here $\ord\mathbf m$ is called the \textsl{order} of $\mathbf m$. \index{tuple of partitions!order} For $\mathbf m,\,\mathbf m'\in\mathcal P$ and a non-negative integer $k$, $\mathbf m+k\mathbf m'\in\mathcal P$ is naturally defined. Note that \begin{align}
\idx(\mathbf m+\mathbf m') &=
\idx\mathbf m+\idx\mathbf m'+2\idx(\mathbf m,\mathbf m'),\\
\Pidx(\mathbf m+\mathbf m') &=
\Pidx\mathbf m+\Pidx\mathbf m'-\idx(\mathbf m,\mathbf m')-1. \end{align} \index{00Pidx@$\Pidx$} For $\mathbf m\in{{\mathcal P}}_{p+1}^{(n)}$ we choose integers $n_0,\dots,n_k$ so that $m_{j,\nu}=0$ for $\nu>n_j$ and $j=0,\dots,p$ and we will sometimes express $\mathbf m$ as \begin{align*}
\mathbf m&=(\mathbf m_0,\mathbf m_1,\dots,\mathbf m_p)\\
&=m_{0,1},\dots,m_{0,n_0};\ldots;m_{k,1},\dots,m_{p,n_p}\\
&=m_{0,1}\cdots m_{0,n_0},m_{1,1}\cdots m_{1,n_1},\dots,
m_{k,1}\cdots m_{p,n_p} \end{align*} if there is no confusion. Similarly $\mathbf m=(m_{0,1},\dots,m_{0,n_0})$ if $\mathbf m\in\mathcal P_1$. Here \begin{equation*}
\mathbf m_j = (m_{j,1},\dots,m_{j,n_j}) \text{ \ and \ }
\ord\mathbf m=m_{j,1}+\cdots+m_{j,n_j}\quad(0\le j\le p). \end{equation*} For example $\mathbf m=(m_{j,\nu})\in{{\mathcal P}}_{3}^{(4)}$ with $m_{1,1}=3$ and $m_{0,\nu}=m_{2,\nu}=m_{1,2}=1$ for $\nu=1,\dots,4$ will be expressed by \begin{equation*}
\mathbf m=1,1,1,1;3,1;1,1,1,1=1111,31,1111=1^4,31,1^4. \end{equation*} Let $\mathfrak S_\infty$ be the restricted permutation group of the set of indices $\{0,1,2,3,\ldots\}=\mathbb Z_{\ge 0}$, which is generated by the transpositions $(j,j+1)$ with $j\in\mathbb Z_{\ge0}$. Put $\mathfrak S_\infty'=\{\sigma\in\mathfrak S_\infty\,;\,\sigma(0)=0\}$, which is isomorphic to $\mathfrak S_\infty$.
\begin{defn}\label{def:Sinfty} \index{00s@$s$, $s\mathbf m$} \index{00Sinfty@$S_\infty,\ S_\infty'$} The transformation groups $S_\infty$ and $S_\infty'$ of $\mathcal P$ are defined by \begin{equation}\label{eq:S_infty}
\begin{split}
S_\infty :\!&=H\ltimes S_\infty',\\
S_\infty':\!&=\{(\sigma_i)_{i=0,1,\ldots}\,;\,\sigma_i\in \mathfrak S_\infty',\
\sigma_i=1\ \ (i\gg1)\},\quad H\simeq\mathfrak S_\infty,\\
m'_{j,\nu} &= m_{\sigma(j),\sigma_j(\nu)}\qquad(j=0,1,\ldots,\ \nu=1,2,\ldots)
\end{split} \end{equation} for $g = (\sigma,\sigma_1,\ldots) \in S_\infty$, $\mathbf m=(m_{j,\nu})\in \mathcal P$ and $\mathbf m'=g\mathbf m$. A tuple $\mathbf m\in\mathcal P$ is \textsl{isomorphic} to a tuple $\mathbf m'\in\mathcal P$ if there exists $g\in S_\infty$ such that $\mathbf m'=g\mathbf m$. \index{tuple of partitions!isomorphic} We denote by $s\mathbf m$ the unique monotone element in $S'_\infty\mathbf m$. \end{defn} \begin{defn}\label{def:FRLM} For a tuple of partitions $\mathbf m=\Bigl(m_{j,\nu}\Bigr) _{\substack{1\le\nu\le n_j\\0\le j\le p}} \in\mathcal P_{p+1}$ and $\lambda=\Bigl(\lambda_{j,\nu}\Bigr)_{\substack{1\le\nu\le n_j\\0\le j\le p}}$ with $\lambda_{j,\nu}\in\mathbb C$, we define \index{000lambda@$\arrowvert$\textbraceleft$\lambda_{\mathbf m}$\textbraceright$\arrowvert$} \begin{equation}
\bigl|\{\lambda_{\mathbf m}\}\bigr|
:=\sum_{j=0}^{p}\sum_{\nu=1}^{n_j}
m_{j,\nu}\lambda_{j,\nu}-\ord\mathbf m
+\frac{\idx\mathbf m}2. \end{equation} \end{defn} We note that the Fuchs relation \eqref{eq:Fuchs} is equivalent to \begin{equation}\label{eq:Fuchidx}\index{Fuchs relation}
|\{\lambda_{\mathbf m}\}|=0 \end{equation} because \begin{align*}
\sum_{j=0}^p\sum_{\nu=1}^{n_j}\sum_{i=0}^{m_{j,\nu}-1}i
&=\frac12\sum_{j=0}^p\sum_{\nu=1}^{n_j} m_{j,\nu}(m_{j,\nu}-1)
=\frac12\sum_{j=0}^p\sum_{\nu=1}^{n_j}m_{j,\nu}^2-\frac12(p+1)n\\
&=\frac12\Bigl(\idx\mathbf m+(p-1)n^2\Bigr)-\frac12(p+1)n\\
&=\frac12\idx\mathbf m-n+\frac{(p-1)n(n-1)}2. \end{align*} \subsection{Conjugacy classes of matrices} Now we review on the conjugacy classes of matrices. For $\mathbf m=(m_1,\dots,m_N)\in\mathcal P^{(n)}_1$ and $\lambda=(\lambda_1,\dots,\lambda_N)\in\mathbb C^N$ we define a matrix $L(\mathbf m;\lambda)\in M(n,\mathbb C)$ as follows, which is introduced and effectively used by \cite{Os} and \cite{O3}:
If $\mathbf m$ is monotone, then
\begin{equation}\label{eq:OSNF} \begin{split}
L(\mathbf m;\mathbf \lambda)
:\!&= \Bigl(A_{ij}\Bigr)_{\substack{1\le i\le N\\1\le j\le N}},\quad
A_{i,j}\in M(m_i,m_j,\mathbb C),\\
A_{ij} &= \begin{cases}
\lambda_i I_{m_i}&(i=j),\\
I_{m_i,m_j}:=
\Bigl(\delta_{\mu\nu}\Bigr)
_{\substack{1\le \mu\le n_i\\1\le \nu\le n_j}}
=
\begin{pmatrix}
I_{m_j} \\ 0
\end{pmatrix}&(i=j-1),\\
0 &(i\ne j,\ j-1).
\end{cases} \end{split} \end{equation} \index{00Lmlambda@$L(\mathbf m;\lambda)$} \index{00Mm@$M(n,\mathbb C)$, $M(m_1,m_2,\mathbb C)$} Here $I_{m_i}$ denote the identity matrix of size $m_i$ and $M(m_i,m_j,\mathbb C)$ means the set of matrices of size $m_i\times m_j$ with components in $\mathbb C$ and $M(m,\mathbb C):=M(m,m,\mathbb C)$.
For example \begin{equation*}
L(2,1,1;\lambda_1,\lambda_2,\lambda_3):=
\begin{pmatrix}
\lambda_1 & 0 &1\\
0 &\lambda_1& 0\\
& &\lambda_2&1\\
& & &\lambda_3\\
\end{pmatrix}. \end{equation*} If $\mathbf m$ is not monotone, we fix a permutation $\sigma$ of $\{1,\dots,N\}$ so that $(m_{\sigma(1)},\ldots,m_{\sigma(N)})$ is monotone and put $L(\mathbf m;\mathbf \lambda)=L(m_{\sigma(1)},\ldots, m_{\sigma(N)};\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(N)})$.
When $\lambda_1=\cdots=\lambda_N=\mu$, $L(\mathbf m;\lambda)$ may be simply denoted by $L(\mathbf m,\mu)$.
We denote $A\sim B$ for $A$, $B\in M(n,\mathbb C)$ if and only if there exists $g\in GL(n,\mathbb C)$ with $B=gAg^{-1}$.
When $A\sim L(\mathbf m;\lambda)$, $\mathbf m$ is called the \textsl{spectral type} of $A$ and denoted by $\spc A$ with a monotone $\mathbf m$.
\begin{rem}\label{rm:1} {\rm i)\ } If $\mathbf m=(m_1,\dots,m_K)\in\mathcal P_1^{(n)}$ is monotone, we have \begin{equation*}
A\sim L(\mathbf m;\lambda)\ \Leftrightarrow\
\rank\prod_{\nu=1}^j(A-\lambda_\nu)
= n - (m_1+\cdots+m_j)\quad(j=0,1,\dots,K). \end{equation*}
{\rm ii)\ } For $\mu\in\mathbb C$, put \begin{equation}\label{eq:Msub}
(\mathbf m;\lambda)_\mu
=(m_{i_1},\ldots,m_{i_K};\mu) \text{ \ with \ }\{i_1,\dots,i_K\}=\{i\,;\,\lambda_i=\mu\}. \end{equation} Then we have \begin{equation}\label{eq:Leigen}
L(\mathbf m;\lambda) \sim\bigoplus_{\mu\in\mathbb C}
L\bigl((\mathbf m;\lambda)_\mu\bigr). \end{equation}
{\rm iii)\ } Suppose $\mathbf m$ is monotone. Then for $\mu\in\mathbb C$ \begin{equation}\label{eq:LJordan}
\begin{split}
L(\mathbf m,\mu) &\sim
\bigoplus_{j=1}^{m_1} J\bigl(\max\{\nu\,;\,m_\nu\ge j\},\mu\bigr),\\
J(k,\mu) :\!&=L(1^k,\mu)\in M(k,\mathbb C).
\end{split} \end{equation}
{\rm iv)\ } For $A\in M(n,\mathbb C)$, we put $Z(A)=Z_{M(n,\mathbb C)}(A) :=\{X\in M(n,\mathbb C)\,;\,AX=XA\}$. \index{00Z@$Z(A)$, $Z(\mathbf M)$} Then \begin{equation*}
\dim Z_{M(n,\mathbb C)}\bigl(L(\mathbf m,\lambda)\bigr)
= m_1^2+m_2^2+\cdots \end{equation*}
{\rm v)\ } (cf.~\cite[Lemma~3.1]{O5}). Let $\mathbf A(t):\,[0,1)\to M(n,\mathbb C)$ be a continuous function. Suppose there exist a continuous function $\lambda=(\lambda_1,\dots,\lambda_K):\,[0,1)\to\mathbb C^K$ such that $A(t)\sim L(\mathbf m;\lambda(t))$ for $t\in(0,1)$. Then \begin{equation}
A(0)\sim L\bigl(\mathbf m;\lambda(0)\bigr)
\text{ \ if and only if \ }\dim Z\bigl(A(0)\bigr)=m_1^2+\cdots+m_K^2. \end{equation} \end{rem} Note that the Jordan canonical form of $L(\mathbf m;\lambda)$ is easily obtained by \eqref{eq:Leigen} and \eqref{eq:LJordan}. For example, $L(2,1,1;\mu)\simeq J(3,\mu)\oplus J(1,\mu)$. \subsection{Realizable tuples of partitions} \begin{prop} Let $Pu=0$ be a differential equation of order $n$ which has a regular singularity at $0$. Let $\{[\lambda_1]_{(m_1)},\dots,[\lambda_q]_{(m_q)}\}$ be the corresponding set of the characteristic exponents. Here $\mathbf m=(m_1,\dots,m_q)$ a partition of $n$.
{\rm i) } Suppose there exists $k$ such that \begin{gather*}
\lambda_1=\lambda_2=\cdots=\lambda_k,\\
m_1\ge m_2\ge \cdots\ge m_k,\\
\lambda_j-\lambda_1\notin\mathbb Z\qquad(j=k+1,\dots,q). \end{gather*} Let $\mathbf m^\vee=(m_1^\vee,\dots,m^\vee_r)$ be the dual partition of $(m_1,\dots,m_k)$ {\rm (cf.~\eqref{eq:dualP})}. Then for $i=0,\ldots,m_1-1$ and $j=0,\ldots,m^\vee_{i+1}-1$ the equation has the solutions \begin{equation}
u_{i,j}(x) = \sum_{\nu=0}^j x^{\lambda_1+i}\log^\nu x\cdot\phi_{i,j,\nu}(x). \end{equation} Here $\phi_{i,j,\nu}(x)\in\mathcal O_0$ and $\phi_{i,\nu,j}(0)=\delta_{\nu,j}$ for $\nu=0,\dots,j-1$.
{\rm ii) } Suppose \begin{equation}
\lambda_i-\lambda_j\ne\mathbb Z\setminus\{0\}
\qquad(0\le i<j\le q). \end{equation} In this case we say that the set of characteristic exponents $\{[\lambda_1]_{(m_1)},\dots,[\lambda_q]_{(m_q)}\}$ is \textsl{distinguished}. \index{characteristic exponent!distinguished} Then the monodromy generator of the solutions of the equation at $0$ is conjugate to \[
L\bigl(\mathbf m;(e^{2\pi\sqrt{-1}\lambda_1},\dots,
e^{2\pi\sqrt{-1}\lambda_q})\bigr). \] \end{prop} \begin{proof} Lemma~\ref{lem:GRS} ii) shows that there exist $u_{i,j}(x)$ of the form stated in i) which satisfy $Pu_{i,j}(x)\in\mathcal O_0(\lambda_1+m_1,j)$ and then we have $v_{i,j}(x)\in\mathcal O_0(\lambda_1+m_1,j)$ such that $Pu_{i,j}(x)=Pv_{i,j}(x)$ because of \eqref{eq:Pbij}. Thus we have only to replace $u_{i,j}(x)$ by $u_{i,j}(x)-v_{i,j}(x)$ to get the claim in i). The claim in ii) follows from that of i). \end{proof} \begin{rem} {\rm i) } Suppose $P$ is a Fuchsian differential operator with regular singularities at $x=c_0=\infty, c_1,\dots,c_p$ and moreover suppose $P$ has distinct exponents. Then the Riemann scheme of $P$ is \eqref{eq:GRS} if and only if $Pu=0$ has local solutions $u_{j,\nu,i}(x) $ of the form \begin{equation}
u_{j,\nu,i}(x) =
\begin{cases}
(x-c_j)^{\lambda_{j,\nu}+i}\bigl(1+o(|x-c_j|^{m_j,{\nu}-i-1})\bigr)\\
\qquad\qquad(x\to c_j,\ i=0,\dots,m_{j,\nu}-1,\ j=1,\dots,p),\\
x^{-\lambda_{0,\nu}-i}\bigl(1+o(x^{-m_{0,\nu}+i+1})\bigr)\\
\qquad\qquad(x\to\infty,\ i=0,\dots,m_{0,\nu}).
\end{cases} \end{equation} Moreover suppose $\lambda_{j,\nu}-\lambda_{j,\nu'}\notin\mathbb Z$ for $1\le\nu<\nu'\le n_j$ and $j=0,\dots,p$. Then \begin{equation}
u_{j,\nu,i}(x)=
\begin{cases}
(x-c_j)^{\lambda_{j,\nu}+i}\phi_{j,\nu,i}(x) &(1\le j\le p)\\
x^{-\lambda_{0,\nu}-i}\phi_{0,\nu,i}(x) & (j=0)
\end{cases} \end{equation} with $\phi_{j,\nu,i}(x)\in\mathcal O_{c_j}$ satisfying $\phi_{j,\nu,i}(c_j)=1$. In this case $P$ has the Riemann scheme \eqref{eq:GRS} if and only if at the each singular point $x=c_j$, the set of characteristic exponents of the equation $Pu=0$ equals $\Lambda_j$ in \eqref{eq:Lambdaj} and the monodromy generator of its solutions is semisimple.
{\rm ii) } Suppose $P$ has the Riemann scheme \eqref{eq:GRS} and $\lambda_{1,1}=\dots=\lambda_{1,n_1}$. Then the monodromy generator of the solutions of $Pu=0$ at $x=c_1$ has the eigenvalue $e^{2\pi\sqrt{-1}\lambda_{1,1}}$ with multiplicity $n$. Moreover the monodromy generator is conjugate to the matrix $L\bigl((m_{1,1},\dots,m_{1,n_1}),e^{2\pi\sqrt{-1}\lambda_{1,1}}\bigr)$,
which is also conjugate to \begin{align*}
J(m^\vee_{1,1},e^{2\pi\sqrt{-1}\lambda_{1,1}})\oplus\cdots\oplus
J(m^\vee_{1,n'_1},e^{2\pi\sqrt{-1}\lambda_{1,1}}). \end{align*} Here $(m^\vee_{1,1},\dots,m^\vee_{1,n^\vee_1})$ is the dual partition of $(m_{1,1},\dots,m_{1,n_1})$. A little weaker condition for $\lambda_{j,\nu}$ assuring the same conclusion is given in Proposition~\ref{prop:nondeg}. \end{rem}
\begin{defn}[realizable spectral type]\label{defn:real} \index{tuple of partitions!realizable} Let $\mathbf m=(\mathbf m_0,\dots,\mathbf m_p)$ be a $(p+1)$-tuple of partitions of a positive integer $n$. Here $\mathbf m_j = (m_{j,1},\dots,m_{j,n_j})$ and $n=m_{j,1}+\dots+m_{j,n_j}$ for $j=0,\dots,p$ and $m_{j,\nu}$ are non-negative numbers. Fix $p$ different points $c_j$ ($j=1,\dots,p$) in $\mathbb C$ and put $c_0=\infty$.
Then $\mathbf m$ is a \textsl{realizable spectral type} if there exists a Fuchsian operator $P$ with the Riemann scheme \eqref{eq:GRS} for generic $\lambda_{j,\nu}$ satisfying the Fuchs relation \eqref{eq:Fuchs}. Moreover in this case if there exists such $P$ so that the equation $Pu=0$ is irreducible, which is equivalent to say that the monodromy of the equation is irreducible, then $\mathbf m$ is \textsl{irreducibly realizable}. \index{tuple of partitions!realizable} \index{tuple of partitions!irreducibly realizable} \end{defn} \begin{rem}\label{rem:generic}
{\rm i)} \ In the above definition $\{\lambda_{j,\nu}\}$ are generic if, for example, $0<m_{0,1}<\ord\mathbf m$ and $\{\lambda_{j,\nu}\,;\,(j,\nu)\ne (0,1),\ j=0,\dots,p,\ 1\le\nu\le n_j\} \cup\{1\}$ are linearly independent over $\mathbb Q$.
{\rm ii) } It follows from the facts (cf.~\eqref{eq:FC1}) in \S\ref{sec:reg} that if $\mathbf m\in\mathcal P$ satisfies \begin{equation}\label{eq:FC12}
\begin{split}
&|\{\lambda_{\mathbf m'}\}|\notin\mathbb Z_{\le 0}
=\{0,-1,-2,\ldots\}
\text{ for any }\mathbf m',\,\mathbf m''\in\mathcal P\\
&\quad\text{ satisfying }\mathbf m=\mathbf m'+\mathbf m''
\text{ and }0<\ord\mathbf m'<\ord\mathbf m,
\end{split} \end{equation} the Fuchsian differential equation with the Riemann scheme \eqref{eq:GRS} is irreducible. Hence if $\mathbf m$ is indivisible and realizable, $\mathbf m$ is irreducibly realizable. \end{rem} Fix distinct $p$ points $c_1,\dots,c_p$ in $\mathbb C$ and put $c_0=\infty$. The Fuchsian differential operator $P$ with regular singularities at $x=c_j$ for $j=1,\dots,n$ has the \textsl{normal form} \index{Fuchsian differential equation/operator!normal form} \begin{equation}\label{eq:FNF}
P = \Bigl(\prod_{j=1}^p (x-c_j)^n\Bigr)\partial^n
+ a_{n-1}(x)\partial^{n-1}+\cdots+a_1(x)\partial + a_0(x), \end{equation} where $a_i(x)\in\mathbb C[x]$ satisfy \begin{align}
\deg a_i(x)&\le (p-1)n+i,\label{C:RSfin}\\
(\partial^\nu a_i)(c_j) &=0
\quad(0\le\nu \le i-1)\label{C:RSinfin} \end{align} for $i=0,\dots,n-1$.
Note that the condition \eqref{C:RSfin} (resp.~\eqref{C:RSinfin}) corresponds to the fact that $P$ has regular singularities at $x=c_j$ for $j=1,\dots,p$ (resp.~at $x=\infty$).
Since $a_i(x)=b_i(x)\prod_{j=1}^p(x-c_j)^i$ with $b_i(x)=\sum_{r=0}^{(p-1)(n-i)}b_{i,r}x^r \in W[x]$ satisfying $\deg b_i(x)\le (p-1)n+i - pi=(p-1)(n-i)$, the operator $P$ has the parameters $\{b_{i,r}\}$. The numbers of the parameters equals \[
\sum_{i=0}^{n-1}\bigl((p-1)(n-i)+1\bigr)
=\frac{(pn +p-n+1)n}{2}, \] The condition $(x-c_j)^{-k}P\in W[x]$ implies $(\partial^\ell a_i)(c_j)=0$ for $0\le \ell\le k-1$ and $0\le i\le n$, which equals $(\partial^\ell b_i)(c_j)=0$ for $0\le \ell\le k-1-i$ and $0\le i\le k-1$. Therefore the condition \begin{equation}\label{C:eachblock}
(x-c_j)^{-m_{j,\nu}}\Ad\bigl((x-c_j)^{-\lambda_{j,\nu}}\bigr)P\in W[x] \end{equation} gives $\frac{(m_{j,\nu}+1)m_{j,\nu}}{2}$ independent linear equations for $\{b_{\nu,r}\}$ since $\sum_{i=0}^{m_{j,\nu}-1} (m_{j,\nu}-i)= \frac{(m_{j,\nu}+1)m_{j,\nu}}{2}$. If all these equations have a simultaneous solution and they are independent except for the relation caused by the Fuchs relation, the number of the parameters of the solution equals \begin{equation}
\begin{split}
&\frac{(pn +p-n+1)n}{2} -
\sum_{j=0}^p\sum_{\nu=1}^{n_j}\frac{m_{j,\nu}(m_{j,\nu}+1)}{2}+1\\
&=\frac{(pn +p-n+1)n}{2} -
\sum_{j=0}^p\sum_{\nu=1}^{n_j}\frac{m_{j,\nu}^2}{2}-(p+1)\frac{n}{2}+1\\
&=\frac12\Bigl((p-1)n^2 - \sum_{j=0}^p\sum_{\nu=1}^{n_j}m_{j,\nu}^2+1
\Bigr)=\Pidx\mathbf m.
\end{split} \end{equation} \begin{rem}[{cf.~\cite[\S5]{O3}}] Katz \cite{Kz} introduced the \textsl{index of rigidity} of an irreducible local system by the number $\idx\mathbf m$ whose spectral type equals $\mathbf m=(m_{j,\nu})_{\substack{ j=0,\dots,p\ \,\\ \nu=1,\dots,n_{j}}}$ and proves $\idx\mathbf m\le 2$, if the local system is irreducible. \index{tuple of partitions!index of rigidity}
Assume the local system is irreducible. Then Katz \cite{Kz} shows that the local system is uniquely determined by the local monodromies if and only if $\idx\mathbf m=2$ and in this case the local system and the tuple of partition $\mathbf m$ are called \textsl{rigid}. \index{tuple of partitions!rigid} If $\idx\mathbf m>2$, the corresponding system of differential equations of \textsl{Schleginger normal form} \begin{equation}
\frac{du}{dx} = \sum_{j=1}^p\frac{A_j}{x-a_j}u \end{equation} has $2\Pidx\mathbf m$ parameters which are independent from the characteristic exponents and local monodromies. They are called \textsl{accessory parameters}. \index{accessory parameter} Here $A_j$ are constant square matrices of size $n$. The number of accessory parameters of the single Fuchsian differential operator without apparent singularities will be the half of this number $2\Pidx\mathbf m$ (cf.~Theorem~\ref{thm:univmodel} and \cite{Sz}). \end{rem}
Lastly in this subsection we calculate the Riemann scheme of the products and the dual of Fuchsian differential operators. \begin{thm}\label{thm:prod} Let $P$ be a Fuchsian differential operator with the Riemann scheme \eqref{eq:GRS}. Suppose $P$ has the normal form \eqref {eq:FNF}.
{\rm i)} Let $P'$ be a Fuchsian differential operator with regular singularities also at $x=c_0=\infty,c_1,\dots,c_p$. Then if $P'$ has the Riemann scheme \begin{equation}\label{eq:GRSP}
\begin{Bmatrix}
x = c_0=\infty & c_j\quad(j=1,\dots,p)\\
[\lambda_{0,1}+m_{0,1}-(p-1)\ord\mathbf m]_{(m'_{0,1})} &
[\lambda_{j,1}+m_{j,1}]_{(m'_{j,1})} \\
\vdots & \vdots\\
[\lambda_{0,n_0}+m_{0,n_0}-(p-1)\ord\mathbf m]_{(m'_{0,n_0})}
& [\lambda_{j,n_j}+m_{j,n_j}]_{(m'_{j,n_j})}
\end{Bmatrix}, \end{equation} the Fuchsian operator $P'P$ has the spectral type $\mathbf m+\mathbf m'$ and the Riemann scheme \begin{equation}\label{eq:GRSA}
\begin{Bmatrix}
x = c_0 =\infty & c_1 & \cdots & c_p\\
[\lambda_{0,1}]_{(m_{0,1}+m'_{0,1})}
& [\lambda_{1,1}]_{(m_{1,1}+m'_{1,1})}&\cdots
&[\lambda_{p,1}]_{(m_{p,1}+m'_{p,1})}\\
\vdots & \vdots & \vdots & \vdots\\
[\lambda_{0,n_0}]_{(m_{0,n_0}+m'_{0,n_0})} &
[\lambda_{1,n_1}]_{(m_{1,n_1}+m'_{1,n_1})}&\cdots
&[\lambda_{p,n_p}]_{(m_{p,n_p}+m'_{1,n_p})}
\end{Bmatrix}. \end{equation} Suppose the Fuchs relation \eqref{eq:Fuchidx} for \eqref{eq:GRS}. Then the Fuchs relation for \eqref{eq:GRSP} is valid if and only if so is the Fuchs relation for \eqref{eq:GRSA}.
{\rm ii)} For $Q=\sum_{k\ge0} q_k(x)\partial^k \in W(x)$, we define \begin{equation}
Q^* := \sum_{k\ge 0}(-\partial)^kq_k(x) \end{equation} and the dual operator $P^\vee$ of $P$ by \index{00Pvee@$P^\vee,\ P^*$} \index{differential equation/operator!dual} \index{Fuchsian differential equation/operator!dual} \begin{equation}\label{eq:dualop}
P^\vee := a_n(x)(a_n(x)^{-1}P)^* \end{equation} when $P=\sum_{k=0}^na_k(x)\partial^k$. Then the Riemann scheme of $P^\vee$ equals \begin{equation} \begin{Bmatrix}
x = c_0 =\infty & c_j\quad(j=1,\dots,p)\\
[2-n-m_{0,1}-\lambda_{0,1}]_{(m_{0,1})} &
[n-m_{j,1}-\lambda_{j,1}]_{(m_{j,1})}\\
\vdots & \vdots\\
[2-n-m_{0,n_0}-\lambda_{0,n_0}]_{(m_{0,n_0})} &
[n-m_{j,n_j}-\lambda_{j,n_j}]_{(m_{j,n_j})} \end{Bmatrix}. \end{equation} \end{thm} \begin{proof} i) \ It is clear that $P'P$ is a Fuchsian differential operator of the normal form if so is $P'$ and Lemma~\ref{lem:GRS} iii) shows that the characteristic exponents of $P'P$ at $x=c_j$ for $j=1,\dots,p$ are just as given in the Riemann scheme \eqref{eq:GRSA}. Put $n=\ord\mathbf m$ and $n'=\mathbf m'$. We can also apply Lemma~\ref{lem:GRS} iii) to $x^{-(p-1)n}P$ and $x^{-(p-1)n'}P'$ under the coordinate transformation $x\mapsto\frac1x$, we have the set of characteristic exponents as is given in \eqref{eq:GRSA} because $x^{-(p-1)(n+n')}P'P= \bigl(\Ad(x^{-(p-1)n})x^{-(p-1)n'}P'\bigr)(x^{-(p-1)n})P$.
The Fuchs relation for \eqref{eq:GRSP} equals \[
\sum_{j=0}^p\sum_{\nu=1}^{n_j} m'_{j,\nu}\bigl(\lambda_{j,\nu}+m_{j,\nu}
-\delta_{j,0}(p-1)\ord\mathbf m\bigr) =\ord\mathbf m'-\frac{\idx\mathbf m'}2. \] Since \[
\sum_{j=0}^p\sum_{\nu=1}^{n_j}
m'_{j,\nu}\bigl(m_{j,\nu} - \delta_{j,0}(p-1)\ord\mathbf m\bigr)
=\idx(\mathbf m,\mathbf m'), \] the condition is equivalent to \begin{equation}
\sum_{j=0}^p\sum_{\nu=1}^{n_j} m'_{j,\nu}\lambda_{j,\nu}
=\ord\mathbf m' -\frac{\idx\mathbf m}2 - \idx(\mathbf m,\mathbf m') \end{equation} and also to \begin{equation}
\sum_{j=0}^p\sum_{\nu=1}^{n_j}
(m_{j,\nu}+m'_{j,\nu})\lambda_{j,\nu}
=\ord(\mathbf m + \mathbf m') -\frac{\idx(\mathbf m+\mathbf m')}2 \end{equation} under the condition \eqref{eq:Fuchidx}.
ii) \ We may suppose $c_1=0$. Then \begin{align*}
a_n(x)^{-1}P &= \sum_{\ell\ge 0}x^{\ell-n}q_\ell(\vartheta)
\prod_{\substack{1\le\nu\le n_1\\0\le i< m_{1,\nu}-\ell}}
(\vartheta-\lambda_{1,\nu}-i),\\
a_n(x)^{-1}P^\vee &=
\sum_{\ell\ge 0}q_\ell(-\vartheta-1)
\prod_{\substack{1\le\nu\le n_1\\0\le i< m_{1,\nu}-\ell}}
(-\vartheta-\lambda_{1,\nu}-i-1)x^{\ell-n}\\
&=\sum_{\ell\ge 0}x^{\ell-n}s_\ell(\vartheta)
\prod_{\substack{1\le\nu\le n_1\\0\le i< m_{1,\nu}-\ell}}
(\vartheta+\lambda_{1,\nu}+i+1+\ell-n)\\
&=\sum_{\ell\ge 0}x^{\ell-n}s_\ell(\vartheta)
\prod_{\substack{1\le\nu\le n_1\\0\le j< m_{1,\nu}-\ell}}
(\vartheta+\lambda_{1,\nu}-j+m_{1,\nu}-n) \end{align*} with suitable polynomials $q_\ell$ and $s_\ell$ such that $q_0,\,s_0\in\mathbb C^\times$. Hence the set of characteristic exponents of $P^\vee$ at $c_1$ is $\{[n-m_{1,\nu}-\lambda_{1,\nu}]_{(m_{1,\nu})}\,;\,\nu=1,\dots,n_1\}$.
At infinity we have \begin{align*}
a_n(x)^{-1}P&=\sum_{\ell\ge 0}x^{-\ell-n}
q_\ell(\vartheta)
\prod_{\substack{1\le\nu\le n_1\\0\le i< m_{0,\nu}-\ell}}
(\vartheta+\lambda_{0,\nu}+i),\\ (a_n(x)^{-1}P)^*&=\sum_{\ell\ge 0}x^{-\ell-n}
s_\ell(\vartheta)
\prod_{\substack{1\le\nu\le n_0\\0\le i< m_{0,\nu}-\ell}}
(\vartheta-\lambda_{0,\nu}-i+1-\ell-n)\\
&=\sum_{\ell\ge 0}x^{-\ell-n}
s_\ell(\vartheta)
\prod_{\substack{1\le\nu\le n_1\\0\le j< m_{0,\nu}-\ell}}
(\vartheta-\lambda_{0,\nu}+j+2-n-m_{0,\nu}) \end{align*} with suitable polynomials $q_\ell$ and $s_\ell$ with $q_0,\,s_0\in\mathbb C^\times$ and the set of characteristic exponents of $P^\vee$ at $c_1$ is $\{[2-n - m_{0,\nu}-\lambda_{0,\nu}]_{(m_{0,\nu})}\,;\,\nu=1,\dots,n_0\}$ \end{proof} \begin{exmp} The Riemann scheme of the dual $P_{\lambda_1,\dots,\lambda_p,\mu}^\vee$ of Jordan-Pochhammer operator $P_{\lambda_1,\dots,\lambda_p,\mu}^\vee$ given in Example~\ref{ex:midconv} iii) is \[
\begin{Bmatrix}
\frac1{c_1} & \cdots & \frac1{c_p} & \infty\\
[1]_{(p-1)} & \cdots & [1]_{(p-1)} & [2-2p+\mu]_{(p-1)}\\
\lambda_1-\mu+p-1 & \cdots & -\lambda_p-\mu+p-1 &
\lambda_1+\cdots+\lambda_p+\mu-p+1
\end{Bmatrix}. \] \end{exmp} \section{Reduction of Fuchsian differential equations}\label{sec:reduction} Additions and middle convolutions introduced in \S\ref{sec:frac} are transformations within Fuchsian differential operators and we examine how their Riemann schemes change under the transformations. \begin{prop}\label{prop:invred} {\rm i)} Let $Pu=0$ be a Fuchsian differential equation. Suppose there exists $c\in\mathbb C$ such that $P\in(\partial-c)W[x]$. Then $c=0$.
{\rm ii)} For $\phi(x)\in \mathbb C(x)$, $\lambda\in\mathbb C$, $\mu\in\mathbb C$ and $P\in W[x]$, we have \begin{align}
P&\in\mathbb C[x]\RAdei\bigl(-\phi(x)\bigr)
\circ\RAdei\bigl(\phi(x)\bigr)P,\\
P&\in\mathbb C[\partial]\RAd\bigl(\partial^{-\mu}\bigr)
\circ\RAd\bigl(\partial^\mu\bigr)P. \end{align} In particular, if the equation $Pu=0$ is irreducible and $\ord P>1$, $\RAd\bigl(\partial^{-\mu}\bigr)
\circ\RAd\bigl(\partial^\mu\bigr)P = cP$ with $c\in\mathbb C^\times$. \end{prop} \begin{proof} {\rm i)} Put $P=(\partial-c)Q$. Then there is a function $u(x)$ satisfying $Qu(x)=e^{cx}$. Since $Pu=0$ has at most a regular singularity at
$x=\infty$, there exist $C>0$ and $N>0$ such that $|u(x)|<C|x|^N$ for
$|x|\gg1$ and $0\le \arg x\le 2\pi$, which implies $c=0$.
{\rm ii)} This follows from the fact \begin{align*}
&\Adei\bigl(-\phi(x)\bigr)\circ\Adei\bigl(\phi(x)\bigr)=\id,\\
&\Adei\bigl(\phi(x)\bigr)f(x)P=f(x)\Adei\bigl(\phi(x)\bigr)P
\quad(f(x)\in\mathbb C(x)) \end{align*} and the definition of $\RAdei\bigl(\phi(x)\bigr)$ and $\RAd(\partial^\mu)$. \end{proof} \if0 Now we prepare a lemma related to middle convolutions. \begin{lem}\label{lem:red} Fix a positive integer $k$.
{\rm i) } Let $q(\vartheta)$ be a polynomial of $\vartheta$. Then $q(\vartheta)=\partial^k r(x,\partial)$ with $r\in W[x]$ if and only if $q(\nu)=0$ for $\nu=-1,-2,\dots,-k$.
{\rm ii) } Let $Q$ be a differential operator of the form \begin{align}\label{eq:RedF}
Q = \sum_{j=0}^N \partial^j q_j(\vartheta) \end{align} with polynomials $q_j(\vartheta)$. The condition \begin{align}\label{C:infred}
q_j(-\nu) = 0\ \ &\text{for \ }
\nu=\mu+1,\mu+2,\dots,\mu+k-j
\text{ \ and \ }
j=0,\dots,k-1 \end{align} is necessary and sufficient for \begin{equation}
\Ad(\partial^{-\mu})Q\in \partial^kW[x]. \end{equation}
{\rm iii) } Suppose $Q$ is a Fuchsian differential operator of order $n$ and $\deg q_0=n$. Then the condition \begin{equation}\label{C:redexp} [\mu+1]_{(k)}\text{ is a generalized exponent of }Q\text{ at }x=\infty \end{equation} is necessary for \eqref{C:infred}. The condition \eqref{C:redexp} is also sufficient for \eqref{C:infred} if $k=1$ or $\mu\ne -1,-2,\ldots,1-k$. \end{lem}
\begin{proof} We have $q(\vartheta) =\sum_{k\ge0} c_i\partial^ix^i$ with some $c_k\in\mathbb C$ and \begin{equation} \partial^ix^i = (\vartheta+1)(\vartheta+2)\cdots(\vartheta+i) \end{equation} by the induction on $i$ and the claims i) and ii) follow from this equality and \[
\Ad(\partial^{-\mu})Q=\sum_{j=0}^N\partial^jq_j(\vartheta-\mu). \]
Put $y=x^{-1}$. Then $x\frac{d}{dx}=-y\frac{d}{dy}$ and $\frac{d}{dx}=-y\cdot y\frac{d}{dy}$. Hence $Q$ is normalized (cf.~Definition~\ref{def:exp}) at $y=0$ under the coordinate $y=x^{-1}$ and \begin{equation}\label{eq:infx}
\begin{split}
&\partial^jq_j(\vartheta)x^{-(\mu+i)}\\
&\quad= (-1)^j
(\mu+i)(\mu+i+1)\cdots(\mu+i+j-1)q_j\bigl(-(\mu+i)\bigr)x^{-(\mu+i+j)}
\end{split} \end{equation} and the condition \eqref{C:redexp} is equivalent to the condition that \eqref{eq:infx} equals 0 if $i\ge 1$, $j\ge 0$ and $i+j\le k$ (cf.~\eqref{C:formblock}). Hence we have the claim iii). \end{proof} \begin{cor}\label{cor:mid} Let $Q$ be a Fuchsian differential operator of the form \eqref{eq:RedF} with $q_0\ne0$. Let \eqref{eq:GRS} be the Riemann scheme of $Q$. Suppose $m_{0,1}\le 1$ or $\lambda_{0,1}\ne 0, -1,\ldots,2-m_{0,1}$. Then $\partial^{-m_{0,1}}\!\Ad(\partial^{1-\lambda_{0,1}})Q\in W[x]$.
If $[\lambda_{0,\nu}]_{(m_{0,1}+1)}$ is not a generalized characteristic exponent of $Q$ at $x=\infty$, \begin{equation} \RAd(\partial^{1-\lambda_{0,1}})Q=
\partial^{-m_{0,1}}\!\Ad(\partial^{1-\lambda_{0,1}})Q. \end{equation} \end{cor} \fi
The addition and the middle convolution transform the Riemann scheme of the Fuchsian differential equation as follows.
\begin{thm}\label{thm:GRSmid} Let $Pu=0$ be a Fuchsian differential equation with the Riemann scheme \eqref{eq:GRS}. We assume that $P$ has the normal form \eqref{eq:FNF}.
{\rm i) (addition)} \ The operator $\Ad\bigl((x-c_j)^\tau \bigr)P$ has the Riemann scheme \begin{equation*}
\begin{Bmatrix}
x = c_0=\infty & c_1 & \cdots &c_j&\cdots& c_p\\
[\lambda_{0,1}-\tau]_{(m_{0,1})} & [\lambda_{1,1}]_{(m_{1,1})}&\cdots
& [\lambda_{j,1}+\tau]_{(m_{j,1})}&\cdots &[\lambda_{p,1}]_{(m_{p,1})}\\
\vdots & \vdots & \vdots & \vdots& \vdots& \vdots\\
[\lambda_{0,n_0}-\tau]_{(m_{0,n_0})}
& [\lambda_{1,n_1}]_{(m_{1,n_1})}&\cdots
& [\lambda_{j,n_j}+\tau]_{(m_{j,1})}&\cdots&[\lambda_{p,n_p}]_{(m_{p,n_p})}
\end{Bmatrix}. \end{equation*}
{\rm ii) (middle convolution)} \ Fix $\mu\in\mathbb C$. By allowing the condition $m_{j,1}=0$, we may assume \begin{equation}\label{eq:midgen}
\mu=\lambda_{0,1}-1\text{ \ and \ }
\lambda_{j,1}=0\text{ for }j=1,\dots,p \end{equation} and $\#\{j\,;\,m_{j,1}<n\}\ge 2$ and $P$ is of the normal form \eqref{eq:FNF}. Putting \begin{equation}\label{eq:defd}
d:=\sum_{j=0}^p m_{j,1} - (p-1)n, \end{equation} we suppose \begin{align}
&\quad m_{j,1}\ge d\text{ \ for \ }j=0,\dots,p,\label{eq:redC1}\\
&\begin{cases}
\lambda_{0,\nu}\notin\{0,-1,-2,\ldots,m_{0,1}-m_{0,\nu}-d+2\}\\
\text{if \ }m_{0,\nu}+\cdots+m_{p,1}-(p-1)n\ge 2,\
m_{1,1}\cdots m_{p,1}\ne0\text{ \ and \ }\nu\ge 1,
\end{cases}\label{eq:redC2}\\
&\begin{cases}
\lambda_{0,1}+\lambda_{j,\nu}\notin\{0,-1,-2,\ldots,m_{j,1}-m_{j,\nu}-d+2\}\\
\text{if \ }m_{0,1}+\cdots+m_{j-1,1}+m_{j,\nu}+m_{j+1,1}+\cdots+m_{p,1}
-(p-1)n \ge2,\\
m_{j,1}\ne0,\ 1\le j\le p\text{ \ and \ }\nu\ge 2.
\end{cases}\label{eq:redC3} \end{align} Then $S:=\partial^{-d}\!\Ad(\partial^{-\mu}) \prod_{j=1}^p(x-c_j)^{-m_{j,1}}P \in W[x]$ and the Riemann scheme of $S$ equals \begin{equation}\label{eq:midR}
\begin{Bmatrix}
x = c_0=\infty & c_1 & \cdots & c_p\\
[1-\mu]_{(m_{0,1}-d)}&[0]_{(m_{1,1}-d)}&\cdots
&[0]_{(m_{p,1}-d)}\\
[\lambda_{0,2}-\mu]_{(m_{0,2})} & [\lambda_{1,2}+\mu]_{(m_{1,2})}&\cdots
&[\lambda_{p,2}+\mu]_{(m_{p,2})}\\
\vdots & \vdots & \vdots & \vdots\\
[\lambda_{0,n_0}-\mu]_{(m_{0,n_0})}
& [\lambda_{1,n_1}+\mu]_{(m_{1,n_1})}&\cdots
&[\lambda_{p,n_p}+\mu]_{(m_{p,n_p})}
\end{Bmatrix}. \end{equation} More precisely, the condition \eqref{eq:redC1} and the condition \eqref{eq:redC2} for $\nu=1$ assure $S\in W[x]$. In this case the condition \eqref{eq:redC2} {\rm (resp.~\eqref{eq:redC3} for a fixed $j$)} assures that the sets of characteristic exponents of $P$ at $x=\infty$ {\rm (resp.~$c_j$)} are equal to the sets given in \eqref{eq:midR}, respectively.
Here we have $\RAd(\partial^{-\mu})\Red P=S$, if \begin{equation}
\begin{cases}
\lambda_{j,1}+m_{j,1}\text{ \ are not
characteristic exponents of $P$}\\
\quad
\text{ at $x=c_j$ for }j=0,\dots,p,\text{ respectively}, \end{cases}\label{eq:redC4} \end{equation} and moreover \begin{equation}
m_{0,1}=d\text{ \ or \ }
\lambda_{0,1}\notin\{-d,-d-1,\dots,1-m_{0,1}\}.\label{eq:redC5} \end{equation}
Using the notation in Definition~\ref{def:coord}, we have \begin{equation}\label{eq:redcoord}
\begin{split}
S &= \Ad\bigl((x-c_1)^{\lambda_{0,1}-2}\bigr)(x-c_1)^{d}T_{\frac1{x-c_1}}^*
(-\partial)^{-d}\Ad(\partial^{-\mu})T_{\frac1x+c_1}^*\\
&\qquad\cdot
(x-c_1)^{d}
\prod_{j=1}^p(x-c_j)^{-m_{j,1}}\Ad\bigl((x-c_1)^{\lambda_{0,1}}\bigr)P \end{split} \end{equation} under the conditions \eqref{eq:redC1} and \begin{equation} \begin{cases}
\lambda_{0,\nu}\notin\{0,-1,-2,\ldots,m_{0,1}-m_{0,\nu}-d+2\}\\
\text{if \ }m_{0,\nu}+\cdots+m_{p,1}-(p-1)n\ge 2,\
m_{1,1}\ne0\text{ \ and \ }\nu\ge 1. \end{cases} \end{equation}
{\rm iii)} Suppose $\ord P>1$ and $P$ is irreducible in {\rm ii)}. Then the conditions \eqref{eq:redC1}, \eqref{eq:redC2}, \eqref{eq:redC3} are valid. The condition \eqref{eq:redC5} is also valid if $d\ge 1$.
All these conditions in {\rm ii)} are valid if $\#\{j\,;\,m_{j,1}<n\}\ge 2$ and $\mathbf m$ is realizable and moreover $\lambda_{j,\nu}$ are generic under the Fuchs relation with $\lambda_{j,1}=0$ for $j=1,\dots,p$.
{\rm iv)} Let $\mathbf m=\bigl(m_{j,\nu}\bigr)_{\substack{j=0,\dots,p\\\nu=1,\dots,n_j}} \in\mathcal P^{(n)}_{p+1}$. Define $d$ by \eqref{eq:defd}. Suppose $\lambda_{j,\nu}$ are complex numbers satisfying \eqref{eq:midgen}. Suppose moreover $m_{j,1}\ge d$ for $j=1,\dots,p$. Defining $\mathbf m'\in\mathcal P^{(n)}_{p+1}$ and $\lambda_{j,\nu}'$ by \begin{align}
m'_{j,\nu}&=m_{j,\nu}-\delta_{\nu,1}d\quad(j=0,\dots,p,\ \nu=1,\dots,n_j),\\
\lambda'_{j,\nu}&=
\begin{cases}
2-\lambda_{0,1}&(j=0,\ \nu=1),\\
\lambda_{j,\nu}-\lambda_{0,1}+1&(j=0,\ \nu>1),\\
0 &(j>0,\ \nu=1),\\
\lambda_{j,\nu}+\lambda_{0,1}-1&(j>0,\ \nu>1),
\end{cases} \end{align} we have \begin{equation}\label{eq:midinv}
\idx\mathbf m=\idx\mathbf m',\quad
|\{\lambda_{\mathbf m}\}|=|\{\lambda'_{\mathbf m'}\}|. \end{equation} \end{thm} \begin{proof} The claim i) is clear from the definition of the Riemann scheme.
{\rm ii)} Suppose \eqref{eq:redC1}, \eqref{eq:redC2} and \eqref{eq:redC3}. Then \begin{equation}
P' := \Bigl(\prod_{j=1}^p (x-c_j)^{-m_{j,1}}\Bigr)P\in W[x]. \end{equation} Note that $\Red P=P'$ under the condition \eqref{eq:redC4}. Put $Q:=\partial^{(p-1)n - \sum_{j=1}^pm_{j,1}}P'$. Here we note that \eqref{eq:redC1} assures $(p-1)n - \sum_{j=1}^pm_{j,1}\ge0$.
Fix a positive integer $j$ with $j\le p$. For simplicity suppose $j=1$ and $c_j=0$. Since $P'=\sum_{j=0}^n a_j(x)\partial^j$ with $\deg a_j(x)\le (p-1)n+j-\sum_{j=1}^pm_{j,1}$, we have \begin{align*}
x^{m_{1,1}}P' &=
\sum_{\ell=0}^Nx^{N-\ell}
r_\ell(\vartheta)
\prod_{\substack{1\le \nu\le n_0\\
0\le i< m_{0,\nu}-\ell}}
(\vartheta+\lambda_{0,\nu}+i) \end{align*} and \[
N:=(p-1)n-\sum_{j=2}^pm_{j,1}=m_{0,1}+m_{1,1}-d \] with suitable polynomials $r_\ell$ such that $r_0\in\mathbb C^\times$. Suppose \begin{equation}\label{eq:redC0}
\prod_{\substack{1\le \nu\le n_0\\0\le i< m_{0,\nu}-\ell}}
(\vartheta+\lambda_{0,\nu}+i)
\notin x W[x]
\text{ \ if \ }N-m_{1,1}+1\le \ell\le N. \end{equation} Since $P'\in W[x]$, we have \[
x^{N-\ell}r_\ell(\vartheta) =
x^{N-\ell}x^{\ell-N+m_{1,1}}\partial^{\ell-N+m_{1,1}}s_\ell(\vartheta)
\text{ \ if \ }N-m_{1,1}+1\le \ell\le N \] for suitable polynomials $s_\ell$. Putting $s_\ell=r_\ell$ for $0\le\ell\le N-m_{1,1}$, we have \begin{equation}\begin{split}
P' &=\sum_{\ell=0}^{N-m_{1,1}}
x^{N-m_{1,1}-\ell}
s_\ell(\vartheta)
\prod_{\substack{1\le \nu\le n_0\\0\le i< m_{0,\nu}-\ell}}
(\vartheta+\lambda_{0,\nu}+i)\\
&\quad+\sum_{\ell=N-m_{1,1}+1}^N\partial^{\ell-N+m_{1,1}}
s_\ell(\vartheta)
\prod_{\substack{1\le \nu\le n_0\\0\le i< m_{0,\nu}-\ell}}
(\vartheta+\lambda_{0,\nu}+i). \end{split}\label{eq:Pdash} \end{equation} Note that $s_0\in\mathbb C^\times$ and the condition \eqref{eq:redC0} is equivalent to the condition $
\lambda_{0,\nu}+i\ne 0 $ for any $\nu$ and $i$ such that there exists an integer $\ell$ with $0\le i\le m_{0,\nu}-\ell-1$ and $N-m_{1,1}+1\le \ell\le N$. This condition is valid if \eqref{eq:redC2} is valid, namely, $m_{1,1}=0$ or \[
\lambda_{0,\nu}\notin\{0,-1,\dots,m_{0,1}-m_{0,\nu}-d+2\} \] for $\nu$ satisfying $m_{0,\nu}\ge m_{0,1}-d+2$. Under this condition we have \begin{align*}
Q &= \sum_{\ell=0}^N\partial^\ell
s_\ell(\vartheta)
\prod_{1\le i\le N-m_{1,1}-\ell}(\vartheta+i)\cdot
\prod_{\substack{1\le \nu\le n_0\\0\le i< m_{0,\nu}-\ell}}
(\vartheta+\lambda_{0,\nu}+i),
\allowdisplaybreaks\\
\Ad(\partial^{-\mu})Q&=
\sum_{\ell=0}^N\partial^\ell
s_\ell(\vartheta-\mu)
\prod_{1\le i\le N-m_{1,1}-\ell}(\vartheta-\mu+i)
\\
&\quad\cdot
\prod_{1\le i\le m_{0,1}-\ell}(\vartheta+i)
\cdot
\prod_{\substack{2\le \nu\le n_0\\0\le i< m_{0,\nu}-\ell}}
(\vartheta-\mu+\lambda_{0,\nu}+i) \end{align*} since $\mu=\lambda_{0,1}-1$. Hence $\partial^{-m_{0,1}}\! \Ad(\partial^{-\mu})Q$ equals \[ \begin{split}
&\sum_{\ell=0}^{m_{0,1}-1} x^{m_{0,1}-\ell}s_\ell(\vartheta-\mu)
\prod_{1\le i\le N-m_{1,1}-\ell}(\vartheta-\mu+i)
\prod_{\substack{2\le \nu\le n_0\\ 0\le i< m_{0,\nu}-\ell}}
(\vartheta-\mu+\lambda_{0,\nu}+i)\\
&+\sum_{\ell=m_{0,1}}^N
\partial^{\ell-m_{0,1}}
s_\ell(\vartheta-\mu)
\prod_{1\le i\le N-m_{1,1}-\ell}(\vartheta-\mu+i)
\prod_{\substack{2\le \nu\le n_0\\ 0\le i< m_{0,\nu}-\ell}}
(\vartheta-\mu+\lambda_{0,\nu}+i) \end{split} \] and then the set of characteristic exponents of this operator at $\infty$ is \[\{[1-\mu]_{(m_{0,1}-d)},[\lambda_{0,2}-\mu]_{(m_{0,2})},\dots, [\lambda_{0,n_0}-\mu]_{(m_{0,n_0})}\}.\] Moreover $\partial^{-m_{0,1}-1}\! \Ad(\partial^{-\mu})Q\notin W[x]$ if $\lambda_{0,1}+m_{0,1}$ is not a characteristic exponent of $P$ at $\infty$ and $-\lambda_{0,1}+1+i\ne m_{0,1}+1$ for $1\le i\le N-m_{1,1}=m_{0,1}-d$, which assures $x^{m_{0,1}}s_0
\prod_{1\le i\le N-m_{1,1}}(\vartheta-\mu+i)
\prod_{\substack{2\le \nu\le n_0\\ 0\le i< m_{0,\nu}}}
(\vartheta-\mu+\lambda_{1,\nu}+i)\notin\partial W[x]$.
Similarly we have \begin{align*}
P' &= \sum_{\ell=0}^{m_{1,1}}\partial^{m_{1,1}-\ell}
q_\ell(\vartheta)
\prod_{\substack{2\le \nu\le n_1\\0\le i< m_{1,\nu}-\ell}}
(\vartheta-\lambda_{1,\nu}-i)\\
&\quad+\sum_{\ell=m_{1,1}+1}^Nx^{\ell-m_{1,1}}
q_\ell(\vartheta)
\prod_{\substack{2\le \nu\le n_1\\0\le i< m_{1,\nu}-\ell}}
(\vartheta-\lambda_{1,\nu}-i),\allowdisplaybreaks\\
Q &= \sum_{\ell=0}^{m_{1,1}}
\partial^{N-\ell}q_\ell(\vartheta)
\prod_{\substack{2\le \nu\le n_1\\0\le i< m_{1,\nu}-\ell}}
(\vartheta+\lambda_{1,\nu}-i)\\
&\quad+\sum_{\ell=m_{1,1}+1}^N \partial^{N-\ell}q_\ell(\vartheta)
\prod_{i=1}^{\ell-m_{1,1}}(\vartheta+i)
\prod_{\substack{2\le \nu\le n_1\\0\le i< m_{1,\nu}-\ell}}
(\vartheta-\lambda_{1,\nu}-i).\allowdisplaybreaks\\
\Ad(\partial^{-\mu})Q&=
\sum_{\ell=0}^N\partial^{N-\ell}
q_\ell(\vartheta-\mu)
\prod_{1\le i\le\ell-m_{1,1}}(\vartheta-\mu+i)\\
&\quad\cdot
\prod_{\substack{2\le \nu\le n_1\\0\le i< m_{1,\nu}-\ell}}
(\vartheta-\mu-\lambda_{1,\nu}-i) \end{align*} with $q_0\in\mathbb C^\times$. Then the set of characteristic exponents of $\partial^{-m_{0,1}}\!\Ad(p^{-\mu})Q$ equals \[
\{[0]_{(m_{1,1}-d)},[\lambda_{1,2}+\mu]_{(m_{1,2})},\dots,
[\lambda_{1,n_1}+\mu]_{(m_{1,n_1})}\} \] if \begin{equation*}
\prod_{\substack{2\le \nu\le n_1\\0\le i< m_{1,\nu}-\ell}}
(\vartheta-\mu-\lambda_{1,\nu}-i) \notin \partial W[x] \end{equation*} for any integers $\ell$ satisfying $0\le\ell\le N$ and $N-\ell<m_{0,1}$. This condition is satisfied if \eqref{eq:redC3} is valid, namely, $m_{0,1}=0$ or \begin{equation*}
\begin{split}
&\lambda_{0,1}+\lambda_{1,\nu}\notin\{0,-1,\dots,m_{1,1}-m_{1,\nu}-d+2\}\\
&\qquad
\text{ \ for \ }\nu\ge 2\text{ \ satisfying \ }
m_{1,\nu}\ge m_{1,1}-d+2
\end{split} \end{equation*} because $m_{1,\nu}-\ell-1\le m_{1,\nu}+m_{0,1}-N-2=m_{1,\nu}-m_{1,1}+d-2$ and the condition $\vartheta-\mu-\lambda_{1,\nu}-i\in \partial W[x]$ means $-1=\mu+\lambda_{1,\nu}+i=\lambda_{0,1}-1+\lambda_{1,\nu}+i$.
Now we will prove \eqref{eq:redcoord}. Under the conditions, it follows from \eqref{eq:Pdash} that \begin{align*}
\tilde P:\!&=x^{m_{0,1}-N}\Ad\bigl(x^{\lambda_{0,1}}\bigr)\prod_{j=2}^p(x-c_j)^{-{m_{j,1}}}P \allowdisplaybreaks\\
&=x^{m_{0,1}+m_{1,1}-N}
\Ad\bigl(x^{\lambda_{0,1}}\bigr)P' \allowdisplaybreaks\\
&=\sum_{\ell=0}^N x^{m_{0,1}-\ell}\Ad\bigl(x^{\lambda_{0,1}}\bigr)
s_\ell(\vartheta)
\prod_{0\le\nu<\ell-N+m_{1,1}}\!\!\!(\vartheta-\nu)
\prod_{\substack{1\le\nu\le n_0\\0\le i<m_{0,\nu}-\ell}}
(\vartheta+\lambda_{0,\nu}+i), \allowdisplaybreaks\\
\tilde Q:\!&=(-\partial)^{N-m_{0,1}}T^*_{\frac1x}\tilde P \allowdisplaybreaks\\
&=(-\partial)^{N-m_{0,1}}\sum_{\ell=0}^N x^{\ell-m_{0,1}}
s_\ell(-\vartheta-\lambda_{0,1})
\prod_{0\le\nu<\ell-N+m_{1,1}}\!\!\!(-\vartheta-\lambda_{0,1}-\nu)
\\&\quad\cdot
\prod_{\substack{2\le\nu\le n_0\\0\le i<m_{0,\nu}-\ell}}
(-\vartheta+\lambda_{0,\nu}-\lambda_{0,1}+i)
\prod_{0\le i\le m_{0,1}-\ell}(-\vartheta+i) \allowdisplaybreaks\\
&=\sum_{\ell=0}^N
(-\partial)^{N-\ell}
s_\ell(-\vartheta-\lambda_{0,1})
\!\!\!\prod_{1\le i\le \ell-m_{0,1}}(-\vartheta-i)
\\&\quad\cdot
\prod_{0\le\nu<\ell-N+m_{1,1}}\!\!\!\!\!(-\vartheta-\lambda_{0,1}-\nu)
\prod_{\substack{2\le\nu\le n_0\\0\le i<m_{0,\nu}-\ell}}\!\!\!
(-\vartheta+\lambda_{0,\nu}-\lambda_{0,1}+i) \end{align*} and therefore \begin{align*}
\Ad(\partial^{-\mu})\tilde Q
&=\sum_{\ell=0}^{N}
(-\partial)^{N-\ell}s_\ell(-\vartheta-1)
\prod_{1\le i\le \ell-m_{0,1}}\!\!\!
(-\vartheta+\lambda_{0,1}-1-i) \\
&
\cdot\prod_{0\le\nu<\ell-N+m_{1,1}}\!\!\!\!\!(-\vartheta-1-\nu)
\prod_{\substack{2\le\nu\le n_0\\0\le i<m_{0,\nu}-\ell}}\!\!\!
(-\vartheta+\lambda_{0,\nu}-1+i). \end{align*} Since \begin{align*}
(-\partial)^{N-\ell-m_{1,1}}\prod_{0\le\nu<\ell-N+m_{1,1}}
\!\!\!\!\!(-\vartheta-1-\nu)
&=\begin{cases}
x^{\ell-N+m_{1,1}}&(N-\ell<m_{1,1}),\\
(-\partial)^{N-\ell-m_{1,1}}&(N-\ell\ge m_{1,1}),
\end{cases} \allowdisplaybreaks\\
&=x^{\ell-N+m_{1,1}}\prod_{0\le\nu< N-\ell-m_{1,1}}
(-\vartheta+\nu), \end{align*} we have \begin{align*}
\tilde Q'&:=(-\partial)^{-m_{1,1}}\Ad(\partial^{-\mu})
\tilde Q =\sum_{\ell=0}^N x^{\ell-N+m_{1,1}}\!\!\!\!\!
\prod_{0\le\nu< N-\ell-m_{1,1}}\!\!\!\!\!\!\!\!\!(-\vartheta+\nu)\\
&\qquad\cdot
s_\ell(-\vartheta-1) \prod_{0\le \nu< \ell-m_{0,1}}\!\!\!\!\!
(-\vartheta+\lambda_{0,1}-2-\nu)
\prod_{\substack{2\le\nu\le n_0\\0\le i<m_{0,\nu}-\ell}}\!\!\!
(-\vartheta+\lambda_{0,\nu}-1+i) \intertext{and} &x^{m_{0,1}+m_{1,1}-N}\Ad(x^{\lambda_{0,1}-2})T^*_{\frac1x} \tilde Q' = \sum_{\ell=0}^N x^{m_{0,1}-\ell}\!\!\!\!\!
\prod_{0\le \nu< \ell-m_{0,1}}\!\!\!\!\!
(\vartheta-\nu)\cdot
s_\ell(\vartheta-\lambda_{0,1}+1)\\
&\quad
\cdot\prod_{0\le\nu< N-m_{1,1}-\ell}\!\!\!\!\!\!\!\!\!
(\vartheta-\lambda_{0,1}+2+\nu)
\prod_{\substack{2\le\nu\le n_0\\0\le i<m_{0,\nu}-\ell}}\!\!\!
(\vartheta+\lambda_{0,\nu}-\lambda_{0,1}+1+i), \end{align*} which equals $\partial^{-m_{0,1}}\! \Ad(\partial^{-\mu})Q$ because $\prod_{0\le \nu<k}(\vartheta-\nu) =x^k\partial^k$ for $k\in\mathbb Z_{\ge 0}$.
iv) (Cf.~Remark~\ref{rem:KacGRS} ii) for another proof.) \ \ Since \begin{align*}
\idx\mathbf m-\idx\mathbf m'&=\sum_{j=0}^p m_{j,1}^2 - (p-1)n^2 -
\sum_{j=0}^p(m_{j,1}-d)^2+(p-1)(n-d)^2 \allowdisplaybreaks\\
&=2d\sum_{j=0}^pm_{j,1}-(p+1)d^2-2(p-1)nd+(p-1)d^2 \allowdisplaybreaks\\
&=d\Bigl(2\sum_{j=0}^pm_{j,1}-2d -2(p-1)n\Bigr)=0 \intertext{and}
\sum_{j=0}^p\sum_{\nu=1}^{n_j}m_{j,\nu}\lambda_{j,\nu}&
-\sum_{j=0}^p\sum_{\nu=1}^{n_j}m'_{j,\nu}\lambda'_{j,\nu} \allowdisplaybreaks\\
&=m_{0,1}(\mu+1)-(m_{0,1}-d)(1-\mu)+\mu(n-m_{0,1}-\sum_{j=1}^p(n-m_{j,1})) \allowdisplaybreaks\\
&=\Bigl(\sum_{j=0}^pm_{j,1} - d - (p-1)n\Bigr)\mu
-m_{0,1}d-(m_{0,1}-d) =d, \end{align*} we have the claim.
The claim iii) follows from the following lemma when $P$ is irreducible.
Suppose $\lambda_{j,\nu}$ are generic in the sense of the claim iii). Put $\mathbf m=\gcd(\mathbf m)\overline{\mathbf m}$. Then an irreducible subspace of the solutions of $Pu=0$ has the spectral type $\ell'\overline{\mathbf m}$ with $1\le \ell'\le \gcd(\mathbf m)$ and the same argument as in the proof of the following lemma shows iii). \end{proof} The following lemma is known which follows from Scott's lemma (cf.~\S\ref{eq:rigididx}).
\begin{lem}\label{lem:irrred} Let $P$ be a Fuchsian differential operator with the Riemann scheme \eqref{eq:GRS}. Suppose $P$ is irreducible. Then \begin{equation}\label{eq:rigididx}
\idx\mathbf m\le 2. \end{equation}
Fix $\ell =(\ell_0,\dots,\ell_p)\in\mathbb Z_{>0}^{p+1}$ and suppose $\ord P>1$. Then \begin{equation}\label{eq:SL0}
m_{0,\ell_0}+m_{1,\ell_1}+\cdots+m_{p,\ell_p}-(p-1)\ord\mathbf m \le m_{k,\ell_k}
\text{ \ for \ }k=0,\dots,p. \end{equation} Moreover the condition \begin{equation}
\lambda_{0,\ell_0}+\lambda_{1,\ell_1}+\cdots+\lambda_{p,\ell_p}\in\mathbb Z \end{equation} implies \begin{equation}\label{eq:SL1}
m_{0,\ell_0}+m_{1,\ell_1}+\cdots+m_{p,\ell_p}\le (p-1)\ord\mathbf m. \end{equation}\end{lem} \begin{proof} Let $M_j$ be the monodromy generators of the solutions of $Pu=0$ at $c_j$, respectively. Then $\dim Z(M_j)\ge \sum_{\nu=1}^{n_j}m_{j,\nu}^2$ and therefore $\sum_{j=0}^p\codim Z(M_j)\le (p+1)n^2 -\bigl(\idx\mathbf m +(p-1)n^2) =2n^2-\idx\mathbf m$. Hence Corollary \ref{cor:katz} (cf.~\eqref{eq:iridx}) proves \eqref{eq:rigididx}. \index{00Z@$Z(A)$, $Z(\mathbf M)$}
We may assume $\ell_j=1$ for $j=0,\dots,p$ and $k=0$ to prove the lemma. By the map $u(x)\mapsto \prod_{j=1}^p(x-c_j)^{-\lambda_{j,1}}u(x)$ we may moreover assume $\lambda_{j,\ell_j}=0$ for $j=1,\dots,p$. Suppose $\lambda_{0,1}\in\mathbb Z$. We may assume $M_p\cdots M_1M_0=I_n$. Since $\dim\ker M_j\ge m_{j,1}$, Scott's lemma (Lemma~\ref{lem:Scott}) assures \eqref{eq:SL1}.
The condition \eqref{eq:SL0} is reduced to \eqref{eq:SL1} by putting $m_{0,\ell_0}=0$ and $\lambda_{0,\ell_0}=-\lambda_{1,\ell_1}-\cdots -\lambda_{p,\ell_p}$ because we may assume $k=0$ and $\ell_0=n_0+1$. \end{proof}
\begin{rem}\label{rem:midisom} {\rm i) } Retain the notation in Theorem~\ref{thm:GRSmid}. The operation in Theorem~\ref{thm:GRSmid} i) corresponds to the \textsl{addition} and the operation in Theorem~\ref{thm:GRSmid} ii) corresponds to Katz's \textsl{middle convolution} (cf.~\cite{Kz}), which are studied by \cite{DR} for the systems of Schlesinger canonical form.
The operation $c(P):=\Ad(\partial^{-\mu})\partial^{(p-1)n}P$ is always well-defined for the Fuchsian differential operator of the normal form which has $p+1$ singular points including $\infty$. This corresponds to the \textsl{convolution} defined by Katz. Note that the equation $Sv=0$ is a quotient of the equation $c(P)\tilde u=0$. \index{convolution}
{\rm ii) } Retain the notation in the previous theorem. Suppose the equation $Pu=0$ is irreducible and $\lambda_{j,\nu}$ are generic complex numbers satisfying the assumption in Theorem~\ref{thm:GRSmid}. Let $u(x)$ be a local solution of the equation $Pu=0$ corresponding to the characteristic exponent $\lambda_{i,\nu}$ at $x=c_i$. Assume $0\le i\le p$ and $1<\nu\le n_i$. Then the irreducible equations $\bigl(\Ad\bigl((x-c_j)^r\bigr)P\bigr)u_1=0$ and $\bigl(\RAd(\partial^{-\mu})\circ\Red P\bigr)u_2=0$ are characterized by the equations satisfied by $u_1(x)=(x-c_j)^ru(x)$ and $u_2(x)=I_{c_i}^\mu(u(x))$, respectively.
Moreover for any integers $k_0,k_1,\dots,k_p$ the irreducible equation $Qu_3=0$ satisfied by $u_3(x)=I_{c_i}^{\mu+k_0}\bigl(\prod_{j=1}^p(x-c_j)^{k_j}u(x)\bigr)$ is isomorphic to the equation $\bigl(\RAd(\partial^{-\mu})\circ\Red P\bigr)u_2=0$ as $W(x)$-modules (cf.~\S\ref{sec:ODE} and \S\ref{sec:contig}). \end{rem}
\begin{exmp}[exceptional parameters]\label{ex:outuniv} \index{Jordan-Pochhammer!exceptional parameter} \index{tuple of partitions!rigid!21,21,21,21}
The Fuchsian differential equation with the Riemann scheme \begin{equation*}
\begin{Bmatrix}
x=\infty & 0 & 1 & c\\
[\delta]_{(2)} & [0]_{(2)} & [0]_{(2)} & [0]_{(2)}\\
2-\alpha-\beta-\gamma-2\delta & \alpha & \beta & \gamma
\end{Bmatrix} \end{equation*} is a Jordan-Pochhammer equation (cf.~Example~\ref{ex:midconv} ii)) if $\delta\ne0$, which is proved by the reduction using the operation $\RAd(\partial^{1-\delta})\Red$ given in Theorem~\ref{thm:GRSmid} ii).
The Riemann scheme of the operator \begin{align*}
P_r&=x(x-1)(x-c)\partial^3\\
&\quad
-\bigl((\alpha+\beta+\gamma-6)x^2-((\alpha+ \beta-4)c+\alpha+\gamma-4)x
+(\alpha-2)c \bigr)\partial^2\\ &\quad
-\bigl(2(\alpha+\beta+\gamma-3)x+(\alpha+\beta-2)c+\alpha+\gamma-2+r\bigr)\partial \end{align*} equals \begin{equation*}
\begin{Bmatrix}
x=\infty & 0 & 1 & c\\
[0]_{(2)} & [0]_{(2)} & [0]_{(2)} & [0]_{(2)}\\
2-\alpha-\beta-\gamma & \alpha & \beta & \gamma
\end{Bmatrix}, \end{equation*} which corresponds to a Jordan-Pochhammer operator when $r=0$. If the parameters are generic, $\RAd(\partial)P_r$ is Heun's operator \eqref{eq:Heun} with the Riemann scheme \index{Heun's equation} \[ \begin{Bmatrix}
x=\infty & 0 & 1 & c\\
2& 0 & 0 & 0\\
3-\alpha-\beta-\gamma&\alpha-1&\beta-1&\gamma-1 \end{Bmatrix}, \] which contains the accessory parameter $r$. This transformation doesn't satisfy \eqref{eq:redC2} for $\nu=1$.
The operator $\RAd(\partial^{1-\alpha-\beta-\gamma})P_r$ has the Riemann scheme \[ \begin{Bmatrix}
x=\infty & 0 & 1 & c\\
\alpha+\beta+\gamma-1& 0 & 0 & 0\\
\alpha+\beta+\gamma&1-\beta-\gamma&1-\gamma-\alpha&1-\alpha-\beta \end{Bmatrix} \] and the monodromy generator at $\infty$ is semisimple if and only if $r=0$. This transformation doesn't satisfy \eqref{eq:redC2} for $\nu=2$. \end{exmp}
\begin{defn}\label{def:pell} Let \[
P=a_n(x)\partial^n+a_{n-1}(x)\partial^{n-1}+\cdots+a_0(x) \] be a Fuchsian differential operator with the Riemann scheme \eqref{eq:GRS}. Here some $m_{j,\nu}$ may be 0. Fix $\ell=(\ell_0,\dots,\ell_p)\in\mathbb Z^{p+1}_{>0}$ with $1\le\ell_j\le n_j$. Suppose \begin{equation}\label{eq:redok}
\#\{j\,;\,m_{j,\ell_j}\ne n\text{ and }0\le j\le p\}\ge 2. \end{equation} Put \index{00dm@$d_\ell(\mathbf m)$} \begin{equation}
d_{\mathbf\ell}(\mathbf m):=m_{0,\ell_0}+\cdots+m_{p,\ell_p}
- (p-1)\ord\mathbf m\label{eq:dm} \end{equation} and \begin{equation}
\begin{split}
\partial_\ell P:=&\,
\Ad\bigl(\prod_{j=1}^p(x-c_j)^{\lambda_{j,\ell_j}})
\prod_{j=1}^p(x-c_j)^{m_{j,\ell_j}-d_\ell(\mathbf m)}\partial^{-m_{0,\ell_0}}
\Ad(\partial^{1-\lambda_{0,\ell_0}-\cdots-\lambda_{p,\ell_p}})\\
&\
\cdot\partial^{(p-1)n-m_{1,\ell_1}-\cdots-m_{p,\ell_p}}
a_n^{-1}(x)\prod_{j=1}^n(x-c_j)^{n-m_{j,\ell_j}}
\Ad\bigl(\prod_{j=1}^p(x-c_j)^{-\lambda_{j,\ell_j}})
P.
\end{split}\label{eq:opred} \end{equation} \index{000deltaell@$\partial,\ \partial_\ell,\ \partial_{max}$} If $\lambda_{j,\nu}$ are generic under the Fuchs relation or $P$ is irreducible, $\partial_\ell P$ is well-defined as an element of $W[x]$ and \begin{align}
\partial_\ell^2 P &= P\text{ \ with $P$ of the form \eqref{eq:FNF}},\label{eq:pellP2}\\
\begin{split}
\partial_\ell P&\in W(x)\RAd\bigl(\prod_{j=1}^p(x-c_j)^{\lambda_{j,\ell_j}})
\RAd(\partial^{1-\lambda_{0,\ell_0}-\cdots-\lambda_{p,\ell_p}})\\
&\quad
\cdot\RAd\bigl(\prod_{j=1}^p(x-c_j)^{-\lambda_{j,\ell_j}})P
\end{split}\label{eq:defpell} \end{align} and $\partial_\ell$ gives a correspondence between differential operators of normal form \eqref{eq:FNF}. Here the spectral type $\partial_\ell\mathbf m$ of $\partial_\ell P$ is given by \begin{align}
\partial_\ell\mathbf m
&:=\bigr(m'_{j,\nu}\bigr)_{\substack{0\le j\le p\\ 1\le\nu\le n_j}}
\text{ \ and \ }
m_{j,\nu}'=
m_{j,\nu}-\delta_{\ell_j,\nu}\cdot d_{\mathbf\ell}(\mathbf m)
\label{eq:redm} \end{align} and the Riemann scheme of $\partial_\ell P$ equals \begin{equation}\label{eq:pellGRS}
\partial_\ell\bigl\{\lambda_{\mathbf m}\bigr\}
:=\bigl\{\lambda'_{\mathbf m'}\bigr\}\text{ \ with \ }
\lambda'_{j,\nu}=
\begin{cases}
\lambda_{0,\nu} - 2\mu_\ell&(j=0,\ \nu=\ell_0)\\
\lambda_{0,\nu} -\mu_\ell&(j=0,\ \nu\ne\ell_0)\\
\lambda_{j,\nu} &(1\le j\le p,\ \nu=\ell_j)\\
\lambda_{0,\nu} +\mu_\ell&(1\le j\le p,\ \nu\ne\ell_j)
\end{cases} \end{equation} by putting \begin{equation}
\mu_\ell := \sum_{j=0}^p \lambda_{j,\ell_j} -1. \end{equation} It follows from Theorem~\ref{thm:GRSmid} that the above assumption is satisfied if \begin{equation}\label{eq:non-neg}
m_{j,\ell_j}\ge d_\ell(\mathbf m)\qquad(j=0,\dots,p) \end{equation} and \begin{equation}\label{eq:dless2} \begin{split} &\sum_{j=0}^p\lambda_{j,\ell_j+(\nu-\ell_j)\delta_{j,k}}
\notin\bigl\{i\in\mathbb Z\,;\,
(p-1)n-\sum_{j=0}^p m_{j,\ell_j+(\nu-\ell_j)\delta_{j,k}}+2\le i\le 0\bigr\}\\ &\qquad\text{for }k=0,\dots,p\text{ and }\nu=1,\dots,n_k. \end{split} \end{equation}
Note that $\partial_\ell \mathbf m\in\mathcal P_{p+1}$ is \textsl{well-defined} for a given $\mathbf m\in\mathcal P_{p+1}$ if \eqref{eq:non-neg} is valid. Moreover we define \begin{align}
\partial\mathbf m&:=\partial_{(1,1,\ldots)}\mathbf m,\\
\begin{split}
\partial_{max}\mathbf m&:=\partial_{\ell_{max}(\mathbf m)}\mathbf m\text{ \ with \ }\\
\ell_{max}(\mathbf m)_j&:=\min\bigl\{\nu\,;\,m_{j,\nu}
=\max\{m_{j,1},m_{j,2},\ldots\}\bigr\},
\end{split}\label{eq:pmax}\\
d_{max}(\mathbf m)
&:=\sum_{j=0}^p\max\{m_{j,1},m_{j,2},\dots,m_{j,n_j}\}-(p-1)\ord\mathbf m. \end{align} \index{000deltaell@$\partial,\ \partial_\ell,\ \partial_{max}$} \index{00dmax@$d_{max},\ \ell_{max}$} For a Fuchsian differential operator $P$ with the Riemann scheme \eqref{eq:GRS} we define \begin{equation}
\partial_{max}P:=\partial_{\ell_{max}(\mathbf m)}P
\text{ \ and \ }\partial_{max}\bigl\{\lambda_{\mathbf m}\bigr\}
=\partial_{\ell_{max}(\mathbf m)}\bigl\{\lambda_{\mathbf m}\bigr\}. \end{equation} A tuple $\mathbf m\in\mathcal P$ is called \textsl{basic} if $\mathbf m$ is indivisible and $d_{max}(\mathbf m)\le 0$. \index{tuple of partitions!basic} \end{defn}
\begin{prop}[linear fractional transformation]\label{prop:coordf} \index{linear fractional transformation} \index{coordinate transformation!linear fractional} Let $\phi$ be a linear fractional transformation of $\mathbb P^1(\mathbb C)$, namely there exists $\left(\begin{smallmatrix}\alpha&\beta\\ \gamma&\delta\end{smallmatrix} \right)\in GL(2,\mathbb C)$ such that $\phi(x)=\frac{\alpha x+\beta}{\gamma x +\delta}$. Let $P$ be a Fuchsian differential operator with the Riemann scheme \eqref{eq:GRS}. We may assume $-\frac\delta\gamma=c_j$ with a suitable $j$ by putting $c_{p+1}=-\frac\delta\gamma$, $\lambda_{p+1,1}=0$ and $m_{p+1,1}=n$ if necessary. Fix $\ell=(\ell_0,\dotsm\ell_p)\in\mathbb Z_{>0}^{p+1}$. If \eqref{eq:non-neg} and \eqref{eq:dless2} are valid, we have \begin{equation}
\begin{split}
\partial_\ell P&\in W(x)\Ad\bigl((\gamma x+\delta)^{2\mu}\bigr)
T^*_{\phi^{-1}}\partial_\ell T^*_{\phi}P,\\
\mu&=\lambda_{0,\ell_0}+\cdots+\lambda_{p,\ell_p}-1.
\end{split} \end{equation} \end{prop} \begin{proof} The claim is clear if $\gamma=0$. Hence we may assume $\phi(x)=\frac1x$ and the claim follows from \eqref{eq:redcoord}. \end{proof}
\begin{rem}\label{rem:idxFuchs} {\rm i) \ } Fix $\lambda_{j,\nu}\in\mathbb C$. If $P$ has the Riemann scheme $\{\lambda_{\mathbf m}\}$ with $d_{max}(\mathbf m)=1$, $\partial_\ell P$ is well-defined and $\partial_{max}P$ has the Riemann scheme $\partial_{max}\{\lambda_{\mathbf m}\}$. This follows from the fact that the conditions \eqref{eq:redC1}, \eqref{eq:redC2} and \eqref{eq:redC3} are valid when we apply Theorem~\ref{thm:GRSmid} to the operation $\partial_{max}:P\mapsto \partial_{max}P$.
{\rm ii) \ } We remark that \begin{align}
\idx\mathbf m &= \idx\partial_\ell \mathbf m,\\
\ord\partial_{max}\mathbf m&=\ord\mathbf m-d_{max}(\mathbf m). \end{align} Moreover if $\idx\mathbf m > 0$, we have \begin{equation}
d_{max}(\mathbf m)> 0 \end{equation} because of the identity \begin{equation}\label{eq:idxd}
\Bigl(\sum_{j=0}^km_{j,\ell_j}-(p-1)\ord\mathbf m\Bigr)
\cdot\ord\mathbf m
= \idx\mathbf m + \sum_{j=0}^p\sum_{\nu=1}^{n_j}
(m_{j,\ell_j}-m_{j,\nu})\cdot m_{j,\nu}. \end{equation} If $\idx\mathbf m=0$, then $d_{max}(\mathbf m)\ge 0$ and the condition $d_{max}(\mathbf m)=0$ implies $m_{j,\nu}=m_{j,1}$ for $\nu=2,\dots,n_j$ and $j=0,1,\dots,p$ (cf.~Corollary~\ref{cor:idx0}).
{\rm iii)} The set of indices $\ell_{max}(\mathbf m)$ is defined in \eqref{eq:pmax} so that it is uniquely determined. It is sufficient to impose only the condition \begin{equation}\label{eq:lmaxe}
m_{j,\ell_{max}(\mathbf m)_j}=\max\{m_{j,1},m_{j,2},\ldots\}
\qquad(j=0,\dots,p) \end{equation} on $\ell_{max}(\mathbf m)$ for the arguments in this paper. \end{rem}
Thus we have the following result. \begin{thm}\label{thm:realizable} A tuple $\mathbf m\in\mathcal P$ is realizable if and only if $s\mathbf m$ is trivial {\rm (cf.~Definitions~\ref{def:tuples} and \ref{def:Sinfty})} or $\partial_{max} \mathbf m$ is well-defined and realizable. \end{thm} \begin{proof} We may assume $\mathbf m\in\mathcal P_{p+1}^{(n)}$ is monotone.
Suppose $\#\{j\,;\,m_{j,1}<n\}<2$. Then $\partial_{max}\mathbf m$ is not well-defined. We may assume $p=0$ and the corresponding equation $Pu=0$ has no singularities in $\mathbb C$ by applying a suitable addition to the equation and then $P\in W(x)\partial^n$. Hence $\mathbf m$ is realizable if and only if $\#\{j\,;\,m_{j,1}<n\}=0$, namely, $\mathbf m$ is trivial.
Suppose $\#\{j\,;\,m_{j,1}<n\}\ge 2$. Then Theorem~\ref{thm:GRSmid} assures that $\partial_{max}\mathbf m$ is realizable if and only if $\partial_{max}\mathbf m$ is realizable. \end{proof}
In the next section we will prove that $\mathbf m$ is realizable if $d_{max}(\mathbf m)\le 0$. Thus we will have a criterion whether a given $\mathbf m\in\mathcal P$ is realizable or not by successive applications of $\partial_{max}$.
\begin{exmp} There are examples of successive applications of $s\circ\partial$ to monotone elements of $\mathcal P$:
$\underline411,\underline411,\underline42,\underline33 \overset{15-2\cdot6=3}\longrightarrow\underline111,\underline111,\underline21 \overset{4-3=1}\longrightarrow{\underline1}1,{\underline1}1,{\underline1}1 \overset{3-2=1}\longrightarrow1,1,1$ (rigid)
$\underline211,\underline211,\underline1111\overset{5-4=1}\longrightarrow \underline111,\underline111,\underline111\overset{3-3=0}\longrightarrow 111,111,111$ (realizable, not rigid)
$\underline211,\underline211,\underline211,\underline31 \overset{9-8=1}\longrightarrow \underline111,\underline111,\underline111,\underline21 \overset{5-6=-1}\longrightarrow$ (realizable, not rigid)
${\underline2}2,{\underline2}2,{\underline1}111\overset{5-4=1}\longrightarrow {\underline2}1,{\underline2}1,{\underline1}11 \overset{5-3=2}\longrightarrow\times$ (not realizablej\\ The numbers on the above arrows are $d_{(1,1,\dots)}(\mathbf m)$. We sometimes delete the trivial partition as above. \end{exmp}
The transformation of the generalized Riemann scheme of the application of $\partial_{max}^k$ is described in the following definition.
\begin{defn}[Reduction of Riemann schemes]\label{def:redGRS} Let $\mathbf m=\bigl(m_{j,\nu}\bigr)_{\substack{j=0,\dots,p\\\nu=1,\dots,n_j}} \in\mathcal P_{p+1}$ and $\lambda_{j,\nu}\in\mathbb C$ for $j=0,\dots,p$ and $\nu=1,\dots,n_j$. Suppose $\mathbf m$ is realizable. Then there exists a positive integer $K$ such that \begin{equation} \begin{split}
&\ord\mathbf m > \ord\partial_{max}\mathbf m > \ord\partial_{max}^2\mathbf m > \cdots
> \ord\partial_{max}^K\mathbf m\\
&\qquad\text{ and }
s\partial_{\max}^K\mathbf m\text{ \ is trivial or \ } d_{max}\bigl(\partial_{max}^K\mathbf m\bigr)\le 0. \end{split} \end{equation} Define $\mathbf m(k)\in\mathcal P_{p+1}$, $\ell(k)\in\mathbb Z$, $\mu(k)\in\mathbb C$ and $\lambda(k)_{j,\nu\in\mathbb C}$ for $k=0,\dots,K$ by \index{00mk@$\mathbf m(k),\ \lambda(k),\ \mu(k),\ \ell(k)$} \begin{align}
\mathbf m(0)&=\mathbf m\text{ \ and \ }
\mathbf m(k)=\partial_{max}\mathbf m(k-1)\quad(k=1,\dots,K),\\
\ell(k) &= \ell_{max}\bigl(\mathbf m(k)\bigr)
\text{ \ and \ }
d(k) = d_{max}\bigl(\mathbf m(k)\bigr)
,\\
\bigl\{\lambda(k)_{\mathbf m(k)}\bigr\}
&=\partial_{max}^k\bigl\{\lambda_{\mathbf m}\bigr\}
\text{ \ and \ }\mu(k)=\lambda(k+1)_{1,\nu} - \lambda(k)_{1,\nu}
\quad(\nu\ne\ell(k)_1). \end{align} Namely, we have \begin{align}
\lambda(0)_{j,\nu}&=\lambda_{j,\nu}\quad(j=0,\dots,p,\ \nu=1,\dots,n_j), \allowdisplaybreaks\\
\mu(k) &= \sum_{j=0}^p\lambda(k)_{j,\ell(k)_j}-1, \allowdisplaybreaks\\ \begin{split}
\lambda(k+1)_{j,\nu}
&=\begin{cases}
\lambda(k)_{0,\nu}-2\mu(k)&(j=0,\ \nu=\ell(k)_0),\\
\lambda(k)_{0,\nu}-\mu(k)&(j=0,\ 1\le\nu\le n_0,\ \nu\ne \ell(k)_0),\\
\lambda(k)_{j,\nu}&(1\le j\le p,\ \nu=\ell(k)_j),\\
\lambda(k)_{j,\nu}+\mu(k)&(1\le j\le p,\ 1\le\nu\le n_j,\ \nu\ne \ell(k)_j)
\end{cases}\\
&=\lambda(k)_{j,\nu}+\bigl((-1)^{\delta_{j,0}}-\delta_{\nu,\ell(k)_j}\bigr)
\mu(k), \end{split}\\ \bigl\{\lambda_{\mathbf m}\bigr\}\xrightarrow{\partial_{\ell(0)}}&\cdots \longrightarrow \bigl\{\lambda(k)_{\mathbf m(k)}\bigr\}\xrightarrow{\partial_{\ell(k)}} \bigl\{\lambda(k+1)_{\mathbf m(k+1)}\bigr\}\xrightarrow{\partial_{\ell(k+1)}}\cdots. \end{align} \end{defn}
\section{Deligne-Simpson problem}\label{sec:DS}
In this section we give an answer for the existence and the construction of Fuchsian differential equations with given Riemann schemes and examine the irreducibility for generic spectral parameters.
\subsection{Fundamental lemmas} First we prepare two lemmas to construct Fuchsian differential operators with a given spectral type. \begin{defn}\label{def:Nnu} For $\mathbf m=\bigl(m_{j,\nu}\bigr)_{\substack{j=0,\dots,p\\ 1\le \nu\le n_j}} \in\mathcal P^{(n)}_{p+1}$, we put \index{00Nnu@$N_\nu(\mathbf m)$} \begin{align}
\begin{split}
N_\nu(\mathbf m)&:=(p-1)(\nu+1)+1\\
&\qquad-\#\{(j,i)\in\mathbb Z^2\,;\,
i\ge 0,\ 0\le j\le p,\ \widetilde m_{j,i}\ge n-\nu\},\label{eq:Nnu}
\end{split}\\
\widetilde m_{j,i}&:=\sum_{\nu=1}^{n_j}\max\bigl\{m_{j,\nu}-i,0\bigr\}.
\label{eq:tildem} \end{align} \end{defn} See the Young diagram in \eqref{eq:young} and its explanation for an interpretation of the number $\widetilde m_{j,i}$.
\begin{lem}\label{lem:ex1} We assume that $\mathbf m=\bigl(m_{j,\nu}\bigr)_{\substack{j=0,\dots,p\\ 1\le \nu\le n_j}} \in\mathcal P^{(n)}_{p+1}$ satisfies \begin{equation}\label{eq:NTP}
m_{j,1}\ge m_{j,2}\ge \cdots\ge m_{j,n_j}>0\text{ \ and \ }
n>m_{0,1}\ge m_{1,1}\ge \cdots\ge m_{p,1} \end{equation} and \begin{equation}\label{eq:NRed}
m_{0,1}+\cdots+m_{p,1}\le (p-1)n. \end{equation} Then \begin{equation}\label{eq:LDS}
N_\nu(\mathbf m)\ge 0
\qquad(\nu=2,3,\dots,n-1) \end{equation} if and only if $\mathbf m$ is not any one of \begin{equation}\label{eq:MAF} \begin{gathered} (k,k;k,k;k,k;k,k),\quad (k,k,k;k,k,k;k,k,k),\\ (2k,2k;k,k,k,k;k,k,k,k)\\ \text{and \ } (3k,3k;2k,2k,2k;k,k,k,k,k,k) \text{ \ with \ }k\ge 2. \end{gathered} \end{equation} \end{lem} \begin{proof} Put \begin{align*}
\phi_j(t)&:=\sum_{\nu=1}^{n_j}\max\{m_{j,\nu}-t,0\}\text{ \ and \ }
\bar \phi_j(t):=n\Bigl(1-\frac{t}{m_{j,1}}\Bigr)
\text{ \ for \ }j=0,\dots,p. \end{align*} Then $\phi_j(t)$ and $\bar\phi_j(t)$ are strictly decreasing continuous functions of $t\in[0,m_{j,1}]$ and \begin{align*} \phi_j(0)&=\bar \phi_j(0)=n,\\ \phi_j(m_{j,1})&=\bar \phi_j(m_{j,1})=0,\\ 2\phi_j(\tfrac{t_1+t_2}2)&\le \phi_j(t_1)+\phi_j(t_2) &(0\le t_1\le t_2\le m_{j,1}),\\ \phi_j'(t)&=-n_j\le -\tfrac{n}{m_{j,1}}=\bar \phi_j'(t) &(0<t<1). \end{align*} Hence we have \begin{align*}
\phi_j(t) &= \bar\phi_j(t) &&(0<t<m_{j,1},\ n=m_{j,1}n_j),\\
\phi_j(t) &< \bar\phi_j(t) &&(0<t<m_{j,1},\ n<m_{j,1}n_j) \end{align*} and for $\nu=2,\dots,n-1$ \begin{equation*}
\begin{split} \sum_{j=0}^p \#\{i\in\mathbb Z_{\ge 0}\,;\,
\phi_j(i)\ge n-\nu\}
&=\sum_{j=0}^p\bigl[\phi_j^{-1}(n-\nu)+1\bigr]\\
&\le \sum_{j=0}^p\bigl(\phi_j^{-1}(n-\nu)+1\bigr)\\
&\le \sum_{j=0}^p\bigl(\bar\phi_j^{-1}(n-\nu)+1\bigr)
=\sum_{j=0}^p \Bigl(\frac{\nu m_{j,1}}n+1\Bigr)\\
&\le (p-1)\nu+(p+1)=(p-1)(\nu+1)+2.
\end{split} \end{equation*} Here $[r]$ means the largest integer which is not larger than a real number $r$.
Suppose there exists $\nu$ with $2\le \nu \le n-1$ such that \eqref{eq:LDS} doesn't hold. Then the equality holds in the above each line, which means \begin{equation}\label{eq:LDSs}
\begin{aligned}
\phi_j^{-1}(n-\nu)&\in\mathbb Z&(j=0,\dots,p),\\
n&=m_{j,1}n_j&(j=0,\dots,p),\\
(p-1)n&=m_{0,1}+\cdots+m_{p,1}.
\end{aligned} \end{equation} Note that $n=m_{j,1}n_j$ implies $m_{j,1}=\cdots=m_{j,n_j}=\frac{n}{n_j}$ and $p-1=\frac{1}{n_0}+\cdots+\frac{1}{n_p}\le \frac{p+1}2$. Hence $p=3$ with $n_0=n_1=n_2=n_3=2$ or $p=2$ with $1=\frac{1}{n_0}+\frac{1}{n_1}+\frac{1}{n_2}$. If $p=2$, $\{n_0,n_1,n_2\}$ equals $\{3,3,3\}$ or $\{2,4,4\}$ or $\{2,3,6\}$. Thus we have \eqref{eq:MAF} with $k=1,2,\ldots$. Moreover since \[
\phi_j^{-1}(n-\nu)=\bar\phi_j^{-1}(n-\nu)
=\frac{\nu m_{j,1}}{n}=\frac{\nu}{n_j}\in\mathbb Z
\quad(j=0,\dots,p), \] $\nu$ is a common multiple of $n_0,\dots,n_p$ and thus $k\ge 2$. If $\nu$ is the least common multiple of $n_0,\dots,n_p$ and $k\ge 2$, then \eqref{eq:LDSs} is valid and the equality holds in the above each line and hence \eqref{eq:LDS} is not valid. \end{proof}
\begin{cor}[Kostov \cite{Ko3}]\label{cor:idx0} \index{tuple of partitions!basic!index of rigidity $=0$} Let $\mathbf m\in\mathcal P$ satisfying $\idx\mathbf m=0$ and $d_{max}(\mathbf m)\le 0$. Then $\mathbf m$ is isomorphic to one of the tuples in \eqref{eq:MAF} with $k=1,2,3,\ldots$. \end{cor} \begin{proof} Remark~\ref{rem:idxFuchs} assures that $d_{max}(\mathbf m)=0$ and $n=m_{j,1}n_j$. Then the proof of the final part of Lemma~\ref{lem:ex1} shows the corollary. \end{proof} \begin{lem}\label{lem:polydef} Let $c_0,\dots,c_p$ be $p+1$ distinct points in\/ $\mathbb C\cup\{\infty\}$. Let $n_0,n_1,\dots,n_p$ be non-negative integers and let $a_{j,\nu}$ be complex numbers for $j=0,\dots,p$ and $\nu=1,\dots,n_j$. Put $\tilde n:=n_0+\cdots+n_p$. Then there exists a unique polynomial $f(x)$ of degree $\tilde n-1$ such that \begin{equation}\label{eq:PCond}
\begin{split}
f(x)&=a_{j,1}+a_{j,2}(x-c_j)+\cdots
+a_{j,n_j}(x-c_j)^{n_j-1}\\
&\quad{}+o(|x-c_j|^{n_j-1})
\qquad(x\to c_j,\ c_j\ne\infty),\\
x^{1-\tilde n}f(x)&=a_{j,1} + a_{j,2}x^{-1}
+a_{j,n_j}x^{1-n_j}+o(|x|^{1-n_j})\\
&\qquad(x\to \infty,\ c_j=\infty).
\end{split} \end{equation} Moreover the coefficients of $f(x)$ are linear functions of the $\tilde n$ variables $a_{j,\nu}$. \end{lem} \begin{proof} We may assume $c_p=\infty$ with allowing $n_p=0$. Put $\tilde n_i=n_0+\cdots+n_{i-1}$ and $\tilde n_0=0$. For $k=0,\dots,\tilde n-1$ we define \[
f_k(x):=
\begin{cases}
(x-c_i)^{k-\tilde n_i}\prod_{\nu=0}^{i-1}(x-c_\nu)^{n_\nu}
&(\tilde n_i\le k<\tilde n_{i+1},\ 0\le i< p),\\
x^{k-\tilde n_p}\prod_{\nu=0}^{n_{p-1}}(x-c_\nu)^{n_\nu}
&(\tilde n_p\le k< \tilde n).
\end{cases} \] Since $\deg f_k(x)=k$, the polynomials $f_0(x),f_1(x),\dots,f_{\tilde n-1}(x)$ are linearly independent over $\mathbb C$. Put $f(x)=\sum_{k=0}^{\tilde n-1} u_k f_k(x)$ with $c_k\in\mathbb C$ and \[
v_k = \begin{cases}
a_{i,k-\tilde n_i+1}&(\tilde n_i\le k<\tilde n_{i+1},\ 0\le i< p),\\
a_{p,\tilde n-k}&(\tilde n_p\le k<\tilde n)
\end{cases} \] by \eqref{eq:PCond}. The correspondence which maps the column vectors $u:=(u_k)_{k=0,\ldots,\tilde n-1}\in\mathbb C^{\tilde n}$ to the column vectors $v:=(v_k)_{k=0,\ldots,\tilde n-1} \in\mathbb C^{\tilde n}$ is given by $v=Au$ with a square matrix $A$ of size $\tilde n$. Then $A$ is an upper triangular matrix of size $\tilde n$ with non-zero diagonal entries and therefore the lemma is clear. \end{proof} \subsection{Existence theorem} \begin{defn}[top term] \index{00Top@$\Top$} Let \[
P = a_n(x)\tfrac{d^n}{dx^n}+a_{n-1}(x)\tfrac{d^{n-1}}{dx^{n-1}}
+ \cdots+a_1(x)\tfrac{d}{dx} + a_0(x) \] be a differential operator with polynomial coefficients. Suppose $a_n\ne0$. If $a_n(x)$ is a polynomial of degree $k$ with respect to $x$, we define $\Top P := a_{n,k}x^k\partial^n$ with the coefficient $a_{n,k}$ of the term $x^k$ of $a_n(x)$. We put $\Top P=0$ when $P=0$. \end{defn} \index{00Nnu@$N_\nu(\mathbf m)$} \begin{thm}\label{thm:ExF} Suppose $\mathbf m\in\mathcal P^{(n)}_{p+1}$ satisfies \eqref{eq:NTP}. Retain the notation in Definition~\ref{def:Nnu}.
{\rm i) } We have $N_1(\mathbf m) = p-2$ and \begin{equation}\label{eq:sumN}
\sum_{\nu=1}^{n-1} N_\nu(\mathbf m) = \Pidx\mathbf m. \end{equation}
{\rm ii) } Suppose $p\ge 2$ and $N_\nu(\mathbf m)\ge 0$ for $\nu=2,\dots,n-1$. Put \begin{align}
q^0_\nu&:=\#\{i\,;\,\widetilde m_{0,i}\ge n-\nu,\ i\ge 0\},\\
I_{\mathbf m}&:=
\{(j,\nu)\in\mathbb Z^2\,;\,
q^0_\nu\le j< q^0_\nu+N_\nu(\mathbf m)\text{ and }
1\le \nu\le n-1\}. \end{align} Then there uniquely exists a Fuchsian differential operator $P$ of the normal form \eqref{eq:FNF} which has the Riemann scheme \eqref{eq:GRS} with $c_0=\infty$ under the Fuchs relation \eqref{eq:Fuchs} and satisfies \begin{equation}
\frac{1}{(\deg P-j-\nu)!}
\frac{d^{\deg P-j-\nu} a_{n-\nu-1}}{dx^{\deg P-j-\nu}}(0)
= g_{j,\nu}\qquad(\forall(j,\nu)\in I_{\mathbf m}). \end{equation} Here $\bigl(g_{j,\nu}\bigr)_{(j,\nu)\in I_{\mathbf m}}
\in\mathbb C^{\Pidx\mathbf m}$ is arbitrarily given. Moreover the coefficients of $P$ are polynomials of $x$, $\lambda_{j,\nu}$ and $g_{j,\nu}$ and satisfy \begin{equation}\label{eq:TopAcc}
x^{j+\nu}\Top
\Bigl(\frac{\partial P}{\partial g_{j,\nu}}\Bigr)\partial^{\nu+1}=\Top P\text{ \ and \ }
\frac{\partial^2 P}{\partial g_{j,\nu}^2}=0. \end{equation}
Fix the characteristic exponents $\lambda_{j,\nu}\in\mathbb C$ satisfying the Fuchs relation. Then all the Fuchsian differential operators of the normal form with the Riemann scheme \eqref{eq:GRS} are parametrized by $(g_{j,\nu})\in\mathbb C^{\Pidx \mathbf m}$. Hence the operators are unique if and only if\/ $\Pidx\mathbf m=0$. \end{thm} \begin{proof} i) Since $\widetilde m_{j,1}=n-n_j\le n-2$, $N_1(\mathbf m)=2(p-1)+1-(p+1)=p-2$ and \begin{align*}
\sum_{\nu=1}^{n-1}
&\#\{(j,i)\in\mathbb Z^2\,;\,i\ge 0,\ 0\le j\le p,\
\widetilde m_{j,i}\ge n-\nu\} \allowdisplaybreaks\\
&=\sum_{j=0}^p\Bigl(\sum_{\nu=0}^{n-1}\#\{i\in\mathbb Z_{\ge 0}\,;\,
\widetilde m_{j,i}\ge n-\nu\}-1\Bigr) \allowdisplaybreaks\\
&=\sum_{j=0}^p\Bigl(\sum_{i=0}^{m_{j,1}}\widetilde m_{j,i} -1\Bigr) =
\sum_{j=0}^p\Bigl(\sum_{i=0}^{m_{j,1}}\sum_{\nu=1}^{n_j}
\max\{m_{j,\nu}-i,0\}-1\Bigr) \allowdisplaybreaks\\
&=\sum_{j=0}^p
\Bigl(\sum_{\nu=1}^{n_j}\frac{m_{j,\nu}(m_{j,\nu}+1)}{2}-1\Bigr)\\
&=\frac12\Bigl(\sum_{j=0}^p\sum_{\nu=1}^{n_j}m_{j,\nu}^2 + (p+1)(n-2) \Bigr), \allowdisplaybreaks\\
\sum_{\nu=1}^{n-1}N_\nu(\mathbf m) &=
(p-1)\Bigl(\frac{n(n+1)}2-1\Bigr)+(n-1)
-\frac12\Bigl(\sum_{j=0}^p\sum_{\nu=1}^{n_j}m_{j,\nu}^2 +(p+1)(n-2)\Bigr) \allowdisplaybreaks\\
&= \frac12\Bigl((p-1)n^2+2-\sum_{j=0}^p\sum_{\nu=1}^{n_j}m_{j,\nu}^2\Bigr)
= \Pidx\mathbf m. \end{align*}
{\rm ii) } Put \begin{align*}
P &= \sum_{\ell=0}^{pn}x^{pn-\ell}p^P_{0,\ell}(\vartheta)\\
&= \sum_{\ell=0}^{pn}(x-c_j)^\ell p^P_{j,\ell}\bigl(
(x-c_j)\partial\bigr)\qquad(1\le j\le n),\\
h_{j,\ell}(t):\!&=
\begin{cases}
\prod_{\nu=1}^{n_0}\prod_{0\le i<m_{0,\nu}-\ell}
\bigl(t+\lambda_{0,\nu}+i\bigr)&(j=0),\\
\prod_{\nu=1}^{n_j}\prod_{0\le i<m_{j,\nu}-\ell}
\bigl(t-\lambda_{j,\nu}-i\bigr)
&(1\le j\le p),
\end{cases}\\
p^P_{j,\ell}(t)&=q^P_{j,\ell}(t)h_{j,\ell}(t)+r^P_{j,\ell}(t)
\qquad(\deg r^P_{j,\ell}(t) < \deg h_{j,\ell}(t)). \end{align*} Here $p^P_{j,\ell}(t)$, $q^P_{j,\ell}(t)$, $r^P_{j,\ell}(t)$ and $h_{j,\ell}(t)$ are polynomials of $t$ and \begin{equation}
\deg h_{j,\ell} = \sum_{\nu=1}^{n_j}\max\{m_{j,\nu}-\ell,0\}. \end{equation} The condition that $P$ of the form \eqref{eq:FNF} have the Riemann scheme \eqref{eq:GRS} if and only if $r^P_{j,\ell}=0$ for any $j$ and $\ell$. Note that $a_{n-k}(x)\in\mathbb C[x]$ should satisfy \begin{equation}
\deg a_{n-k}(x)\le pn-k\text{ \ and \ }
a_{n-k}^{(\nu)}(c_j)=0\quad(0\le\nu\le n-k-1,\ 1\le k\le n), \end{equation} which is equivalent to the condition that $P$ is of the Fuchsian type.
Put $P(k):=\bigl(\prod_{j=1}^p(x-c_j)^n\bigr)\frac{d^n}{dx^n}+a_{n-1}(x)
\frac{d^{n-1}}{dx^{n-1}}+\cdots
+a_{n-k}(x)\frac{d^{n-k}}{dx^{n-k}}$.
Assume that $a_{n-1}(x),\ldots,a_{n-k+1}(x)$ have already defined so that $\deg r_{j,\ell}^{P(k-1)}<n-k+1$ and we will define $a_{n-k}(x)$ so that $\deg r_{j,\ell}^{P(k)}<n-k$.
When $k=1$, we put \[
a_{n-1}(x) = -a_n(x)\sum_{j=1}^p(x-c_j)^{-1}
\sum_{\nu=1}^{n_j}\sum_{i=0}^{m_{j,\nu}-1}(\lambda_{j,\nu}+i) \] and then we have $\deg r_{j,\ell}^{P(1)}<n-1$ for $j=1,\dots,p$. Moreover we have $\deg r_{0,\ell}^{P(1)}<n-1$ because of the Fuchs relation.
Suppose $k\ge 2$ and put \[
a_{n-k}(x) =
\begin{cases}
\sum_{\ell\ge 0}c_{0,k,\ell}x^{pn-k-\ell},\\
\sum_{\ell\ge 0}c_{j,k,\ell}(x-c_j)^{n-k+\ell} &(j=1,\dots,p)
\end{cases} \] with $c_{i,j,\ell}\in\mathbb C$. Note that \begin{align*}
a_{n-k}(x)\partial^{n-k}
&=\sum_{\ell\ge 0}c_{0,k,\ell}x^{(p-1)n-\ell}\prod_{i=0}^{n-k-1}
(\vartheta-i)\\
&=\sum_{\ell\ge0}c_{j,k,\ell}(x-c_j)^{\ell}\prod_{i=0}^{n-k-1}
\bigl((x-c_j)\partial-i\bigr). \end{align*} Then $\deg r_{j,\ell}^{P(k)}<n-k$ if and only if $\deg h_{j,\ell}\le n-k$ or \begin{equation}\label{eq:cfc1}
c_{j,k,\ell} = -\frac1{(n-k)!}
\Bigl(\frac{d^{n-k}}{dt^{n-k}}r_{j,\ell}^{P(k-1)}(t)\Bigr)\Bigl|_{t=0}. \end{equation} Namely, we impose the condition \eqref{eq:cfc1} for all $(j,\ell)$ satisfying \[
{\widetilde m}_{j,\ell}=\sum_{\nu=1}^{n_j}\max\{m_{j,\nu}-\ell,0\} > n-k. \] The number of the pairs $(j,\ell)$ satisfying this condition equals $(p-1)k+1- N_{k-1}(\mathbf m)$. Together with the conditions $a_{n-k}^{(\nu)}(c_j)=0$ for $j=1,\dots,p$ and $\nu=0,\dots,n-k-1$, the total number of conditions imposing to the polynomial $a_{n-k}(x)$ of degree $pn-k$ equals \[
p(n-k)+(p-1)k+1- N_{k-1}(\mathbf m)=(pn-k+1)-N_{k-1}(\mathbf m). \] Hence Lemma~\ref{lem:polydef} shows that $a_{n-k}(x)$ is uniquely defined by giving $c_{0,k,\ell}$ arbitrarily for $q_{k-1}^0\le\ell< q_{k-1}^0+N_{k-1}(\mathbf m)$ because $q_{k-1}^0=\#\{\ell\ge0\,;\,\widetilde m_{0,\ell}>n-k\}$. Thus we have the theorem. \end{proof} \begin{rem} The numbers $N_\nu(\mathbf m)$ don't change if we replace a $(p+1)$-tuple $\mathbf m$ of partitions of $n$ by the $(p+2)$-tuple of partitions of $n$ defined by adding a trivial partition $n=n$ of $n$ to $\mathbf m$. \end{rem}
\index{00Nnu@$N_\nu(\mathbf m)$} \begin{exmp}\label{ex:Nj} We will examine the number $N_\nu(\mathbf m)$ in Theorem~\ref{thm:ExF}. In the case of the Simpson's list (cf.~\S\ref{sec:rigidEx}) we have the following. \begin{align*}
\mathbf m&= n-11,1^n,1^n\tag{$H_n$: hypergeometric family}\\
\widetilde{\mathbf m} &= n,n-2,n-3,\ldots1;n;n \allowdisplaybreaks\\
\mathbf m&=mm,mm-11,1^{2m}\tag{$EO_{2m}$: even family}\\
\widetilde{\mathbf m} &= 2m,2m-2,\dots,2;2m,2m-3,\dots,1;2m \allowdisplaybreaks\\
\mathbf m&= m+1m,mm1,1^{2m+1}\tag{$EO_{2m+1}$: odd family}\\
\widetilde{\mathbf m} &=2m+1,2m-1,\dots,1;2m+1,2m-2,\dots,2;2m+1 \allowdisplaybreaks\\
\mathbf m&= 42,222,1^6\tag{$X_6$: extra case}\\
\widetilde{\mathbf m} &=6,4,2,1;6,3;6 \end{align*} In these cases $p=2$ and we have $N_\nu(\mathbf m)=0$ for $\nu=1,2,\dots,n-1$ because \begin{equation}\label{eq:tildebm}
\begin{split}
\widetilde{\mathbf m}
:\!&=\{\widetilde m_{j,\nu}\,;\,\nu=0,\dots,m_{j,1}-1,\ j=0,\dots,p\bigr\}\\
&= \{n,n,n,n-2,n-3,n-4,\dots,2,1\}.
\end{split} \end{equation} \index{00Hn@$H_n$}\index{00EOn@$EO_n$}\index{00X6@$X_6$} See Proposition~\ref{prop:sred} ii) for the condition that $N_\nu(\mathbf m)\ge 0$ for $\nu=1,\dots,\ord\mathbf m-1$.
We give other examples:
\begin{tabular}{|c|c|l|l|} \hline $
\mathbf m
$ & $\Pidx$ & $
\widetilde{\mathbf m}
$
& $N_1,N_2,\dots,N_{\ord\mathbf m-1}$ \\ \hline\hline $221,221,221$ &0& $52,52,52$ & $0,1,-1,0$ \\ \hline $21,21,21,21\,(P_3)$ &0& $31,31,31,31$ & $1,-1$ \\ \hline $22,22,22$ &$-3$& $42,42,42$ & $0,-2,-1$ \\ \hline $11,11,11,11\,(\tilde D_4)$ &1& $2,2,2,2$& $1$ \\ \hline $111,111,111\,(\tilde E_6)$ &1& $3,3,3$ & $0,1$ \\ \hline $22,1111,1111\,(\tilde E_7)$ &1& $42,4,4$ & $0,0,1$ \\ \hline $33,222,111111\,(\tilde E_8)$ &1& $642,63,6$ & $0,0,0,0,1$ \\ \hline $21,21,21,111$ &1& $31,31,31,3$ & $1,0$ \\ \hline $222,222,222$ &1& $63,63,63$ & $0,1,-1,0,1$ \\ \hline $11,11,11,11,11$ &2& $2,2,2,2,2$ & $2$ \\ \hline $55,3331,22222$ & 2 & $10,8,6,4,2;10,6,3;10,5$ & $0,0,1,0,0,0,0,0,1$ \\ \hline $22,22,22,211$ &2& $42,42,42,41$ & $1,0,1$ \\ \hline $22,22,22,22,22$ &5& $42,42,42,42,42$ & $2,0,3$ \\ \hline $32111,3221,2222$ &8& $831,841,84$ & $0,1,2,1,1,2,1$ \\ \hline \end{tabular} \end{exmp} Note that if $\Pidx \mathbf m=0$, in particular, if $\mathbf m$ is rigid, then $\mathbf m$ doesn't satisfy \eqref{eq:NRed}. The tuple $222,222,222$ of partitions is the second case in \eqref{eq:MAF} with $k=2$.
\begin{rem}\label{rem:bas0} Note that \cite[Proposition~8.1]{O3} proves that there exit only finite basic tuples of partitions with a fixed index of rigidity.
\index{tuple of partitions!basic!index of rigidity $=0$} Those with index of rigidity $0$ are of only 4 types, which are $\tilde D_4$, $\tilde E_6$, $\tilde E_7$ and $\tilde E_8$ given in the above (cf.~Corollary~\ref{cor:idx0}, Kostov \cite{Ko3}). Namely, those are in the $S_\infty$-orbit of \begin{equation}\label{eq:basic0}
\{11,11,11,11\quad111,111,111\quad22,1111,1111\quad33,222,111111\} \end{equation} and the operator $P$ in Theorem~\ref{thm:ExF} with any one of this spectral type has one accessory parameter in its $0$-th order term.
The equation corresponding to $11,11,11,11$ is called Heun's equation (cf.~\cite{SW,WW}), which is given by the operator \index{Heun's equation} \begin{equation}\label{eq:Heun}
\begin{split}
P_{\alpha,\beta,\gamma,\delta,\lambda} &= x(x-1)(x-c)\partial^2
+\bigl(\gamma(x-1)(x-c)+\delta x(x-c)\\
&\quad{}+(\alpha+\beta+1-\gamma-\delta)x(x-1)\bigr)\partial
+\alpha\beta x-\lambda
\end{split} \end{equation} with the Riemann scheme \begin{equation}
\begin{Bmatrix}
x = 0 & 1 & c & \infty\\
0 & 0 & 0 & \alpha &;\,x\\
1-\gamma & 1-\delta & \gamma+\delta-\alpha-\beta & \beta&;\,\lambda
\end{Bmatrix}. \end{equation} Here $\lambda$ is an accessory parameter. Our operation cannot decrease the order of $P_{\alpha,\beta,\gamma,\delta,\lambda}$ but gives the following transformation. \begin{equation}
\begin{split}
&\Ad(\partial^{1-\alpha})P_{\alpha,\beta,\gamma,\delta,\lambda}
=P_{\alpha',\beta',\gamma',\delta',\lambda'},\\
&\begin{cases}
\alpha'=2-\alpha,\ \beta'=\beta-\alpha+1,\ \gamma'=\gamma-\alpha+1,
\ \delta'=\delta-\alpha+1,\\
\lambda'=\lambda+(1-\alpha)\bigl(\beta-\delta+1+(\gamma+\delta-\alpha)c\bigr).
\end{cases}
\end{split} \end{equation} \end{rem} \index{tuple of partitions!basic!index of rigidity $=-2$} \begin{prop}{\rm(\cite[Proposition~8.4]{O3})}.\label{prop:bas2} The basic tuples of partitions with index of rigidity $-2$ are in the $S_\infty$-orbit of the set of the $13$ tuples
\begin{align*}
\bigl\{&
11,11,11,11,11
\ \ 21,21,111,111
\ \ 31,22,22,1111
\ \ 22,22,22,211
\\&
211,1111,1111
\ \ \ 221,221,11111
\ \ 32,11111,11111
\ \ 222,222,2211
\\&
33,2211,111111
\ \ 44,2222,22211
\ \ 44,332,11111111
\ \ 55,3331,22222
\\&
66,444,2222211
\bigr\}. \end{align*} \end{prop} \begin{proof} Here we give the proof in \cite{O3}.
Assume that $\mathbf m\in\mathcal P_{p+1}$ is basic and monotone and $\idx\mathbf m=-2$. Note that \eqref{eq:idxd} shows \[
0\le \sum_{j=0}^p\sum_{\nu=2}^{n_j}
(m_{j,1}-m_{j,\nu})\cdot m_{j,\nu}\le -\idx\mathbf m=2. \] Hence \eqref{eq:idxd} implies $\sum_{j=0}^p\sum_{\nu=2}^{n_j} (m_{j,1}-m_{j,\nu})m_{j,\nu}=0$ or $2$ and we have only to examine the following 5 possibilities.
(A) \ $m_{0,1}\cdots m_{0,n_0} = 2\cdots211$
and $m_{j,1}=m_{j,n_j}$ for $1\le j\le p$.
(B) \ $m_{0,1}\cdots m_{0,n_0} = 3\cdots31$
and $m_{j,1}=m_{j,n_j}$ for $1\le j\le p$.
(C) \ $m_{0,1}\cdots m_{0,n_0} = 3\cdots32$
and $m_{j,1}=m_{j,n_j}$ for $1\le j\le p$.
(D) \ $m_{i,1}\cdots m_{i,n_0} = 2\cdots21$
and $m_{j,1}=m_{j,n_j}$
for $0\le i\le 1<j\le p$.
(E) \ $m_{j,1}=m_{j,n_j}$ for $0\le j\le p$ and $\ord\mathbf m=2$.
Case (A). \ If $2\cdots211$ is replaced by $2\cdots22$, $\mathbf m$ is transformed into $\mathbf m'$ with $\idx\mathbf m'=0$. If $\mathbf m'$ is indivisible, $\mathbf m'$ is basic and $\idx\mathbf m'=0$ and therefore $\mathbf m$ is $211,1^4,1^4$ or $33,2211,1^6$. If $\mathbf m'$ is not indivisible, $\frac12\mathbf m'$ is basic and $\idx\frac12\mathbf m'=0$ and hence $\mathbf m$ is one of the tuples in \[\{211,22,22,22\ \ 2211,222,222\ \
22211,2222,44\ \ 2222211,444,66\}.\]
Put $m=n_0-1$ and examine the identity \[ \sum_{j=0}^p\frac{m_{j,1}}{\ord\mathbf m} =p-1 + (\ord\mathbf m)^{-2}\Bigl(\idx\mathbf m
+ \sum_{j=0}^p\sum_{\nu=1}^{n_j}(m_{j,1}-m_{j,\nu})m_{j,\nu}\Bigr) \]
Case (B). \ Note that $\ord\mathbf m=3m+1$ and therefore $\frac3{3m+1}+\frac1{n_1}+\cdots+\frac1{n_p} =p-1$. Since $n_j\ge 2$, we have $\frac12p -1\le\frac{3}{3m+1}<1$ and $p\le 3$.
If $p=3$, we have $m=1$, $\ord\mathbf m=4$, $\frac1{n_1}+\frac1{n_2}+\frac1{n_3}=\frac54$, $\{n_1,n_2,n_3\}=\{2,2,4\}$ and $\mathbf m=31,22,22,1111$.
Assume $p=2$. Then $\frac1{n_1}+\frac1{n_2}=1-\frac{3}{3m+1}$. If $\min\{n_1,n_2\}\ge 3$, $\frac1{n_1}+\frac1{n_2}\le\frac23$ and $m\le2$. If $\min\{n_1,n_2\}=2$, $\max\{n_1,n_2\}\ge 3$ and $\frac3{3m+1}\ge\frac16$ and $m\le 5$. Note that $\frac1{n_1}+\frac1{n_2}=\frac{13}{16}$, $\frac{10}{13}$, $\frac{7}{10}$, $\frac{4}{7}$ and $\frac{1}{4}$ according to $m=5$, $4$, $3$, $2$ and $1$, respectively. Hence we have $m=3$, $\{n_1,n_2\}=\{2,5\}$ and $\mathbf m=3331,55,22222$.
Case (C). \ We have $\frac{3}{3m+2}+\frac1{n_1}+\cdots+\frac1{n_p}=p-1$. Since $n_j\ge 2$, $\frac12p-1\le \frac{3}{3m+2}<1$ and $p\le 3$. If $p=3$, then $m=1$, $\ord\mathbf m=5$ and $\frac1{n_1}+\frac1{n_2}+\frac1{n_3}=\frac75$, which never occurs.
Thus we have $p=2$, $\frac1{n_1}+\frac1{n_2}=1-\frac{3}{3m+2}$ and hence $m\le 5$ as in Case (B). Then $\frac1{n_1}+\frac1{n_2}=\frac{14}{17}$, $\frac{11}{14}$, $\frac{8}{11}$, $\frac{5}{8}$ and $\frac{2}{5}$ according to $m=5$, $4$, $3$, $2$ and $1$, respectively. Hence we have $m=1$ and $n_1=n_2=5$ and $\mathbf m=32,11111,11111$ or $m=2$ and $n_1=2$ and $n_2=8$ and $\mathbf m=332,44,11111111$.
Case (D). \ We have $\frac2{2m+1}+\frac2{2m+1}+\frac1{n_2}+\cdots+\frac1{n_p}=p-1$. Since $n_j\ge3$ for $j\ge2$, we have $p-1\le \frac32\frac{4}{2m+1}=\frac{6}{2m+1}$ and $m\le2$. If $m=1$, then $p=3$ and $\frac1{n_2}+\frac1{n_3}=2-\frac43=\frac23$ and we have $\mathbf m=21,21,111,111$. If $m=2$, then $p=2$, $\frac1{n_2}=1-\frac{4}{5}$ and $\mathbf m=221,221,11111$.
Case (E). \ Since $m_{j,1}=1$ and \eqref{eq:idxd} means $-2=\sum_{j=0}^p 2m_{j,1}-4(p-1)$, we have $p=4$ and $\mathbf m=11,11,11,11,11$. \end{proof}
\subsection{Divisible spectral types} \begin{prop}\label{prop:dividx0}\index{tuple of partitions!divisible} Let $\mathbf m$ be any one of the partition of type $\tilde D_4$, $\tilde E_6$, $\tilde E_7$ or $\tilde E_8$ in Example~\ref{ex:Nj} and put $n=\ord\mathbf m$. Then $k\mathbf m$ is realizable but it isn't irreducibly realizable for $k=2,3,\ldots$. Moreover we have the operator $P$ of order $k\ord\mathbf m$ satisfying the properties in\/ {\rm Theorem~\ref{thm:ExF} ii)} for the tuple $k\mathbf m$. \end{prop} \begin{proof} Let $P(k,c)$ be the operator of the normal form with the Riemann scheme \[
\begin{Bmatrix}
x=c_0=\infty & x=c_j\ (j=1,\dots,p)\\
[\lambda_{0,1}-k(p-1)n+km_{0,1}]_{(m_{0,1})}
& [\lambda_{j,1}+km_{j,1}]_{(m_{j,1})}\\
\vdots&\vdots\\
[\lambda_{0,n_1}-k(p-1)n+km_{0,1}]_{(m_{0,n_1})}
& [\lambda_{j,n_j}+km_{j,n_j}]_{(m_{j,n_j})}
\end{Bmatrix} \] of type $\mathbf m$. Here $\mathbf m=\bigl(m_{j,\nu}\bigr)_{\substack{j=0,\dots,p\\ \nu=1,\dots,n_j}}$, $n=\ord\mathbf m$ and $c$ is the accessory parameter contained in the coefficient of the 0-th order term of $P(k,c)$. Since $\Pidx\mathbf m=0$ means \begin{align*}
\sum_{j=0}^{p}\sum_{\nu=1}^{n_j} m_{j,\nu}^2
=(p-1)n^2=\sum_{\nu=0}^{n_0}(p-1)nm_{0,\nu}, \end{align*} the Fuchs relation \eqref{eq:Fuchs} is valid for any $k$. Then it follows from Lemma~\ref{lem:block} that the Riemann scheme of the operator $P_k(c_1,\dots,c_k)=P(k-1,c_k)P(k-2,c_{k-1})\cdots P(0,c_1)$ equals \begin{equation}\label{eq:MAFGRS}
\begin{Bmatrix}
x=c_0=\infty & x=c_j\ (j=1,\dots,p)\\
[\lambda_{0,1}]_{(km_{0,1})}
& [\lambda_{j,1}]_{(km_{j,1})}\\
\vdots&\vdots\\
[\lambda_{0,n_1}]_{(km_{0,n_1})}
& [\lambda_{j,n_j}]_{(km_{j,n_j})}
\end{Bmatrix} \end{equation} and it contain an independent accessory parameters in the coefficient of $\nu n$-th order term of $P_k(c_1,\dots,c_k)$ for $\nu=0,\dots,k-1$ because for the proof of this statement we may assume $\lambda_{j,\nu}$ are generic under the Fuchs relation.
Note that \[
N_\nu(k\mathbf m) = \begin{cases}
1 & (\nu\equiv n-1 \mod n),\\
-1 & (\nu\equiv 0 \mod n),\\
0 & (\nu\not\equiv 0,\ n-1\mod n)
\end{cases} \] for $\nu=1,\dots,kn-1$ because \[
\widetilde{k\mathbf m}=
\begin{cases}
\{2i,2i,2i,2i\,;\,i=1,2,\ldots,k\}\quad\text{if }\mathbf m\text{ is of type }
\tilde D_4,\\
\{ni,ni,ni,ni-2,ni-3,\ldots,ni-n+1\,;\,i=1,2,\ldots,k\}\\
\qquad\qquad\qquad\qquad\qquad\qquad\quad\
\text{if }\mathbf m \text{ is of type }
\tilde E_6,\,\tilde E_7\text{ or }\tilde E_8
\end{cases} \] under the notation \eqref{eq:tildem} and \eqref{eq:tildebm}. Then the operator $P_k(c_1,\dots,c_k)$ shows that when we inductively determine the coefficients of the operator with the Riemann scheme \eqref{eq:MAFGRS} as in the proof of Theorem~\ref{thm:ExF}, we have a new accessory parameter in the coefficient of the $\bigl((k-j)n\bigr)$-th order term and then the conditions for the coefficients of the $\bigl((k-j)n-1\bigr)$-th order term are overdetermined but they are automatically compatible for $j=1,\dots,k-1$.
Thus we can conclude that the operators of the normal form with the Riemann scheme \eqref{eq:MAFGRS} are $P_k(c_1,\dots,c_k)$, which are always reducible. \end{proof}
\begin{prop}\label{prop:irred} Let $k$ be a positive integer and let\/ $\mathbf m$ be an indivisible $(p+1)$-tuple of partitions of $n$. Suppose $k\mathbf m$ is realizable and\/ $\idx\mathbf m<0$. Then any Fuchsian differential equation with the Riemann scheme \eqref{eq:MAFGRS} is always irreducible if $\lambda_{j,\nu}$ is generic under the Fuchs relation \begin{equation}\label{eq:FCdiv}
\sum_{j=0}^p\sum_{\nu=1}^{n_j} m_{j,\nu}\lambda_{j,\nu}
= \ord \mathbf m - k\frac{\idx\mathbf m}2. \end{equation} \end{prop} \begin{proof} The above Fuchs relation follows from \eqref{eq:Fuchidx} with the identities $\ord k\mathbf m=k\ord\mathbf m$ and $\idx k\mathbf m=k^2\idx\mathbf m$.
Suppose $Pu=0$ is reducible. Then Remark~\ref{rem:generic} ii) says that there exist $\mathbf m'$, $\mathbf m''\in\mathcal P$ such that $k\mathbf m=\mathbf m'+\mathbf m''$ and $0<\ord\mathbf m'<k\ord\mathbf m$ and
$|\{\lambda_{\mathbf m'}\}|\in\{0,-1,-2,\ldots\}$. Suppose $\lambda_{j,\nu}$ are generic under \eqref{eq:FCdiv}. Then the condition $|\{\lambda_{\mathbf m'}\}|\in\mathbb Z$ implies $\mathbf m'=\ell\mathbf m$ with a positive integer satisfying $\ell<k$ and \begin{align*}
|\{\lambda_{\ell\mathbf m}\}|
&=\sum_{j=0}^p\sum_{\nu=1}^{n_j}\ell m_{j,\nu}\lambda_{j,\nu}-\ord\ell\mathbf m
+\ell^2\idx\mathbf m\\
&= \ell\Bigl(\ord\mathbf m - k\frac{\idx\mathbf m}2\Bigr)
- \ell \ord\mathbf m +\ell^2\idx\mathbf m\\
&= \ell(\ell-k)\idx\mathbf m>0. \end{align*}
Hence $|\{\lambda_{\mathbf m'}\}|>0$. \end{proof}
\subsection{Universal model}\index{universal model} \index{Fuchsian differential equation/operator!universal operator} Now we have a main result in \S\ref{sec:DS} which assures the existence of Fuchsian differential operators with given spectral types.
\begin{thm}\label{thm:univmodel} Fix a tuple\/ $\mathbf m=\bigl(m_{j,\nu}\bigr) _{\substack{0\le j\le p\\ 1\le \nu\le n_j}}\in\mathcal P^{(n)}_{p+1}$.
{\rm i) } Under the notation in Definitions~\ref{def:tuples}, \ref{defn:real} and \ref{def:pell}, the tuple\/ $\mathbf m$ is realizable if and only if there exists a non-negative integer\/ $K$ such that\/ $\partial_{max}^i \mathbf m$ are well-defined for $i=1,\dots,K$ and \begin{equation}\label{eq:decord}
\begin{split}
&\ord\mathbf m>\ord\partial_{max}\mathbf m>\ord\partial_{max}^2\mathbf m>\cdots
>\ord\partial_{max}^K\mathbf m,\\
&d_{max}(\partial_{max}^K\mathbf m)=2\ord \partial_{max}^K\mathbf m\text{ \ or \ }
d_{max}(\partial_{max}^K\mathbf m)\le0.
\end{split} \end{equation}
{\rm ii) } Fix complex numbers $\lambda_{j,\nu}$. If there exists an irreducible Fuchsian operator with the Riemann scheme \eqref{eq:GRS} such that it is locally non-degenerate {\rm (cf.~Definition~\ref{def:locnondeg}),} then\/ $\mathbf m$ is irreducibly realizable.
Here we note that if $P$ is irreducible and\/ $\mathbf m$ is rigid, $P$ is locally non-degenerate {\rm (cf.~Definition~\ref{def:locnondeg}).}
Hereafter in this theorem we assume\/ $\mathbf m$ is realizable.
{\rm iii) } $\mathbf m$ is irreducibly realizable if and only if\/ $\mathbf m$ is indivisible or $\idx\mathbf m<0$.
{\rm iv) } There exists a \textsl{universal model} $P_{\mathbf m}u=0$ associated with\/ $\mathbf m$ which has the following property.
Namely, $P_{\mathbf m}$ is the Fuchsian differential operator of the form \begin{equation}\label{eq:uinvPm} \begin{split}
P_{\mathbf m}&=\Bigl(\prod_{j=1}^p(x-c_j)^n\Bigr)\frac{d^n}{dx^n}
+ a_{n-1}(x)\frac{d^{n-1}}{dx^{n-1}}+\cdots+a_1(x)\frac{d}{dx}+a_0(x),\\
&a_j(x)\in\mathbb C[\lambda_{j,\nu},g_1,\dots,g_N]
\end{split} \end{equation} such that $P_{\mathbf m}$ has regular singularities at $p+1$ fixed points $x=c_0=\infty, c_1,\dots,c_p$ and the Riemann scheme of $P_{\mathbf m}$ equals \eqref{eq:GRS} for any $g_i\in\mathbb C$ and $\lambda_{j,\nu}\in\mathbb C$ under the Fuchs relation \eqref{eq:Fuchs}. Moreover the coefficients $a_j(x)$ are polynomials of $x$, $\lambda_{j,\nu}$ and $g_i$ with the degree at most $(p-1)n+j$ for $j=0,\dots,n$, respectively. Here $g_i$ are called accessory parameters and we call $P_{\mathbf m}$ the \textsl{universal operator} of type\/ $\mathbf m$.
The non-negative integer $N$ will be denoted by\/ $\Ridx\mathbf m$ and given by \begin{equation}
N=\Ridx\mathbf m:=\begin{cases}
0&(\idx\mathbf m > 0),\\
\gcd \mathbf m&(\idx\mathbf m=0),\\
\Pidx\mathbf m&(\idx\mathbf m<0).
\end{cases} \end{equation}\index{00Ridx@$\Ridx$} Put\/ $\overline{\mathbf m}=\bigl({\overline m}_{j,\nu}\bigr)_ {\substack{0\le j\le p\\ 1\le \nu\le n_j}}:=\partial_{max}^K\mathbf m$ with the non-negative integer $K$ given in\/ {\rm i)}.
When\/ $\idx\mathbf m\le 0$, we define \begin{align*}
q^0_\ell &:= \#\{i\,;\,\sum_{\nu=1}^{\bar n_0}
\max\{{\overline m}_{0,\nu}-i,0\}\ge\ord\overline{\mathbf m}-\ell,\ i\ge 0\},\\
I_{\mathbf m}&:=
\{(j,\nu)\in\mathbb Z^2\,;\,
q^0_\nu\le j\le q^0_\nu+N_\nu-1,\
1\le \nu\le\ord\overline{\mathbf m}-1\}. \end{align*} When\/ $\idx\mathbf m> 0$, we put $I_{\mathbf m}=\emptyset$.
Then\/ $\#I_{\mathbf m}=\Ridx\mathbf m$ and we can define\/ $I_i$ such that $I_{\mathbf m}=\{I_i\,;\,i=1,\dots,N\}$ and $g_i$ satisfy \eqref{eq:TopAcc} by putting $g_{I_i}=g_i$ for $i=1,\dots,N$.
{\rm v) } Retain the notation in Definition~\ref{def:redGRS}. If $\lambda_{j,\nu}\in\mathbb C$ satisfy \begin{equation}\label{eq:inUniv}
\begin{cases}
\sum_{j=0}^p
\lambda(k)_{j,\ell(k)_j+\delta_{j,j_o}(\nu_o-\ell(k)_j)}\\
\
\notin\{0,-1,-2,-3,\ldots,m(k)_{j_o,\ell(k)_{j_o}}-m(k)_{j_o,\nu_o}-d(k)+2\}\\
\quad\text{for any }k=0,\dots,K-1\text{ and }(j_0,\nu_o)\text{ satisfying}\\
\quad m(k)_{j_o,\nu_o}\ge m(k)_{j_o,\ell(k)_{j_o}} - d(k)+2,
\end{cases} \end{equation} any Fuchsian differential operator $P$ of the normal form which has the Riemann scheme \eqref{eq:GRS} belongs to $P_{\mathbf m}$ with a suitable $(g_1,\dots,g_N)\in\mathbb C^N$. \begin{align} &\begin{cases} \text{If\/ $\mathbf m$ is a scalar multiple of a fundamental tuple or simply reducible, }\\ \text{\eqref{eq:inUniv} is always valid for any $\lambda_{j,\nu}$.} \end{cases}\allowdisplaybreaks\label{eq:CSR}\\ &\begin{cases} \text{Fix $\lambda_{j,\nu}\in\mathbb C$. Suppose there is an irreducible Fuchsian differential}\\ \text{operator with the Riemann scheme \eqref{eq:GRS} such that the operator is}\\ \text{locally non-degenerate or $K\le 1$, then \eqref{eq:inUniv} is valid.} \end{cases}\label{eq:irinuniv} \end{align}
Suppose\/ $\mathbf m$ is monotone. Under the notation in {\rm\S\ref{sec:KM}}, the condition \eqref{eq:inUniv} is equivalent to \begin{equation}\label{eq:KinUniv}
\begin{split}
&(\Lambda(\lambda)|\alpha)+1\notin
\{0,-1,\ldots,2-(\alpha|\alpha_{\mathbf m})\}\\
&\qquad
\text{ for any }\alpha\in\Delta(\mathbf m)
\text{ satisfying }(\alpha|\alpha_{\mathbf m})>1.
\end{split} \end{equation} \end{thm}
Example~\ref{ex:outuniv} gives a Fuchsian differential operator with the rigid spectral type $21,21,21,21$ which doesn't belong to the corresponding universal operator.
The fundamental tuple and the simply reducible tuple are defined as follows. \begin{defn}\label{def:fund} {\rm i) }(fundamental tuple) \index{tuple of partitions!fundamental}\index{00fm@$f\mathbf m$} An irreducibly realizable tuple $\mathbf m\in\mathcal P$ is called \textsl{fundamental} if $\ord\mathbf m=1$ or $d_{\max}(\mathbf m)\le 0$.
For an irreducibly realizable tuple $\mathbf m\in\mathcal P$, there exists a non-negative integer $K$ such that $\partial^K_{max}\mathbf m$ is fundamental and satisfies \eqref{eq:decord}. Then we call $\partial^K_{max}\mathbf m$ is a fundamental tuple corresponding to $\mathbf m$ and define $f\mathbf m:=\partial^K_{max}\mathbf m$.
{\rm ii) }(simply reducible tuple) \index{tuple of partitions!simply reducible} A tuple $\mathbf m$ is \textsl{simply reducible} if there exists a positive integer $K$ satisfying \eqref{eq:decord} and $\ord\partial_{max}^K\mathbf m=\ord\mathbf m-K$. \end{defn} \begin{proof}[Proof of Theorem~\ref{thm:univmodel}] {\rm i)} We have proved that $\mathbf m$ is realizable if $d_{max}(\mathbf m)\le 0$. Note that the condition $d_{max}(\mathbf m)=2\ord\mathbf m$ is equivalent to the fact that $s\mathbf m$ is trivial. Hence Theorem~\ref{thm:realizable} proves the claim.
{\rm iv) } Now we use the notation in Definition~\ref{def:redGRS}. The existence of the universal operator is clear if $s\mathbf m$ is trivial. If $d_{max}(\mathbf m)\le 0$, Theorem~\ref{thm:ExF} and Proposition~\ref{prop:dividx0} with Corollary~\ref{cor:idx0} assure the existence of the universal operator $P_{\mathbf m}$ claimed in iii). Hence iii) is valid for the tuple $\mathbf m(K)$ and we have a universal operator $P_K$ with the Riemann scheme $\{\lambda(K)_{\mathbf m(K)}\}$.
The universal operator $P_k$ with the Riemann scheme $\{\lambda(k)_{\mathbf m(k)}\}$ are inductively obtained by applying $\partial_{\ell(k)}$ to the universal operator $P_{k+1}$ with the Riemann scheme $\{\lambda(k+1)_{\mathbf m(k+1)}\}$ for $k=K-1,K-2,\dots,0$. Since the claims in iii) such as \eqref{eq:TopAcc} are kept by the operation $\partial_{\ell(k)}$, we have iv).
{\rm iii) } Note that $\mathbf m$ is irreducibly realizable if $\mathbf m$ is indivisible (cf.~Remark~\ref{rem:generic} ii)). Hence suppose $\mathbf m$ is not indivisible. Put $k=\gcd\mathbf m$ and $\mathbf m=k\mathbf m'$. Then $\idx\mathbf m =k^2\idx\mathbf m'$.
If $\idx\mathbf m>0$, then $\idx\mathbf m>2$ and the inequality \eqref{eq:rigididx} in Lemma~\ref {lem:irrred} implies that $\mathbf m$ is not irreducibly realizable. If $\idx\mathbf m<0$, Proposition~\ref{prop:irred} assures that $\mathbf m$ is irreducibly realizable.
Suppose $\idx\mathbf m=0$. Then the universal operator $P_{\mathbf m}$ has $k$ accessory parameters. Using the argument in the first part of the proof of Proposition~\ref{prop:dividx0}, we can construct a Fuchsian differential operator $\tilde P_{\mathbf m}$ with the Riemann scheme $\bigl\{\lambda_{\mathbf m}\bigr\}$. Since $\tilde P_{\mathbf m}$ is a product of $k$ copies of the universal operator $P_{\overline{\mathbf m}}$ and it has $k$ accessory parameters, the operator $P_{\mathbf m}$ coincides with the reducible operator $\tilde P_{\mathbf m}$ and hence $\mathbf m$ is not irreducibly realizable.
{\rm v) } Fix $\lambda_{j,\nu}\in\mathbb C$. Let $P$ be a Fuchsian differential operator with the Riemann scheme $\{\lambda_{\mathbf m}\}$. Suppose $P$ is of the normal form.
Theorem~\ref{thm:ExF} and Proposition~\ref{prop:dividx0} assure that $P$ belongs to $P_{\mathbf m}$ if $K=0$.
Theorem~\ref{thm:GRSmid} proves that if $\partial_{max}^kP$ has the Riemann scheme $\{\lambda(k)_{\mathbf m(k)}\}$ and \eqref{eq:inUniv} is valid, then $\partial_{max}^{k+1}P=\partial_{\ell(k)}\partial_{max}^kP$ is well-defined and has the Riemann scheme $\{\lambda(k+1)_{\mathbf m(k+1)}\}$ for $k=0,\dots,K-1$ and hence it follows from \eqref{eq:pellP2} that $P$ belongs to the universal operator $P_{\mathbf m}$ because $\partial_{max}^K P$ belongs to the universal operator $P_{\mathbf m(K)}$.
If $\mathbf m$ is simply reducible, $d(k)=1$ and therefore \eqref{eq:inUniv} is valid because $m(k)_{j,\nu}\le m(k)_{j,\ell(k)_\nu}<m(k)_{j,\ell(k)_\nu}-d(k)+2$ for $j=0,\dots,p$ and $\nu=1,\dots,n_j$ and $k=0,\dots,K-1$.
The equivalence of the conditions \eqref{eq:inUniv} and \eqref{eq:KinUniv} follows from the argument in \S\ref{sec:KM}, Proposition~\ref{prop:wm} and Theorem~\ref{thm:irrKac}.
{\rm ii) } Suppose there exists an irreducible operator $P$ with the Riemann scheme \eqref{eq:GRS}. Let $\mathbf M=(M_0,\dots,M_p)$ be the tuple of monodromy generators of the equation $Pu=0$ and put $\mathbf M(0)=\mathbf M$. Let $\mathbf M(k+1)$ be the tuple of matrices applying the operations in \S\ref{sec:MM} to $\mathbf M(k)$ corresponding to the operations $\partial_{\ell(k)}$ for $k=0,1,2,\ldots$.
Comparing the operations on $\mathbf M(k)$ and $\partial_{\ell(k)}$, we can conclude that there exists a non-negative integer $K$ satisfying the claim in i). In fact Theorem~\ref{thm:Mmid} proves that $\mathbf M(k)$ are irreducible, which assures that the conditions \eqref{eq:redC2} and \eqref{eq:redC3} corresponding to the operations $\partial_{\ell(k)}$ are always valid (cf.~Corollary~\ref{cor:irred}). Therefore $\mathbf m$ is realizable and moreover we can conclude that \eqref{eq:irinuniv} implies \eqref{eq:inUniv}. If $\idx\mathbf m$ is divisible and $\idx\mathbf m=0$, then $P_{\mathbf m}$ is reducible for any fixed parameters $\lambda_{j,\nu}$ and $g_i$. Hence $\mathbf m$ is irreducibly realizable. \end{proof} \begin{rem}\label{rem:inuniv} {\rm i) } The uniqueness of the universal operator in Theorem~\ref{thm:univmodel} is obvious. But it is not valid in the case of systems of Schlesinger canonical form (cf.~Example~\ref{ex:univSch}).
{\rm ii)} The assumption that $Pu=0$ is locally non-degenerate seems to be not necessary in Theorem~\ref{thm:univmodel} ii) and \eqref{eq:irinuniv}. When $K=1$, this is clear from the proof of the theorem. For example, the rigid irreducible operator with the spectral type $31,31,31,31,31$ belongs to the universal operator of type $211,31,31,31,31$. \end{rem}
\subsection{Simply reducible spectral type}\label{sec:simpred} \index{tuple of partitions!simply reducible} In this subsection we characterize the tuples of the simply reducible spectral type. \index{00Nnu@$N_\nu(\mathbf m)$} \begin{prop}\label{prop:sred} {\rm i)} A realizable tuple\/ $\mathbf m\in\mathcal P^{(n)}$ satisfying $m_{0,\nu}=1$ for $\nu=1,\ldots,n$ is simply reducible if\/ $\mathbf m$ is not fundamental.
{\rm ii)} The simply reducible rigid tuple corresponds to the tuple in Simpson's list {\rm(cf.~\S\ref{sec:rigidEx})} or it is isomorphic to\/ $21111,222,33$. \index{tuple of partitions!rigid!21111,222,33}
{\rm iii)} Suppose\/ $\mathbf m\in\mathcal P_{p+1}$ is not fundamental. Then\/ $\mathbf m$ satisfies the condition $N_\nu(\mathbf m)\ge 0$ for $\nu=2,\dots,\ord\mathbf m-1$ in Definition~\ref{def:Nnu} if and only if\/ $\mathbf m$ is realizable and simply reducible.
{\rm iv)} Let $\mathbf m\in\mathcal P_{p+1}$ be a realizable monotone tuple. Suppose\/ $\mathbf m$ is not fundamental. Then under the notation in {\rm\S\ref{sec:KM}}, $\mathbf m$ is simply reducible if and only if \begin{equation}
(\alpha|\alpha_{\mathbf m})=1\quad(\forall\alpha\in\Delta(\mathbf m)), \end{equation} namely\/ $[\Delta(\mathbf m)]=1^{\#\Delta(\mathbf m)}$ {\rm(cf.~Remark~\ref{rem:length}~ii)).} \end{prop} \begin{proof} {\rm i)} The claim is obvious from the definition.
{\rm ii)} Let $\mathbf m'$ be a simply reducible rigid tuple. We have only to prove that $\mathbf m=\partial_{max}\mathbf m'$ is in the Simpson's list or $21111,222,33$ and $\ord\mathbf m'=\ord\mathbf m+1$ and $d_{max}(\mathbf m)=1$, then $\mathbf m'$ is in Simpson's list or $21111,222,33$. The condition $\ord\mathbf m'=\ord\mathbf m+1$ implies $\mathbf m\in\mathcal P_3$. We may assume $\mathbf m$ is monotone and $\mathbf m'=\partial_{\ell_0,\ell_1,\ell_2}\mathbf m$. The condition $\ord\mathbf m'=\ord\mathbf m+1$ also implies \[
(m_{0,1}-m_{0,\ell_0})+(m_{1,1}-m_{1,\ell_0})+(m_{2,1}-m_{2,\ell_0})=2. \] Since $\partial_{max}\mathbf m'=\mathbf m$, we have $m_{j,\ell_j}\ge m_{j,1}-1$ for $j=0,1,2$. Hence there exists an integer $k$ with $0\le k\le 2$ such that $m_{j,\ell_j} = m_{j,1}-1+\delta_{j,k}$ for $j=0,1,2$. Then the following claims are easy, which assures the proposition.
If $\mathbf m=11,11,11$, $\mathbf m'$ is isomorphic to $1^3,1^3,21$.
If $\mathbf m=1^3,1^3,21$, $\mathbf m'$ is isomorphic to $1^4,1^4,31$ or $1^4,211,22$.
If $\mathbf m=1^n,1^n,n-11$ with $n\ge 4$, $\mathbf m'=1^{n+1},1^{n+1},n1$.
If $\mathbf m=1^{2n},nn-11,nn$ with $n\ge 2$, $\mathbf m'=1^{2n+1},nn1,n+1n$.
If $\mathbf m=1^5,221,32$, then $\mathbf m'=1^6,33,321$ or $1^6,222,42$ or $21111,222,33$.
If $\mathbf m=1^{2n+1},n+1n,nn1$ with $n\ge 3$, $\mathbf m'=1^{2n+2},n+1n+1,n+1n1$.
If $\mathbf m=1^6,222,42$ or $\mathbf m=21111,222,33$, $\mathbf m'$ doesn't exists.
{\rm iii)} Note that Theorem~\ref{thm:ExF} assures that the condition $N_\nu(\mathbf m)\ge 0$ for $\nu=1,\dots,\ord\mathbf m-1$ implies that $\mathbf m$ is realizable.
We may assume $\mathbf m\in\mathcal P_{p+1}^{(n)}$ is standard. Put $d=m_{0,1}+\cdots+m_{p,1}-(p-1)n>0$ and $\mathbf m'=\partial_{max}\mathbf m$. Then $m'_{j,\nu}=m_{j,\nu}-\delta_{\nu,1}d$ for $j=0,\dots,p$ and $\nu\ge 1$. Under the notation in Definition~\ref{def:Nnu} the operation $\partial_{max}$ transforms the sets \begin{equation*}
\mathfrak m_j:=\{\widetilde m_{j,k}\,;\,
k=0,1,2,\ldots\text{ and }\widetilde m_{j,k}>0\} \end{equation*} into \begin{equation*}
\mathfrak m_j'
=\bigl\{\widetilde m_{j,k} - \min\{d,m_{j,1}-k\}\,;\,k=0,\dots,
\max\{m_{j,1}-d,m_{j,2}-1\}\bigr\}, \end{equation*} respectively because $\widetilde m_{j,i}=\sum_\nu\max\{m_{j,\nu}-i,0\}$. Therefore $N_\nu(\mathbf m')\le N_\nu(\mathbf m)$ for $\nu=1,\dots,n-d-1=\ord\mathbf m'-1$. Here we note that \begin{equation*}
\sum_{\nu=1}^{n-1}N_\nu(\mathbf m)
= \sum_{\nu=1}^{n-d-1}N_\nu(\mathbf m') =\Pidx\mathbf m. \end{equation*} Hence $N_\nu(\mathbf m)\ge 0$ for $\nu=1,\dots,n-1$ if and only if $N_\nu(\mathbf m')=N_\nu(\mathbf m)$ for $\nu=1,\dots,(n-d)-1$ and moreover $N_\nu(\mathbf m)=0$ for $\nu=n-d,\dots,n-1$. Note that the condition that $N_\nu(\mathbf m')= N_\nu(\mathbf m)$ for $\nu=1,\dots,(n-d)-1$ equals \begin{equation}\label{eq:young}
m_{j,1}-d\ge m_{j,2}-1\text{ \ for \ }j=0,\dots,p.\qquad\qquad \raisebox{-15pt}{\scalebox{.5}{ \begin{Young}
& &+&+&+&$-$&$-$&$-$\cr
& &+&+&+&+\cr
& &+&+&+&+\cr
& &+\cr
\cr \end{Young}}} \end{equation} This is easy to see by using a Young diagram. For example, when $\{8,6,6,3,1\}= \{m_{0,1},m_{0,2},m_{0,3},m_{0,4},m_{0,5}\}$ is a partition of $n=24$, the corresponding Young diagram is as above and then $\widetilde m_{0,2}$ equals 15, namely, the number of boxes with the sign + or $-$. Moreover when $d=3$, the boxes with the sign $-$ are deleted by $\partial_{max}$ and the number $\widetilde m_{0,2}$ changes into 12. In this case $m_0=\{24,19,15,11,8,5,2,1\}$ and $m_0'=\{21,16,12,8,5,2\}$.
If $d\ge2$, then $1\in\mathfrak m_j$ for $j=0,\dots,p$ and therefore $N_{n-2}(\mathbf m)-N_{n-1}(\mathbf m)=2$, which means $N_{n-1}(\mathbf m)\ne 0$ or $N_{n-2}(\mathbf m)\ne 0$. When $d=1$, we have $N_\nu(\mathbf m)=N_\nu(\mathbf m')$ for $\nu=1,\dots,n-2$ and $N_{n-1}(\mathbf m)=0$. Thus we have the claim.
iv) The claim follows from Proposition~\ref{prop:wm}. \end{proof} \begin{exmp}\label{ex:SR0} \index{tuple of partitions!simply reducible} We show the simply reducible tuples with index 0 whose fundamental tuple is of type $\tilde D_4$, $\tilde E_6$, $\tilde E_7$ or $\tilde E_8$ (cf.~Example~\ref{ex:Nj}).
$\tilde D_4$: $21,21,21,111\quad 22,22,31,211\quad 22,31,31,1111\quad$
$\tilde E_6$: $211,211,1111\ \ 221,221,2111\ \ 221,311,11111\ \ 222,222,3111 \ \ 222,321,2211$
$\qquad\!222,411,111111 \ \ 322,331,2221 \ \ 332,431,2222 \ \ 333,441,3222$
$\tilde E_7$: $11111,2111,32 \ \ 111111,2211,42 \ \ 21111,2211,33 \ \ 111111,3111,33$
$\qquad\!22111,2221,43 \ \ 1111111,2221,52 \ \ 22211,2222,53 \ \ 11111111,2222,62$
$\qquad\!32111,2222,44 \ \ 22211,3221,53$
$\tilde E_8$: $1111111,322,43 \ \ 11111111,332,53 \ \ 2111111, 332,44 \ \ 11111111,422,44$
$\qquad\!2211111,333,54\ \ 111111111,333,63 \ \ 2221111,433,55 \ \ 2222111,443,65$
$\qquad\!3222111,444,66 \ \ 2222211,444,75 \ \ 2222211,543,66 \ \ 2222221,553,76$
$\qquad\!2222222,653,77$
\end{exmp} In general, we have the following proposition. \begin{prop} There exist only a finite number of standard and simply reducible tuples with any fixed non-positive index of rigidity. \end{prop} \begin{proof} First note that $\mathbf m\in\mathcal P_{p+1}$ if $d_{max}(\mathbf m)=1$ and $\ord\mathbf m>3$ and $\partial_{max}\mathbf m\in\mathcal P_{p+1}$. Since there exist only finite basic tuples with any fixed index of rigidity (cf.~Remark~\ref{rem:Fbasic}), we have only to prove the non-existence of the infinite sequence \[
\mathbf m(0)\xleftarrow{\partial_{max}}\mathbf m(1)\xleftarrow{\partial_{max}}\cdots\cdots
\xleftarrow{\partial_{max}}\mathbf m(k)\xleftarrow{\partial_{max}}\mathbf m(k+1)
\xleftarrow{\partial_{max}}\cdots \] such that $d_{max}(\mathbf m(k))=1$ for $k\ge 1$ and $\idx\mathbf m(0)\le 0$.
Put \begin{align*}
\bar m(k)_j&=\max_{\nu}\{m(k)_{j,\nu}\},\\
a(k)_j&=\#\{\nu\,;\,m(k)_{j,\nu}=\bar m(k)_j\},\\
b(k)_j&=\begin{cases}
\#\{\nu\,;\,m(k)_{j,\nu}=\bar m(k)_j-1\} &(\bar m(k)_j>1),\\
\infty&(\bar m(k)_j=1).
\end{cases} \end{align*} The assumption $d_{max}(\mathbf m(k))=d_{max}(\mathbf m(k+1))=1$ implies that there exist indices $0\le j_k<j'_k$ such that \begin{equation}\label{eq:SRP1}
(a(k+1)_j,b(k+1)_j)
=\begin{cases}
(a(k)_j+1,b(k)_j-1) &(j=j_k\text{ or }j'_k),\\
(1,a(k)_j-1) &(j\ne j_k\text{ and }j'_k)
\end{cases} \end{equation} and \begin{equation}\label{eq:RSM1}
\bar m(k)_0+\cdots+\bar m(k)_p=(p-1)\ord\mathbf m(k)+1\qquad(p\gg 1) \end{equation} for $k=1,2,\dots$. Since $a(k+1)_j+b(k+1)_j\le a(k)_j+b(k)_j$, there exists a positive integer $N$ such that $a(k+1)_j+b(k+1)_j = a(k)_j+b(k)_j$ for $k\ge N$, which means \begin{equation}\label{eq:SRP2}
b(k)_j\begin{cases}
>0&(j=j_k\text{ or }j'_k),\\
=0&(j\ne j_k\text{ and }j'_k).
\end{cases} \end{equation} Putting $(a_j,b_j)=(a(N)_j,b(N)_j)$, we may assume $b_0\ge b_1>b_2=b_3=\cdots=0$ and $a_2\ge a_3\ge\cdots$. Moreover we may assume $j'_{N+1}\le 3$, which means $a_j=1$ for $j\ge 4$. Then the relations \eqref{eq:SRP1} and \eqref{eq:SRP2} for $k=N, N+1, N+2$ and $N+3$ prove that $\bigl((a_0,b_0),\cdots,(a_3,b_3)\bigr)$ is one of the followings: \begin{align} &((a_0,\infty),(a_1,\infty),(1,0),(1,0)),\label{eq:RSC1}\\ &((a_0,\infty),(1,1),(2,0),(1,0)),\label{eq:RSC2}\\ &((2,2),(1,1),(4,0),(1,0)),\ ((1,3),(3,1),(2,0),(1,0)),\label{eq:RSC3}\\ &((1,2),(2,1),(3,0),(1,0)),\label{eq:RSC4}\\ &((1,1),(1,1),(2,0),(2,0)).\label{eq:RSC5} \end{align} In fact if $b_1>1$, $a_2=a_3=1$ and we have \eqref{eq:RSC1}. Thus we may assume $b_1=1$. If $b_0=\infty$, $a_3=1$ and we have \eqref{eq:RSC2}. If $b_0=b_1=1$, we have easily \eqref{eq:RSC5}. Thus we may moreover assume $b_1=1<b_0<\infty$ and $a_3=1$. In this case the integers $j''_k$ satisfying $b(k)_{j''_k}=0$ and $0\le j''_k\le 2$ for $k\ge N$ are uniquely determined and we have easily \eqref{eq:RSC3} or \eqref{eq:RSC4}.
Put $n=\ord\mathbf m(N)$. We may suppose $\mathbf m(N)$ is standard. Let $p$ be an integer such that $m_{j,0}<n$ if and only if $j\le p$. Note that $p\ge 2$. Then if $\mathbf m(N)$ satisfies \eqref{eq:RSC1} (resp.~\eqref{eq:RSC2}), \eqref{eq:RSM1} implies $\mathbf m(N)=1^n,1^n,n-11$ (resp.~$1^n,mm-11,mm$ or $1^n,m+1m,mm1$) and $\mathbf m(N)$ is rigid.
Suppose one of \eqref{eq:RSC3}--\eqref{eq:RSC5}. Then it is easy to check that $\mathbf m(N)$ doesn't satisfy \eqref{eq:RSM1}. For example, suppose \eqref{eq:RSC4}. Then $3m_{0,1}-2\le n$, $3m_{1,1}-1\le n$ and $3m_{2,1}\le n$ and we have $m_{0,1}+m_{1,1}+m_{2,1}\le [\frac {n+2}3]+[\frac{n+1}3]+[\frac{n}3]=n$, which contradicts to \eqref{eq:RSM1}. The relations $[\frac{n+2}{4}]+[\frac n2]+[\frac{n}4]\le n$ and $2[\frac {n+1}2]+2[\frac n2]=2n$ assure the same conclusion in the other cases. \end{proof}
\section{A Kac-Moody root system}\label{sec:KacM} \subsection{Correspondence with a Kac-Moody root system}\label{sec:KM} \index{Kac-Moody root system} We review a Kac-Moody root system to describe the combinatorial structure of middle convolutions on the spectral types. Its relation to Deligne-Simpson problem is first clarified by \cite{CB}.
Let \begin{equation}
I:=\{0,\,(j,\nu)\,;\,j=0,1,\ldots,\ \nu=1,2,\ldots\}. \end{equation} \index{00I@$I,\ I'$} be a set of indices and let $\mathfrak h$ be an infinite dimensional real vector space with the set of basis $\Pi$, where \begin{equation}
\Pi=\{\alpha_i\,;\,i\in I\}
=\{\alpha_0,\ \alpha_{j,\nu}\,;\,j=0,1,2,\ldots,\ \nu=1,2,\ldots\}. \end{equation} \index{000Pi@$\Pi,\ \Pi',\ \Pi(\mathbf m)$} Put \begin{align}
I' &:= I\setminus\{0\},\qquad \Pi':=\Pi\setminus\{\alpha_0\},\\
Q &:=\sum_{\alpha\in\Pi}\mathbb Z\alpha\ \supset\
Q_+:=\sum_{\alpha\in\Pi}\mathbb Z_{\ge0}\alpha.\index{00Qp@$Q_+$} \end{align} We define an indefinite symmetric bilinear form on $\mathfrak h$ by \begin{equation}\label{eq:PIKac} \index{000alpha0@$\alpha_0$, $\alpha_\ell$, $\alpha_{\mathbf m}$}
\begin{split}
(\alpha|\alpha)
&= 2\qquad\ \,(\alpha\in\Pi),\\
(\alpha_0|\alpha_{j,\nu})
&=-\delta_{\nu,1},\\
(\alpha_{i,\mu}|\alpha_{j,\nu})
&=\begin{cases}
0 &(i\ne j\text{ \ \ or \ \ }|\mu-\nu|>1),\\
-1&(i=j\text{ \ and \ }|\mu-\nu|=1).
\end{cases}
\end{split}\quad\ \ \raisebox{10pt}{\text{\small \begin{xy} \ar@{-} *++!D{\text{$\alpha_0$}} *\cir<4pt>{}="O";
(10,0) *+!L!D{\text{$\alpha_{1,1}$}} *\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *+!L!D{\text{$\alpha_{1,2}$}} *\cir<4pt>{}="B", \ar@{-} "B"; (30,0) *{\cdots}, \ar@{-} "O"; (10,-7) *+!L!D{\text{$\alpha_{2,1}$}} *\cir<4pt>{}="C", \ar@{-} "C"; (20,-7) *+!L!D{\text{$\alpha_{2,2}$}} *\cir<4pt>{}="E", \ar@{-} "E"; (30,-7) *{\cdots} \ar@{-} "O"; (10,8) *+!L!D{\text{$\alpha_{0,1}$}} *\cir<4pt>{}="D", \ar@{-} "D"; (20,8) *+!L!D{\text{$\alpha_{0,2}$}} *\cir<4pt>{}="F", \ar@{-} "F"; (30,8) *{\cdots} \ar@{-} "O"; (10,-13) *+!L!D{\text{$\alpha_{3,1}$}} *\cir<4pt>{}="G", \ar@{-} "G"; (20,-13) *+!L!D{\text{$\alpha_{3,2}$}} *\cir<4pt>{}="H", \ar@{-} "H"; (30,-13) *{\cdots}, \ar@{-} "O"; (7, -13), \ar@{-} "O"; (4, -13), \end{xy}}} \end{equation}
\index{Weyl group} \index{simple root} \index{simple reflection} The element of $\Pi$ is called the \textsl{simple root} of a Kac-Moody root system and the \textsl{Weyl group} $W_{\!\infty}$ of this Kac-Moody root system is generated by the \textsl{simple reflections} $s_i$ with $i\in I$. Here the \textsl{reflection} with respect to an element
$\alpha\in\mathfrak h$ satisfying $(\alpha|\alpha)\ne0$ is the linear transformation \index{00salpha@$s_\alpha,\ s_i$}\index{reflection} \begin{equation}
s_\alpha\,:\,\mathfrak h\ni x\mapsto
x-2\frac{(x|\alpha)}{(\alpha|\alpha)}\alpha\in\mathfrak h \end{equation} and \begin{equation}\label{eq:Kzri}
s_i=s_{\alpha_i} \text{ \ for \ } i\in I. \end{equation}
In particular $s_i(x)=x-(\alpha_i|x)\alpha_i$ for $i\in I$ and the subgroup of $W_{\!\infty}$ generated by $s_i$ for $i\in I\setminus\{0\}$ is denoted by $W'_{\!\infty}$.
The Kac-Moody root system is determined by the set of simple roots $\Pi$ and its Weyl group $W_{\!\infty}$ and it is denoted by $(\Pi,W_{\!\infty})$. \index{00Winfty@$W_{\!\infty},\ W'_{\!\infty},\ \widetilde W_{\!\infty}$}
Denoting $\sigma(\alpha_0)=\alpha_0$ and $\sigma(\alpha_{j,\nu})= \alpha_{\sigma(j),\nu}$ for $\sigma\in\mathfrak S_\infty$, we put \begin{equation}\label{eq:Kzouter}
\widetilde W_{\!\infty}:=\mathfrak S_\infty\ltimes W_{\!\infty}, \end{equation} which is an automorphism group of the root system. \begin{rem}[\cite{Kc}]\label{rem:Kac} \index{000Delta@$\Delta,\ \Delta_+,\ \Delta_-$} \index{000Delta0@$\Delta^{re},\ \Delta^{re}_+,\ \Delta^{re}_-,\ \Delta^{im}, \Delta^{im}_+,\ \Delta^{im}_-$} \index{00B@$B$} The set $\Delta^{re}$ of \textsl{real roots} equals the $W_{\!\infty}$-orbit of $\Pi$, which also equals $W_{\!\infty}\alpha_0$. Denoting \begin{equation}
B:=\{\beta\in Q_+\,;\,\supp\beta\text{ is connected and }
(\beta,\alpha)\le 0\quad(\forall\alpha\in\Pi)\}, \end{equation} the set of \textsl{positive imaginary roots} $\Delta^{im}_+$ equals $W_{\!\infty} B$. Here \begin{equation} \supp\beta:=\{\alpha\in\Pi\,;\,n_\alpha\ne0\}\text{ \ if \ } \beta=\sum_{\alpha\in\Pi} n_\alpha\alpha. \end{equation}
The set $\Delta$ of roots equals $\Delta^{re}\cup\Delta^{im}$ by denoting $\Delta_-^{im}=-\Delta_+^{im}$ and $\Delta^{im}=\Delta_+^{im}\cup\Delta_-^{im}$. Put $\Delta_+=\Delta\cap Q_+$, $\Delta_-=-\Delta_+$, $\Delta^{re}_+=\Delta^{re}\cap Q_+$ and $\Delta^{re}_-=-\Delta^{re}_+$. Then $\Delta=\Delta_+\cup\Delta_-$, $\Delta^{im}_+\subset\Delta_+$ and $\Delta^{re}=\Delta^{re}_+\cup\Delta^{re}_-$. The root in $\Delta$ is called \textsl{positive} if and only if $\alpha\in Q_+$. \index{real root}\index{imaginary root}\index{positive root}
A subset $L\subset\Pi$ is called \textsl{connected} if the decomposition $L_1\cup L_2= L$ with $L_1\ne\emptyset$ and $L_2\ne\emptyset$ always implies the
existence of $v_j\in L_j$ satisfying $(v_1|v_2)\ne0$. Note that $\supp\alpha\ni\alpha_0$ for $\alpha\in\Delta^{im}$.
The subset $L$ is called classical if it corresponds to the classical Dynkin diagram, which is equivalent to the condition that the group generated by the reflections with respect to the elements in $L$ is a finite group.
The connected subset $L$ is called \textsl{affine} if it corresponds to affine Dynkin diagram and in our case it corresponds to $\tilde D_4$ or $\tilde E_6$ or $\tilde E_7$ or $\tilde E_8$ with the following Dynkin diagram, respectively.
\index{Dynkin diagram} \index{affine} \index{00D4@$\tilde D_4,\ \tilde E_6,\ \tilde E_7,\ \tilde E_8$} {\small \begin{equation}\index{Dynkin diagram}\label{eq:Dynkinidx0} \begin{gathered} \begin{xy}
\ar@{-} *++!D{1} *\cir<4pt>{};
(10,0) *+!L+!D{2}*\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *++!D{1} *\cir<4pt>{}, \ar@{-} "A"; (10,-10) *++!L{1} *\cir<4pt>{}, \ar@{-} "A"; (10,10) *++!L{1} *\cir<4pt>{}, \ar@{} (10,-14) *{11,11,11,11} \end{xy} \quad \begin{xy}
\ar@{-} *++!D{2} *\cir<4pt>{};
(10,0) *++!D{4} *\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *+!L+!D{6}*\cir<4pt>{}="B", \ar@{-} "B"; (30,0) *++!D{5} *\cir<4pt>{}="C", \ar@{-} "C"; (40,0) *++!D{4} *\cir<4pt>{}="D", \ar@{-} "D"; (50,0) *++!D{3} *\cir<4pt>{}="E", \ar@{-} "E"; (60,0) *++!D{2} *\cir<4pt>{}="F", \ar@{-} "F"; (70,0) *++!D{1} *\cir<4pt>{}, \ar@{-} "B"; (20,10) *++!L{3} *\cir<4pt>{} \ar@{} (20,-4) *{33,222,111111} \end{xy} \allowdisplaybreaks\\[-.8cm] \begin{xy}
\ar@{-} *++!D{1} *\cir<4pt>{};
(10,0) *++!D{2} *\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *++!D{3} *\cir<4pt>{}="B", \ar@{-} "B"; (30,0) *+!L+!D{4}*\cir<4pt>{}="C", \ar@{-} "C"; (40,0) *++!D{3} *\cir<4pt>{}="D", \ar@{-} "D"; (50,0) *++!D{2} *\cir<4pt>{}="E", \ar@{-} "E"; (60,0) *++!D{1} *\cir<4pt>{}, \ar@{-} "C"; (30,10) *++!L{2} *\cir<4pt>{}, \ar@{} (30,-4) *{22,1111,1111} \end{xy} \quad \begin{xy}
\ar@{-} *++!D{1} *\cir<4pt>{};
(10,0) *++!D{2} *\cir<4pt>{}="O", \ar@{-} "O"; (20,0) *+!L+!D{3}*\cir<4pt>{}="A", \ar@{-} "A"; (30,0) *++!D{2} *\cir<4pt>{}="B", \ar@{-} "B"; (40,0) *++!D{1} *\cir<4pt>{} \ar@{-} "A"; (20,10) *++!L{2} *\cir<4pt>{}="C", \ar@{-} "C"; (20,20) *++!L{1} *\cir<4pt>{}, \ar@{} (20,-4) *{111,111,111} \end{xy} \end{gathered}\end{equation}} \end{rem}
\noindent Here the circle correspond to simple roots and the numbers attached to simple roots are the coefficients $n$ and $n_{j,\nu}$ in the expression \eqref{eq:root2part} of a root $\alpha$.
For a tuple of partitions $\mathbf m
=\bigl(m_{j,\nu}\bigr)_{j\ge 0,\ \nu\ge 1} \in\mathcal P^{(n)}$, we define \index{000alpha0@$\alpha_0$, $\alpha_\ell$, $\alpha_{\mathbf m}$} \begin{equation}\label{eq:Kazpart}
\begin{split}
n_{j,\nu}&:=m_{j,\nu+1}+m_{j,\nu+2}+\cdots,\\
\alpha_{\mathbf m}&:=n\alpha_0
+ \sum_{j=0}^\infty\sum_{\nu=1}^\infty n_{j,\nu}\alpha_{j,\nu}
\in Q_+,\\
\kappa(\alpha_{\mathbf m})&:=\mathbf m.
\end{split} \end{equation} As is given in \cite[Proposition~2.22]{O3} we have \begin{prop}\label{prop:Kac} {\rm i)} \
$\idx(\mathbf m,\mathbf m')=
(\alpha_{\mathbf m}|\alpha_{\mathbf m'})$.
\phantom{ABCDEFG}
{\rm ii)} \ Given $i\in I$, we have $\alpha_{\mathbf m'} = s_i(\alpha_\mathbf m)$ with \begin{equation*}\index{tuple of partitions!index}
\mathbf m'=
\begin{cases}
\partial\mathbf m&(i=0),\\
(m_{0,1}\dots,m_{j,1}\dots
\overset{\underset{\smallsmile}\nu}{m_{j,\nu+1}}
\overset{\underset{\smallsmile}{\nu+1}}{m_{j,\nu}}\dots,\dots)
&\bigl(i=(j,\nu)\bigr).
\end{cases} \end{equation*} Moreover for $\ell=(\ell_0,\ell_1,\ldots)\in\mathbb Z_{>0}^\infty$ satisfying $\ell_\nu=1$ for $\nu\gg1$ we have \index{000alpha0@$\alpha_0$, $\alpha_\ell$, $\alpha_{\mathbf m}$} \begin{align}
\alpha_\ell:=\alpha_{\mathbf 1_\ell} &=\alpha_0
+\sum_{j=0}^\infty\sum_{\nu=1}^{\ell_j-1}\alpha_{j,\nu}
=\biggl(\prod_{j\ge 0}
s_{j,\ell_j-1}\cdots s_{j,2}s_{j,1}\biggr)(\alpha_0),\label{eq:alpl}\\
\alpha_{\partial_\ell(\mathbf m)} &=s_{\alpha_\ell}(\alpha_{\mathbf m})
= \alpha_{\mathbf m}
- 2\frac{(\alpha_{\mathbf m}|\alpha_\ell)}
{(\alpha_\ell|\alpha_\ell)}\alpha_\ell
= \alpha_{\mathbf m}-(\alpha_{\mathbf m}|\alpha_\ell)\alpha_\ell.\label{eq:M2K} \end{align} \end{prop}
Note that \begin{equation}
\begin{split}\alpha&=n\alpha_0+\sum_{j\ge 0}
\sum_{\nu\ge 1}n_{j,\nu}\alpha_{j,\nu}\in\Delta^+ \text{ with }n>0\\
&\quad
\Rightarrow \
n\ge n_{j,1}\ge n_{j,2}\ge \cdots\qquad(j=0,1,\ldots).
\end{split}\label{eq:root2part} \end{equation} In fact, for a sufficiently large $K\in\mathbb Z_{>0}$, we have $n_{j,\mu}=0$ for $\mu\ge K$ and \[
s_{\alpha_{j,\nu}+\alpha_{j,\nu+1}+\cdots+\alpha_{j,K}}\alpha
= \alpha + (n_{j,\nu-1}-n_{j,\nu})
(\alpha_{j,\nu}+\alpha_{j,\nu+1}+\cdots+\alpha_{j,K})\in\Delta^+ \] for $\alpha\in\Delta_+$ in \eqref{eq:root2part}, which means $n_{j,\nu-1}\ge n_{j,\nu}$ for $\nu\ge1$. Here we put $n_{j,0}=n$ and $\alpha_{j,0}=\alpha_0$. Hence for $\alpha\in\Delta_+$ with $\supp\alpha\ni\alpha_0$, there uniquely exists $\mathbf m\in\mathcal P$ satisfying $\alpha=\alpha_{\mathbf m}$.
It follows from \eqref{eq:M2K} that under the identification $\mathcal P\subset Q_+$ with \eqref{eq:Kazpart}, our operation $\partial_\ell$ corresponds to the reflection with respect to the root $\alpha_\ell$. Moreover the rigid (resp.\ indivisible realizable) tuple of partitions corresponds to the positive real root (resp.\ indivisible positive root) whose support contains $\alpha_0$, which were first established by \cite{CB} in the case of Fuchsian systems of Schlesinger canonical form (cf.~\cite{O3}).
The corresponding objects with this identification are as follows, which will be clear in this subsection. Some of them are also explained in \cite{O3}.
\noindent
\begin{longtable}{|c|c|}\hline $\mathcal P$
& Kac-Moody root system\rule{0pt}{11.5pt}\\ \hline\hline $\mathbf m$
& $\alpha_{\mathbf m}$ \ (cf.~\eqref{eq:Kazpart})\rule{0pt}{11.5pt}\\ \hline {$\mathbf m$ : monotone}
& $\alpha\in Q_+$\,:\, $(\alpha|\beta)\le 0
\ \ (\forall\beta\in\Pi')$\rule{0pt}{11.5pt}\\ \hline $\mathbf m$ : realizable
& $\alpha\in\overline\Delta_+$ \rule{0pt}{11.5pt}\\ \hline $\mathbf m$ : rigid
& $\alpha\in\Delta_+^{re}\,:\,\supp\alpha\ni\alpha_0$ \rule{0pt}{11.5pt}\\ \hline $\mathbf m$ : monotone and fundamental
& $\alpha\in Q_+$\,:$\,\alpha=\alpha_0\text{ or }
(\alpha|\beta)\le 0
\ \ (\forall\beta\in\Pi)$\rule{0pt}{11.5pt}\\ \hline \raisebox{-6.5pt}{$\mathbf m$ : irreducibly realizable}
& $\alpha\in\Delta_+,\ \,\,\supp\alpha\ni\alpha_0$\\[-4pt]
& indivisible or $(\alpha|\alpha)<0$\\ \hline
\raisebox{-6.5pt}{$\mathbf m$ : basic and monotone}
& $\alpha\in Q_+$\,:\, $(\alpha|\beta)\le 0
\ \ (\forall\beta\in\Pi)$\rule{0pt}{11.5pt}\\[-2pt]
& indivisible \\ \hline
\raisebox{-6.5pt}{$\mathbf m$\,:\! simply reducible and monotone}
& $\alpha\in\Delta_+$\,:\,$(\alpha|\alpha_{\mathbf m})=1\quad(\forall\alpha\in \Delta(\mathbf m))$\rule{0pt}{11.5pt}\\[-2pt]
& $\alpha_0\in\Delta(\mathbf m),\ \ (\alpha|\beta)\le 0\ \ (\forall\beta\in\Pi')$ \\ \hline $\ord\mathbf m$ & $n_0\,:\,\alpha=n_0\alpha_0+\sum_{i,\nu}n_{i,\nu}\alpha_{i,\nu}$ \rule{0pt}{11.5pt}\\ \hline $\idx(\mathbf m,\mathbf m')$
& $(\alpha_{\mathbf m}|\alpha_{\mathbf m'})$ \rule{0pt}{11.5pt}\\ \hline $\idx\mathbf m$
& $(\alpha_{\mathbf m}|\alpha_{\mathbf m})$ \rule{0pt}{11.5pt}\\ \hline $d_{\ell}(\mathbf m)$ \ (cf.~\eqref{eq:dm})
& $(\alpha_\ell|\alpha_{\mathbf m})$ \ (cf.~\eqref{eq:alpl}) \rule{0pt}{11.5pt}\\ \hline $\Pidx\mathbf m+\Pidx\mathbf m'=\Pidx(\mathbf m+\mathbf m')$
&$(\alpha_{\mathbf m}|\alpha_{\mathbf m'})=-1$ \rule{0pt}{11.5pt}\\ \hline $(\nu,\nu+1)\in G_j\subset S_\infty'$ \ (cf.~\eqref{eq:S_infty})
&
$s_{j,\nu}\in W_{\!\infty}'$ \ (cf.~\eqref{eq:Kzri}) \rule[-4.5pt]{0pt}{16pt}\\ \hline $H\simeq \mathfrak S_\infty$ \ (cf.~\eqref{eq:S_infty})
& $\mathfrak S_\infty$ in \eqref{eq:Kzouter} \rule{0pt}{11.5pt}\\ \hline $\partial_{\bf 1}$ & $s_0$\rule{0pt}{11.5pt}\\ \hline $\partial_{\ell}$ & $s_{\alpha_\ell}$ \ (cf.~\eqref{eq:alpl}) \rule{0pt}{11.5pt}\\ \hline $\langle\partial_{\bf 1},\,S_\infty\rangle$
& $\widetilde W_{\!\infty}\rule{0pt}{11.5pt}$ \ (cf.~\eqref{eq:Kzouter}) \rule{0pt}{11.5pt}\\ \hline $\{\lambda_{\mathbf m}\}$
& $(\Lambda(\lambda),\alpha_{\mathbf m})$ \ (cf.~\eqref{eq:defKacGRS}) \rule{0pt}{11.5pt}\\ \hline
$|\{\lambda_{\mathbf m}\}|$
&
$(\Lambda(\lambda)+\frac12\alpha_{\mathbf m}|\alpha_{\mathbf m})$ \rule[-4.5pt]{0pt}{16pt}\\ \hline $\Ad\bigl((x-c_j)^\tau\bigr)$
& $+\tau\Lambda^0_{0,j}$ \ (cf.~\eqref{eq:defKacGRS}) \rule[-4.5pt]{0pt}{16pt}\\ \hline \end{longtable}\index{000lambda@$\arrowvert$\textbraceleft$\lambda_{\mathbf m}$\textbraceright$\arrowvert$}
Here\index{000Delta00@$\overline\Delta_+$} \begin{equation} \overline\Delta_+:=\{k\alpha\,;\,\alpha\in\Delta_+,\ k\in\mathbb Z_{>0}, \ \supp\alpha\ni\alpha_0\}, \end{equation} $\Delta(\mathbf m)\subset \Delta_+^{re}$ is given in \eqref{def:Deltam} and $\Lambda(\lambda)\in{\overline{\mathfrak h}}_p$ is defined as follows.
\begin{defn} Fix a positive integer $p$ which may be $\infty$. Put \begin{equation}
I_p:=\{0,\ (j,\nu)\,;\,j=0,1,\dots,p,\ \nu=1,2,\ldots\}\subset I \index{00Ip@$I_p$} \end{equation} for a positive integer $p$ and $I_\infty=I$.
Let $\mathfrak h_p$ be the $\mathbb R$-vector space of finite linear combinations the elements of $\Pi_p:=\{\alpha_i\,;\,i\in\Pi_p\}$ and let $\mathfrak h_p^\vee$ be the $\mathbb C$-vector space whose elements are linear combinations of infinite or finite elements of $\Pi_p$, which is identified with $\Pi_{i\in I_p}\mathbb C\alpha_i$ and contains ${\mathfrak h}_p$. \index{00hp@$\mathfrak h_p$, $\mathfrak h_p^\vee$, $\overline{\mathfrak h}_p$}
The element $\Lambda\in\mathfrak h_p^\vee$ naturally defines a linear form of $\mathfrak h_p$ by
$(\Lambda|\ \cdot\ )$ and the group $\widetilde W_{\!\infty}$ acts on $\mathfrak h_p^\vee$. If $p=\infty$, we assume that the element $\Lambda=\xi_0\alpha_0+ \sum \xi_{j,\nu}\alpha_{j,\nu}\in\mathfrak h_\infty^\vee$ always satisfies $\xi_{j,1}=0$ for sufficiently large $j\in\mathbb Z_{\ge0}$. Hence we have naturally $\mathfrak h_p^\vee\subset\mathfrak h_{p+1}^\vee$ and $\mathfrak h_\infty^\vee=\bigcup_{j\ge0}\mathfrak h_j^\vee$.
Define the elements of $\mathfrak h_p^\vee$: \begin{equation}
\begin{split}
\Lambda_0&:=\frac12\alpha_0+
\frac12\sum_{j=0}^p\sum_{\nu=1}^\infty(1-\nu)\alpha_{j,\nu},\\
\Lambda_{j,\nu}&:=\sum_{i=\nu+1}^\infty(i-\nu)\alpha_{j,i}
\quad(j=0,\dots,p,\ \nu=0,1,2,\ldots)\\
\Lambda^0&:=2\Lambda_0-2\Lambda_{0,0}
=\alpha_0+\sum_{\nu=1}^\infty(1+\nu)\alpha_{0,\nu}+
\sum_{j=1}^p\sum_{\nu=1}^\infty(1-\nu)\alpha_{j,\nu},\\
\Lambda^0_{j,k}&:=\Lambda_{j,0}-\Lambda_{k,0}=
\sum_{\nu=1}^\infty\nu(\alpha_{k,\nu}-\alpha_{j,\nu})
\quad(0\le j<k\le p),\\
\Lambda(\lambda)&:=-\Lambda_0-\sum_{j=0}^p\sum_{\nu=1}^\infty
\Bigl(\sum_{i=1}^\nu\lambda_{j,i}
\Bigr)\alpha_{j,\nu}\\
&\;=-\Lambda_0+\sum_{j=0}^p\sum_{\nu=1}^\infty\lambda_{j,\nu}
(\Lambda_{j,\nu-1}-\Lambda_{j,\nu}).
\end{split}\label{eq:defKacGRS} \end{equation} \end{defn} Under the above definition we have \begin{align}\index{000lambda0@$\Lambda_0,\ \Lambda_{j,\nu},\ \Lambda^0,\ \Lambda^0_{j,\nu},\ \Lambda(\lambda)$}
(\Lambda^0|\alpha)&=(\Lambda^0_{j,k}|\alpha)=0
\quad(\forall\alpha\in\Pi_p),\\
(\Lambda_{j,\nu}|\alpha_{j',\nu'})&=\delta_{j,j'}\delta_{\nu,\nu'}
\quad(j,\,j'=0,1,\ldots,\ \nu,\,\nu'=1,2,\ldots)\allowdisplaybreaks\\
(\Lambda_0|\alpha_i)&=(\Lambda_{j,0}|\alpha_i)=
\delta_{i,0} \ \quad(\forall i\in \Pi_p),
\allowdisplaybreaks\\
|\{\lambda_{\mathbf m}\}|&=
(\Lambda(\lambda)+\tfrac12 \alpha_{\mathbf m}|\alpha_{\mathbf m}),
\allowdisplaybreaks\index{Fuchs relation}\\
\begin{split}
s_0(\Lambda(\lambda))&=-\Bigl(\sum_{j=0}^p\lambda_{j,1}-1\Bigr)\alpha_0
+\Lambda(\lambda)\\
&=-\mu\Lambda^0-\Lambda_0-
\sum_{\nu=1}^\infty\Bigl(\sum_{i=1}^\nu(\lambda_{0,i}-(1+\delta_{i,0})\mu
\Bigr)\alpha_{0,\nu}\\
&\quad{}-
\sum_{j=1}^p\sum_{\nu=1}^\infty
\Bigl(\sum_{i=1}^\nu(\lambda_{j,i}+(1-\delta_{i,0})\mu)
\Bigr)\alpha_{j,\nu}
\end{split}\label{eq:s0Kac} \end{align} with $\mu=\sum_{j=0}^p\lambda_{j,1}-1$.
We identify the elements of $\mathfrak h_p^\vee$ if their difference are in $\mathbb C\Lambda^0$, namely, consider them in $\overline{\mathfrak h}_p:=\mathfrak h_p^\vee/\mathbb C\Lambda^0$. Then the elements have the unique representatives in $\mathfrak h_p^\vee$ whose coefficients of $\alpha_0$ equal $-\frac12$. \index{00hp@$\mathfrak h_p$, $\mathfrak h_p^\vee$, $\overline{\mathfrak h}_p$}
\begin{rem}\label{rem:KacGRS} i) \ If $p<\infty$, we have \begin{equation}
\{\Lambda\in\mathfrak h_p^\vee\,;\,(\Lambda|\alpha)=0\quad(\forall \alpha\in\Pi_p)\}
=\mathbb C\Lambda^0+\sum_{j=1}^p\mathbb C\Lambda^0_{0,j}. \end{equation}
ii) The invariance of the bilinear form $(\ |\ )$ under the Weyl group $W_{\!\infty}$ proves \eqref{eq:midinv}.
iii) The addition given in Theorem~\ref{thm:GRSmid} i) corresponds to the map $\Lambda(\lambda)\mapsto\Lambda(\lambda) +\tau\Lambda^0_{0,j}$ with $\tau\in\mathbb C$ and $1\le j\le p$.
iii) Combining the action of $s_{j,\nu}$ on $\mathfrak h_p^\vee$ with that of $s_0$, we have \begin{equation}\label{eq:KacGRS}
\Lambda(\lambda') - s_{\alpha_\ell}
\Lambda(\lambda)\in\mathbb C\Lambda^0\text{ \ and \ }
\alpha_{\mathbf m'}=s_{\alpha_\ell}\alpha_{\mathbf m}
\quad\text{when \ }\{\lambda'_{\mathbf m'}\}=\partial_\ell\{\lambda_\mathbf m\} \end{equation} because of \eqref{eq:pellGRS} and \eqref{eq:s0Kac}.
\end{rem} Thus we have the following theorem. \begin{thm}\label{thm:KatzKac} Under the above notation we have the commutative diagram \begin{equation*}
\begin{matrix} \bigl\{P_{\mathbf m}:\,\text{Fuchsian differential operators with } \{\lambda_{\mathbf m}\}\bigr\}
&\rightarrow
&\bigl\{(\Lambda(\lambda),\alpha_{\mathbf m})\,;\,\alpha_{\mathbf m}\in\overline\Delta_+\bigr\}
\\
\\%[-10pt]
\downarrow \text{fractional operations}
&\circlearrowright&\quad \downarrow {W_{\!\infty}\text{-action},\
+\tau\Lambda^0_{0,j}}\\
\\%[-10pt] \bigl\{P_{\mathbf m}:\,\text{Fuchsian differential operators with } \{\lambda_{\mathbf m}\}\bigr\}
& \rightarrow
& \bigl\{(\Lambda(\lambda), \alpha_{\mathbf m})\,;\,\alpha_{\mathbf m}
\in\overline\Delta_+\bigr\}.
\end{matrix} \end{equation*} Here the defining domain of $w\in W_{\!\infty}$ is $\{\alpha\in\overline\Delta_+\,;\,w\alpha\in\overline\Delta_+\}$. \end{thm} \begin{proof} Let $T_i$ denote the corresponding operation on $\{(P_{\mathbf m},\{\lambda_{\mathbf m}\})\}$ for $s_i\in W_\infty$ with $i\in I$. Then $T_0$ corresponds to $\partial_{\mathbf 1}$ and when $i\in I'$, $T_i$ is naturally defined and it doesn't change $P_{\mathbf m}$. The fractional transformation of the Fuchsian operators and their Riemann schemes corresponding to an element $w\in W_{\!\infty}$ is defined through the expression of $w$ by the product of simple reflections. It is clear that the transformation of their Riemann schemes do not depend on the expression.
Let $i\in I$ and $j\in I$. We want to prove that $(T_iT_j)^k=id$ if $(s_is_j)^k=id$ for a non-negative integer $k$. Note that $T_i^2=id$ and the addition commutes with $T_i$. Since $T_i=id$ if $i\in I'$, we have only to prove that $(T_{j,1}T_0)^3=id$. Moreover Proposition~\ref{prop:coordf} assures that we may assume $j=0$.
Let $P$ be a Fuchsian differential operator with the Riemann scheme \eqref{eq:GRS}. Applying suitable additions to $P$, we may assume $\lambda_{j,1}=0$ for $j\ge 1$ to prove $(T_{0,1}T_0)^3P=P$ and then this easily follows from the definition of $\partial_{\mathbf 1}$ (cf.~\eqref{eq:opred}) and the relation \begin{align*} &\begin{Bmatrix} \infty & c_j\ (1\le j\le p)\\ [\lambda_{0,1}]_{(m_{0,1})}&[0]_{(m_{j,1})}\\ [\lambda_{0,2}]_{(m_{0,2})}&[\lambda_{j,2}]_{(m_{j,2})}\\ [\lambda_{0,\nu}]_{(m_{0,\nu})}&[\lambda_{j,\nu}]_{(m_{j,\nu})} \end{Bmatrix}\quad(d=m_{0,1}+\cdots+m_{p,1}-\ord\mathbf m)
\allowdisplaybreaks\\ &\quad\xrightarrow[\partial^{1-\lambda_{0,1}}]{T_{0,1}T_0} \begin{Bmatrix} \infty & c_j\ (1\le j\le p)\\ [\lambda_{0,2}-\lambda_{0,1}+1]_{(m_{0,1})}&[0]_{(m_{j,1}-d)}\\ [-\lambda_{0,1}+2]_{(m_{0,2}-d)}&[\lambda_{j,2}+\lambda_{0,1}-1]_{(m_{j,2})}\\ [\lambda_{0,\nu}-\lambda_{0,1}+1]_{(m_{0,\nu})}&[\lambda_{j,\nu}+\lambda_{0,1}-1]_{(m_{j,\nu})} \end{Bmatrix}
\allowdisplaybreaks\\ &\quad \xrightarrow[\partial^{\lambda_{0,1}-\lambda_{0,2}}]{T_{0,1}T_0} \begin{Bmatrix} \infty & c_j\ (1\le j\le p)\\ [-\lambda_{0,2}+2]_{(m_{0,1}-d)}&[0]_{(m_{j,1}+m_{0,1}-m_{0,2}-d)}\\ [\lambda_{0,1}-\lambda_{0,2}+1]_{(m_{0,1})}&[\lambda_{j,2}+\lambda_{0,2}-1]_{(m_{j,2})}\\ [\lambda_{0,\nu}-\lambda_{0,2}+1]_{(m_{0,\nu})}&[\lambda_{j,\nu}+\lambda_{0,2}-1]_{(m_{j,\nu})} \end{Bmatrix}
\allowdisplaybreaks\\ &\quad \xrightarrow[\partial^{\lambda_{0,2}-1}]{T_{0,1}T_0} \begin{Bmatrix} \infty & c_j\ (1\le j\le p)\\ [\lambda_{0,1}]_{(m_{0,1})}&[0]_{(m_{j,1})}\\ [\lambda_{0,2}]_{(m_{0,2})}&[\lambda_{j,2}]_{(m_{j,2})}\\ [\lambda_{0,\nu}]_{(m_{0,\nu})}&[\lambda_{j,\nu}]_{(m_{j,\nu})} \end{Bmatrix} \end{align*} and $(T_{0,1}T_0)^3P
\in \mathbb C[x]
\Ad(\partial^{\lambda_{0,2}-1})\circ\Ad(\partial^{\lambda_{0,2}-\lambda_{0,1}})\circ
\Ad(\partial^{1-\ \lambda_{0,1}})\Red P=\mathbb C[x]\Red P$. \end{proof}
\begin{defn} For an element $w$ of the Weyl group $W_{\!\infty}$ we put \begin{equation}
\Delta(w):=\Delta^{re}_+\cap w^{-1}\Delta^{re}_-. \end{equation} \index{000Delta0s@$\Delta(\mathbf m)$, $\Delta(w)$} If $w=s_{i_1}s_{i_2}\cdots s_{i_k}$ with $i_\nu\in I$ is the \textsl{minimal expression} \index{Weyl group!minimal expression} of $w$ as the products of simple reflections which means $k$ is minimal by definition, we have \index{minimal expression} \begin{equation}\label{eq:Lw}
\Delta(w)=\bigl\{
\alpha_{i_k},s_{i_k}(\alpha_{i_{k-1}}),s_{i_k}s_{i_{k-1}}(\alpha_{i_{k-2}}),
\ldots,s_{i_k}\cdots s_{i_2}(\alpha_{i_1})\bigr\}. \end{equation} \end{defn} The number of the elements of $\Delta(w)$ equals the number of the simple reflections in the minimal expression of $w$, which is called the \textsl{length} of $w$ and denoted by $L(w)$.\index{00Lw@$L(w)$} \index{Weyl group!length} The equality \eqref{eq:Lw} follows from the following lemma.
\begin{lem}\label{lem:minrep} Fix $w\in W_{\!\infty}$ and $i\in I$. If $\alpha_i\in\Delta(w)$, there exists a minimal expression $w=s_{i'_1}s_{i'_2}\cdots s_{i'_k}$ with $s_{i'_k}=s_i$ and $L(ws_i)=L(w)-1$ and $\Delta(ws_i)
=s_i\bigl(\Delta(w)\setminus\{\alpha_i\}\bigr)$. If $\alpha_i\notin\Delta(w)$, $L(ws_i)=L(w)+1$ and $\Delta(ws_i)=s_i\Delta(w)\cup\{\alpha_i\}$. Moreover if $v\in W_{\!\infty}$ satisfies $\Delta(v)=\Delta(w)$, then $v=w$. \end{lem} \begin{proof} The proof is standard as in the case of classical root system, which follows from the fact that the condition $\alpha_i=s_{i_k}\cdots s_{i_{\ell+1}}(\alpha_{i_\ell})$ implies \begin{equation}
s_i=s_{i_k}\cdots s_{i_{\ell+1}}s_{i_\ell}s_{i_{\ell+1}}\cdots s_{i_k} \end{equation} and then $w=ws_is_i=s_{i_1}\cdots s_{i_{\ell-1}}s_{i_{\ell+1}}\cdots s_{i_k}s_i$. \end{proof}
\begin{defn}\label{def:wm} For $\alpha\in Q$, put \begin{equation}
h(\alpha):=n_0+\sum_{j\ge 0}\sum_{\nu\ge 1}n_{j,\nu}
\text{ \ if \ }
\alpha=n_0\alpha_0+\sum_{j\ge 0}\sum_{\nu\ge 1}n_{j,\nu}\alpha_{j,\nu}\in Q. \end{equation}\index{00halpha@$h(\alpha)$} Suppose $\mathbf m\in\mathcal P_{p+1}$ is irreducibly realizable. Note that $sf \mathbf m$ is the monotone fundamental element determined by $\mathbf m$, namely, $\alpha_{sf\mathbf m}$ is the unique element of $W\alpha_{\mathbf m}\cap \bigl(B\cup\{\alpha_0\}\bigr)$. We inductively define $w_{\mathbf m}\in W_{\!\infty}$ satisfying $w_{\mathbf m}\alpha_{\mathbf m}=\alpha_{sf{\mathbf m}}$. We may assume $w_{\mathbf m'}$ has already defined if $h(\alpha_{\mathbf m'}) <h(\alpha_{\mathbf m})$. If $\mathbf m$ is not monotone, there exists $i\in I\setminus\{0\}$
such that $(\alpha_{\mathbf m}|\alpha_i)>0$ and then $w_{\mathbf m}=w_{\mathbf m'}s_i$ with $\alpha_{\mathbf m'}=s_i\alpha_{\mathbf m}$. If $\mathbf m$ is monotone and $\mathbf m\ne f\mathbf m$, $w_{\mathbf m}=w_{\partial\mathbf m}s_0$. \index{00wm@$w_{\mathbf m}$}
We moreover define \begin{align}\label{def:Deltam}
\Delta(\mathbf m)&:= \Delta(w_{\mathbf m}). \end{align} \end{defn} \index{000Delta0s@$\Delta(\mathbf m)$, $\Delta(w)$}
Suppose $\mathbf m$ is monotone, irreducibly realizable and $\mathbf m\ne sf{\mathbf m}$. We define $w_{\mathbf m}$ so that there exists $K\in\mathbb Z_{>0}$ and $v_1,\dots,v_K\in W'_{\!\infty}$ satisfying \begin{equation}\label{eq:vmrp}
\begin{split}
w_{\mathbf m}&=v_Ks_0\cdots v_2s_0v_1s_0,\\
(v_k s_0\cdots v_1s_0\alpha_{\mathbf m}|\alpha)&\le 0
\quad(\forall\alpha\in\Pi\setminus\{0\},\ k=1,\dots,K),
\end{split} \end{equation} which uniquely characterizes $w_{\mathbf m}$. Note that \begin{equation}\label{eq:vmrps}
v_k s_0\cdots v_1s_0\alpha_{\mathbf m}=\alpha_{(s\partial)^k\mathbf m}
\quad(k=1,\dots,K). \end{equation}
The following proposition gives the correspondence between the reduction of realizable tuples of partitions and the minimal expressions of the elements of the Weyl group.
\begin{prop}\label{prop:wm} Definition~\ref{def:wm} naturally gives the product expression $w_{\mathbf m}=s_{i_1}\cdots s_{i_k}$ with $i_\nu\in I\ \ (1\le\nu\le k)$.
{\rm i) } We have \begin{align}
L(w_{\mathbf m})&=k,\\
(\alpha|\alpha_{\mathbf m})&>0\quad(\forall\alpha\in\Delta(\mathbf m)),
\label{eq:amplus}\\
h(\alpha_{\mathbf m})&=h(\alpha_{sf\mathbf m})+
\sum_{\alpha\in\Delta(\mathbf m)}(\alpha|\alpha_{\mathbf m}).
\label{eq:dht} \end{align} Moreover $\alpha_0\in\supp\alpha$ for $\alpha\in\Delta(\mathbf m)$ if\/ $\mathbf m$ is monotone.
{\rm ii)} Suppose\/ $\mathbf m$ is monotone and $f\mathbf m\ne\mathbf m$. Fix maximal integers $\nu_j$ such that $m_{j,1}-d_{max}(\mathbf m)<m_{j,\nu_j+1}$ for $j=0,1,\dots$ Then \begin{gather}
\begin{aligned}
\Delta(\mathbf m)&=s_0\Bigl(\prod_{\substack{j\ge 0\\ \nu_j>0}}
s_{j,1}\cdots s_{j,\nu_j}\Bigr)
\Delta(s\partial\mathbf m)\cup\{\alpha_0\}\label{eq:Ddelta}\\
&\qquad\cup\{\alpha_0+\alpha_{j,1}+\cdots+\alpha_{j,\nu}\,;\,
1\le \nu\le\nu_j\text{ and } j=0,1,\ldots\},
\end{aligned}\\
(\alpha_0+\alpha_{j,1}+\cdots+\alpha_{j,\nu}|\alpha_{\mathbf m})=
d_{max}(\mathbf m)+m_{j,\nu+1}-m_{j,1}\quad(\nu\ge0).
\label{eq:Dd2} \end{gather}
{\rm iii)} Suppose\/ $\mathbf m$ is not rigid. Then
$\Delta(\mathbf m)=\{\alpha\in\Delta^{re}_+\,;\,(\alpha|\alpha_{\mathbf m})>0\}$.
{\rm iv)}
Suppose\/ $\mathbf m$ is rigid. Let $\alpha\in\Delta^{re}_+$ satisfying $(\alpha|\alpha_{\mathbf m})>0$ and $s_{\alpha}(\alpha_{\mathbf m})\in\Delta_+$. Then \begin{equation} \begin{cases} \alpha\in\Delta(\mathbf m)
&\text{if \ }(\alpha|\alpha_{\mathbf m})>1,\\ \#\Bigl(\{\alpha,\,\alpha_{\mathbf m}-\alpha\}\cap\Delta(\mathbf m)\Bigr)
= 1
&\text{if \ }(\alpha|\alpha_{\mathbf m})=1. \end{cases} \end{equation} Moreover if a root $\gamma\in\Delta(\mathbf m)$ satisfies
$(\gamma|\alpha_{\mathbf m})=1$, then $\alpha_{\mathbf m}-\gamma\in\Delta^{re}_+$ and $\alpha_0\in \supp(\alpha_{\mathbf m}-\gamma)$.
{\rm v)} $w_{\mathbf m}$ is the unique element with the minimal length satisfying $w_{\mathbf m}\alpha_{\mathbf m}=\alpha_{sf{\mathbf m}}$. \end{prop}
\begin{proof}
Since $h(s_{i'}\alpha)-h(\alpha)=-(\alpha_{i'}|\alpha)
=(s_{i'}\alpha_{i'}|\alpha)$, we have \[
\begin{split}
h(s_{i'_\ell}\cdots s_{i'_1}\alpha)-h(\alpha)
&=\sum_{\nu=1}^\ell\Bigl(h(s_{i'_\nu}\cdots s_{i'_1}\alpha)
-h(s_{i'_{\nu-1}}\cdots s_{i'_1}\alpha)\Bigr)
\\
&=\sum_{\nu=1}^\ell
(\alpha_{i'_\nu}|s_{i'_\nu}\cdots s_{i'_1}\alpha)
= \sum_{\nu=1}^\ell
(s_{i'_\ell}\cdots s_{i'_{\nu+1}}\alpha_{i'_\nu}|
s_{i'_\ell}\cdots s_{i'_1}\alpha) \end{split} \] for $i',\,i'_{\nu}\in I$ and $\alpha\in\Delta$.
i) We show by the induction on $k$. We may assume $k\ge 1$. Put $w'=s_{i_1}\cdots s_{i_{k-1}}$ and $\alpha_{\mathbf m'}=s_{i_k}\alpha_{\mathbf m}$ and $\alpha(\nu)=s_{i_{k-1}}\cdots s_{i_{\nu+1}}\alpha_{i_\nu}$ for $\nu=1,\dots,k-1$. The hypothesis of the induction assures $L(w')=k-1$, $\Delta(\mathbf m')=\{\alpha(1),\dots,\alpha(k-1)\}$
and $(\alpha(\nu)|\alpha_{\mathbf m'})>0$ for $\nu=1,\dots,k-1$. If $L(w_{\mathbf m})\ne k$, there exists $\ell$ such that $\alpha_{i_k}=\alpha(\ell)$ and $w_{\mathbf m}=s_{i_1}\cdots s_{i_{\ell-1}}s_{i_{\ell+1}}\cdots s_{i_{k-1}}$ is a minimal expression. Then $h(\alpha_{\mathbf m})-h(\alpha_{\mathbf m'})
=-(\alpha_{i_k}|\alpha_{\mathbf m'})=-(\alpha(\ell)|\alpha_{\mathbf m'})<0$, which contradicts to the definition of $w_{\mathbf m}$. Hence we have i). Note that \eqref{eq:amplus} implies $\supp\alpha\ni\alpha_0$ if $\alpha\in\Delta(\mathbf m)$ and $\mathbf m$ is monotone.
ii) The equality \eqref{eq:Ddelta} follows from \[
\Delta(\partial\mathbf m)\cap\!\!
\sum_{\alpha\in\Pi\setminus\{0\}}\!\!\!\mathbb Z\alpha
=\{\alpha_{j,1}+\cdots+\alpha_{j,\nu_j}\,;\,\nu=1,\dots,\nu_j,\ \nu_j>0
\text{ and }j=0,1,\ldots\} \] because $\Delta(\mathbf m)=s_0\Delta(\partial\mathbf m)\cup\{\alpha_0\}$ and $\Bigl(\prod_{\substack{j\ge 0\\ \nu_j>0}}
s_{j,\nu_j}\cdots s_{j,1}\Bigr)
\alpha_{\partial\mathbf m}=\alpha_{s\partial\mathbf m}$.
The equality \eqref{eq:Dd2} is clear because
$(\alpha_0|\alpha_{\mathbf m})=d_{\mathbf 1}(\mathbf m)=d_{max}(\mathbf m)$
and $(\alpha_{j,\nu}|\alpha_{\mathbf m})=m_{j,\nu+1}-m_{j,\nu}$.
iii) Note that $\gamma\in\Delta(\mathbf m)$ satisfies
$(\gamma|\alpha_{\mathbf m})>0$.
Put $w_\nu=s_{i_{\nu+1}}\cdots s_{i_{k-1}}s_{i_k}$ for $\nu=0,\dots,k$. Then $w_{\mathbf m}=w_0$ and $\Delta(\mathbf m)=\{w_\nu^{-1}\alpha_{i_\nu}\,;\,\nu=1,\dots,k\}$. Moreover $w_{\nu'}w_\nu^{-1}\alpha_{i_\nu}\in\Delta^{re}_-$ if and only if $0\le \nu'<\nu$.
Suppose $\mathbf m$ is not rigid. Let $\alpha\in\Delta^{re}_+$ with $(\alpha|\alpha_{\mathbf m})>0$. Since $(w_{\mathbf m}\alpha|\alpha_{\overline{\mathbf m}})>0$, $w_{\mathbf m}\alpha\in\Delta^{re}_-$. Hence there exists $\nu$ such that $w_{\nu}\alpha\in\Delta_+$ and $w_{\nu-1}\alpha\in\Delta_-$, which implies $w_{\nu}\alpha=\alpha_{i_\nu}$ and the claim.
iv)
Suppose $\mathbf m$ is rigid. Let $\alpha\in\Delta^{re}_+$. Put $\ell=(\alpha|\alpha_{\mathbf m})$. Suppose $\ell>0$ and $\beta:=s_{\alpha}\alpha_{\mathbf m}\in\Delta_+$. Then $\alpha_{\mathbf m}=\ell\alpha+\beta$, $\alpha_0=\ell w_{\mathbf m}\alpha+w_{\mathbf m}\beta$ and
$(\beta|\alpha_{\mathbf m})
=(\alpha_{\mathbf m}-\ell\alpha|\alpha_{\mathbf m})=2-\ell^2$. Hence if $\ell\ge 2$, $\mathbb R\beta\cap\Delta(\mathbf m)=\emptyset$ and the same argument as in the proof of iii) assures $\alpha\in\Delta(\mathbf m)$.
Suppose $\ell=1$. There exists $\nu$ such that $w_{\nu}\alpha$ or $w_{\nu}\beta$ equals $\alpha_{i_\nu}$. We may assume $w_{\nu}^{-1}\alpha=\alpha_{i_\nu}$. Then $\alpha\in\Delta(\mathbf m)$.
Suppose there exists $w_{\nu'}\beta=\alpha_{i_{\nu'}}$. We may assume $\nu'< \nu$. Then $w_{\nu'}\alpha_{\mathbf m}= w_{\nu'-1}\alpha+w_{\nu'-1}\beta\in\Delta^{re}_-$, which contradicts to the definition of $w_\nu$. Hence $w_{\nu'}\beta=\alpha_{i_{\nu'}}$ for $\nu'=1,\dots,k$ and therefore $\beta\notin \Delta(\mathbf m)$.
Let $\gamma=w_\nu^{-1}\alpha_{i_\nu}\in\Delta(\mathbf m)$ and
$(\gamma|\alpha_{\mathbf m})=1$. Put $\beta=\alpha_{\mathbf m}-\alpha=s_{\alpha}\alpha_{\mathbf m}$. Then $w_{\nu-1}\alpha_{\mathbf m}=w_{\nu}\beta\in\Delta^{re}_+$. Since $\beta\notin\Delta(\mathbf m)$, we have $\beta\in\Delta^{re}_+$.
Replacing $\mathbf m$ by $s\mathbf m$,
we may assume $\mathbf m$ is monotone to prove $\alpha_0\in\supp\beta$. Since $(\beta|\alpha_{\mathbf m})=1$ and $(\alpha_i|\alpha_{\mathbf m})\le 0$ for $i\in I\setminus\{0\}$, we have $\alpha_0\in\supp\beta$.
v) The uniqueness of $w_{\mathbf m}$ follows from iii) when $\mathbf m$ is not rigid. It follows from \eqref{eq:amplus}, Theorem~\ref{thm:Nuida} and Corollary~\ref{cor:Nuida} when $\mathbf m$ is rigid. \end{proof}
\begin{cor} Let\/ $\mathbf m$, $\mathbf m'$, $\mathbf m''\in\mathcal P$ and $k\in\mathbb Z_{>0}$ such that \begin{equation}
\mathbf m=k\mathbf m'+\mathbf m'',\ \idx\mathbf m=\idx\mathbf m''
\text{ and\/ $\mathbf m'$ is rigid}. \end{equation} Then\/ $\mathbf m$ is irreducibly realizable if and only if so is $\mathbf m''$.
Suppose\/ $\mathbf m$ is irreducibly realizable. If\/ $\idx\mathbf m\le 0$ or $k>1$, then\/ $\mathbf m'\in\Delta(\mathbf m)$. If $\idx\mathbf m=2$, then\/ $\{\alpha_{\mathbf m'},\,\alpha_{\mathbf m''}\}\cap\Delta(\mathbf m) =\{\alpha_{\mathbf m'}\}$ or $\{\alpha_{\mathbf m''}\}$. \end{cor} \begin{proof}
The assumption implies $(\alpha_{\mathbf m}|\alpha_{\mathbf m})=
2k^2+2k(\alpha_{\mathbf m'}|\alpha_{\mathbf m''})+
(\alpha_{\mathbf m''}|\alpha_{\mathbf m''})$ and hence
$(\alpha_{\mathbf m'}|\alpha_{\mathbf m''})=-k$ and $s_{\alpha_{\mathbf m'}}\alpha_{\mathbf m''}=\alpha_{\mathbf m}$. Thus we have the first claim (cf.~Theorem~\ref{thm:KatzKac}). The remaining claims follow from Proposition~\ref{prop:wm}. \end{proof}
\begin{rem}\label{rem:length} i) \ In general, $\gamma\in\Delta(\mathbf m)$ does not always imply $s_\gamma\alpha_{\mathbf m}\in\Delta_+$.
Put $\mathbf m=32,32,32,32$, $\mathbf m'=10,10,10,10$ and $\mathbf m''=01,01,01,01$. Putting $v=s_{0,1}s_{1,1}s_{2,1}s_{3,1}$, we have $\alpha_{\mathbf m'}=\alpha_0$, $\alpha_{\mathbf m''}=v\alpha_0$,
$(\alpha_{\mathbf m'}|\alpha_{\mathbf m''})=-2$, $s_0\alpha_{\mathbf m''}=2\alpha_{\mathbf m'}+\alpha_{\mathbf m''}$, $vs_0\alpha_{\mathbf m''}=\alpha_0+2\alpha_{\mathbf m''}$ and $s_0vs_0v\alpha_0=s_0vs_0\alpha_{\mathbf m''} =3\alpha_{\mathbf m'}+2\alpha_{\mathbf m''}=\alpha_{\mathbf m}$.
Then $\gamma:= s_0v\alpha_0=2\alpha_{\mathbf m'}+\alpha_{\mathbf m''}\in\Delta(\mathbf m)$,
$(\gamma|\alpha_{\mathbf m})=
(s_0v\alpha_{\mathbf m'}|s_0vs_0v\alpha_{\mathbf m'})
=(\alpha_{\mathbf m'}|s_0v\alpha_{\mathbf m'})
=(\alpha_{\mathbf m'}|2\alpha_{\mathbf m'}+\alpha_{\mathbf m''}) =2$ and $s_\gamma(\alpha_{\mathbf m})= (3\alpha_{\mathbf m'}+2\alpha_{\mathbf m''}) -2(2\alpha_{\mathbf m'}+\alpha_{\mathbf m''})= -\alpha_{\mathbf m'}\in\Delta_-$.
{\rm ii)} Define \begin{equation}\label{eq:tDelta}
\index{000Delta1@$[\Delta(\mathbf m)]$}
[\Delta(\mathbf m)]:=
\bigl\{(\alpha|\alpha_{\mathbf m})\,;\,\alpha\in\Delta(\mathbf m)\bigr\}. \end{equation} Then $[\Delta(\mathbf m)]$ gives a partition of the non-negative integer $h(\alpha_{\mathbf m})-h(sf\mathbf m)$, which we call \textsl{the type of}\/ $\Delta(\mathbf m)$. It follows from \eqref{eq:dht} that \begin{equation}
\#\Delta(\mathbf m)\le h(\alpha_{\mathbf m})-h(sf\mathbf m) \end{equation} for a realizable tuple $\mathbf m$ and the equality holds in the above if $\mathbf m$ is monotone and simply reducible. Moreover we have \begin{align}
[\Delta(\mathbf m)]&=[\Delta(s\partial\mathbf m)]\cup\{d(\mathbf m)\}
\cup\bigcup_{j=0}^p\{m_{j,\nu}-m_{j,1}-d(\mathbf m)\in\mathbb Z_{>0}\,;\,
\nu>1\},\label{eq:DmInd}\\
\#\Delta(\mathbf m)&=
\#\Delta(s\partial\mathbf m)
+\sum_{j=0}^p\Bigl(\min\bigl\{\nu\,;\,m_{j,\nu}>m_{j,1}-d(\mathbf m)\bigr\}
-1\Bigr)+1
,\\
h(\mathbf m)&=h(sf\mathbf m)+\sum_{i\in[\Delta(\mathbf m)]}i\label{eq:sumDelta} \end{align} if $\mathbf m\in\mathcal P_{p+1}$ is monotone, irreducibly realizable and not fundamental. Here we use the notation in Definitions~\ref{def:Sinfty}, \ref{def:pell} and \ref{def:fund}. For example, \[
\begin{tabular}{|c|c|c|c|}\hline type&$\mathbf m$ & $h(\alpha_{\mathbf m})$ & $\#\Delta(\mathbf m)$\\ \hline\hline $H_n\rule[-2pt]{0pt}{12pt}$ & $1^n,1^n,n-11$ & $n^2+1$ & $n^2$ \\ \hline $EO_{2m}\rule[-5pt]{0pt}{15pt}$
& $1^{2m},mm,mm-11$ & {\small $2m^2+3m+1$} & $\binom{2m}{2}+4m$
\\ \hline $\!EO_{2m+1}\rule[-5pt]{0pt}{15pt}\!$
& $1^{2m+1},m+1m,mm1$ & {\small $2m^2+5m+3$} & $\binom{2m+1}{2}+4m+2$\\ \hline $X_6\rule[-2pt]{0pt}{12pt}$ & $111111,222,42$ & 29 & 28 \\ \hline \rule[-2pt]{0pt}{12pt}&$21111,222,33$ & 25 & 24\\ \hline $\!P_n\rule[-2pt]{0pt}{12pt}\!$
& {\small $n-11,n-11,\ldots\in\mathcal P_{n+1}^{(n)}$} & $2n+1$ &
{\!\small$[\Delta(\mathbf m)]:$\,}$1^{n+1}\cdot${\small$(n-1)$}\!\\ \hline
$\!P_{4,2m+1}\!\rule[-2pt]{0pt}{12pt}$
& {\small $m+1m,m+1m,m+1m,m+1m$} & $6m+1$ &
{\small$[\Delta(\mathbf m)]:\,$}$1^{4m}\cdot2^m$\\ \hline \end{tabular} \] \end{rem}
Suppose $\mathbf m\in\mathcal P_{p+1}$ is basic. We may assume \eqref{eq:NTP}.
Suppose $(\alpha_{\mathbf m}|\alpha_0)=0$, which is equivalent to $\sum_{j=0}^p m_{j,1}=(p-1)\ord\mathbf m$. Let $k_j$ be positive integers such that \begin{equation}\label{eq:classic}
(\alpha_{\mathbf m}|\alpha_{j,\nu})=0\text{ \ for \ } 1\le \nu<k_j \text{ \ and \ }
(\alpha_{\mathbf m}|\alpha_{j,k_j})<0, \end{equation} which is equivalent to $m_{j,1}=m_{j,2}=\cdots=m_{j,k_j}>m_{j,k_j+1}$ for $j=0,\dots,p$. Then \begin{equation}\label{eq:R2ineq}
\sum_{j=0}^p\frac1{k_j}\ge
\sum_{j=0}^p\frac{m_{j,1}}{\ord\mathbf m}=p-1. \end{equation} If the equality holds in the above, we have $k_j\ge 2$ and $m_{j,k_j+1}=0$ and therefore $\mathbf m$ is of one of the types $\tilde D_4$ or $\tilde E_6$ or $\tilde E_7$ or $\tilde E_8$. Hence if $\idx\mathbf m<0$, the set $\{k_j\,;\,0\le j\le p,\ k_j> 1\}$ equals one of the set $\emptyset$, $\{2\}$, $\{2,\nu\}$ with $2\le \nu\le 5$, $\{3,\nu\}$ with $3\le\nu\le 5$, $\{2,2,\nu\}$ with $2\le\nu\le 5$ and $\{2,3,\nu\}$ with $3\le\nu\le 5$. In this case the corresponding Dynkin diagram of $\{\alpha_0,\alpha_{j,\nu}\,;\,1\le\nu<k_j,\ j=0,\dots,p\}$ is one of the types $A_\nu$ with $1\le \nu\le 6$, $D_\nu$ with $4\le\nu\le 7$ and $E_\nu$ with $6\le\nu\le 8$. Thus we have the following remark. \begin{rem}\label{rem:classinbas} Suppose a tuple $\mathbf m\in\mathcal P_{p+1}^{(n)}$ is basic and monotone. The subgroup of $W_{\!\infty}$ generated by reflections with respect to $\alpha_\ell$ (cf.~\eqref{eq:alpl})
which satisfy $(\alpha_{\mathbf m}|\alpha_\ell)=0$ is infinite if and only if $\idx\mathbf m=0$.
For a realizable monotone tuple $\mathbf m\in\mathcal P$, we define \index{000Pi@$\Pi,\ \Pi',\ \Pi(\mathbf m)$} \begin{equation}\label{eq:Pim}
\Pi(\mathbf m):=
\{\alpha_{j,\nu}\in\supp\alpha_{\mathbf m}\,;\,
m_{j,\nu}=m_{j,\nu+1}\}\cup
\begin{cases}
\{\alpha_0\}&(d_{\mathbf 1}(\mathbf m)=0),\\
\emptyset&(d_{\mathbf 1}(\mathbf m)\ne 0).
\end{cases} \end{equation}
Note that the condition $(\alpha_{\mathbf m}|\alpha_\ell)=0$, which is equivalent to say that $\alpha_\ell$ is a root of the root space with the fundamental system $\Pi(\mathbf m)$, means that the corresponding middle convolution $\partial_\ell$ keeps the spectral type invariant. \end{rem}
\subsection{Fundamental tuples}\label{sec:basic} We will prove some inequalities \eqref{eq:bineq} and \eqref{eq:b4ineq} for fundamental tuples which are announced in \cite{O3}.
\begin{prop}\label{prop:Bineq} Let\/ $\mathbf m\in\mathcal P_{p+1}\setminus\mathcal P_p$ be a fundamental tuple. Then \begin{align}\label{eq:bineq}
\ord\mathbf m &\le 3|\idx\mathbf m| + 6,\allowdisplaybreaks\\
\label{eq:b4ineq}
\ord\mathbf m&\le |\idx\mathbf m|+2\quad\text{if \ }
p\ge 3,\allowdisplaybreaks\\
p&\le\tfrac12|\idx\mathbf m|+3.\label{eq:pineq} \end{align} \end{prop} \begin{exmp}\label{ex:special} For a positive integer $m$ we have special 4 elements \begin{equation}\index{00D4z@$D_4^{(m)},\ E_6^{(m)},\ E_7^{(m)},\ E_8^{(m)}$} \begin{aligned}\label{eq:Qsp}
&D_4^{(m)}: m^2,m^2,m^2,m(m-1)1&&
\quad E_6^{(m)}: m^3,m^3,m^2(m-1)1\\
&E_7^{(m)}: (2m)^2,m^4,m^3(m-1)1&&
\quad E_8^{(m)}: (3m)^2,(2m)^3,m^5(m-1)1 \end{aligned} \end{equation} with orders $2m$, $3m$, $4m$ and $6m$, respectively, and index of rigidity $2-2m$.
Note that $E_8^{(m)}$, $D_4^{(m)}$ and $11,11,11,\cdots\in\mathcal P_{p+1}^{(2)}$ attain the equalities \eqref{eq:bineq}, \eqref{eq:b4ineq} and \eqref{eq:pineq}, respectively. \end{exmp} \begin{rem}\label{rem:Fbasic} It follows from the Proposition~\ref{prop:Bineq} that there exist only finite basic tuples $\mathbf m\in\mathcal P$ with a fixed index of rigidity under the normalization \eqref{eq:NTP}. This result is given in \cite[Proposition~8.1]{O3}.
\index{Fuchsian differential equation/operator!universal operator!fundamental} Hence there exist only finite \textsl{fundamental universal Fuchsian differential operators} with a fixed number of accessory parameters. Here a fundamental universal Fuchsian differential operator means a universal operator given in Theorem~\ref{thm:univmodel} whose spectral type is fundamental (cf.~Definition~\ref{def:fund}). \end{rem} Now we prepare a lemma. \begin{lem}\label{lem:abck} Let $a\ge 0$, $b>0$ and $c>0$ be integers such that $a+c-b>0$. Then \[
\frac{b+kc-6}{(a+c-b)b}
\begin{cases}
< k+1&(0\le k\le 5),\\
\le 7&(0\le k\le 6).
\end{cases} \] \end{lem} \begin{proof} Suppose $b\ge c$. Then \begin{align*} \frac{b+kc-6}{(a+c-b)b}
&\le \frac{b+kb-6}{b} < k+1. \end{align*} Next suppose $b<c$. Then \begin{align*} (k+1)(a+c-b)b - (b+kc-6)
&\ge (k+1)(c-b)b - b-kc+6\\
&\ge (k+1)b - b - k(b+1)+6 = 6-k. \end{align*} Thus we have the lemma. \end{proof} \begin{proof}[Proof of Proposition~\ref{prop:Bineq}] Since $\idx k\mathbf m=k^2\idx\mathbf m$ for a basic tuple $\mathbf m$ and $k\in\mathbb Z_{>0}$, we may assume that $\mathbf m$ is basic and $\idx\mathbf m\le -2$ to prove the proposition.
Fix a basic monotone tuple $\mathbf m$. Put $\alpha=\alpha_{\mathbf m}$ under the notation \eqref{eq:Kazpart} and $n=\ord\mathbf m$. Note that \begin{equation}\label{eq:DynNorm}
(\alpha|\alpha)=n(\alpha|\alpha_0)+\sum_{j=0}^p\sum_{\nu=1}^{n_j}
n_{j,\nu}(\alpha|\alpha_{j,\nu}),\quad
(\alpha|\alpha_0)\le 0,\quad(\alpha|\alpha_{j,\nu})\le0. \end{equation} We first assume that \eqref{eq:bineq} is not valid, namely, \begin{equation}\label{eq:beq}
3|(\alpha|\alpha)|+6 < n. \end{equation}
In view of \eqref{eq:basic0}, we have $(\alpha|\alpha)<0$
and the assumption implies $|(\alpha|\alpha_0)|=0$
because $|(\alpha|\alpha)|\ge n|(\alpha|\alpha_0)|$.
Let $\Pi_0$ be the connected component of
$\{\alpha_i\in\Pi\,;\,(\alpha|\alpha_i)=0 \text{ and }\alpha_i\in\supp\alpha\}$ containing $\alpha_0$. Note that $\supp\alpha$ generates a root system which is neither classical nor affine but $\Pi_0$ generates a root system of finite type.
Put $J=\{j\,;\,\exists\alpha_{j,\nu}\in\supp\alpha_{\mathbf m}
\text{ such that }(\alpha|\alpha_{j,\nu})<0\}\ne\emptyset$ and for each $j\in J$ define $k_j$ with the condition \eqref{eq:classic}. Then we note that \[
(\alpha|\alpha_{j,\nu})
=\begin{cases}
0 & (1\le \nu < k_j),\\
2n_{j,k_j}-n_{j,k_{j-1}}-n_{j,k_{j+1}}\le -1& (\nu=k_j).
\end{cases} \] Applying the above lemma to $\mathbf m$ by putting $n=b+k_jc$ and $n_{j,\nu}=b+(k_j-\nu)c$ $(1\le \nu \le k_j)$ and $n_{j,k_j+1}=a$, we have \begin{equation}\label{eq:Dynk1}
\frac{n-6}{(n_{j,k_{j-1}}+n_{j,k_{j+1}}-2n_{j,k_j})n_{j,k_j}}
\begin{cases}
< k_j+1 &(1\le k_j\le 5),\\
\le 7 &(1\le k_j\le 6).
\end{cases} \end{equation}
Here $(\alpha|\alpha_{j,k_j})=b-c-a\le -1$ and
we have $|(\alpha|\alpha)| \ge |(\alpha|\alpha_{j,\nu})| > \frac{n-6}{k_j+1}$ if $k_j<6$ and therefore $k_j\ge 3$.
It follows from the condition $k_j\ge 3$ that $\mathbf m\in\mathcal P_3$ because $\Pi_0$ is of finite type and moreover that $\Pi_0$ is of exceptional type, namely, of type $E_6$ or $E_7$ or $E_8$ because $\supp\alpha$ is not of finite type.
Suppose $\#J\ge 2$. We may assume $\{0,1\}\subset J$ and $k_0\le k_1$. Since $\Pi_0$ is of exceptional type and $\supp\alpha$ is not of finite type, we may assume $k_0=3$ and $k_1\le 5$. Owing to \eqref{eq:DynNorm} and \eqref{eq:Dynk1}, we have \begin{align*}
|(\alpha|\alpha)|&\ge
n_{0,3}(n_{0,2}+n_{0,4}-2n_{0,3})
+n_{1,k_1}(n_{1,k_1-1}+n_{1,k_1+1}-2n_{1,k_1})\\
&>\tfrac{n-6}{3+1}+\tfrac{n-6}{5+1}
>\tfrac{n-6}3, \end{align*} which contradicts to the assumption.
Thus we may assume $J=\{0\}$. For $j=1$ and $2$ let $n_j$ be the positive integer such that $\alpha_{j,n_j}\in\supp\alpha$ and $\alpha_{j,n_j+1}\notin\supp\alpha$. We may assume $n_1\ge n_2$.
Fist suppose $k_0=3$. Then $(n_1,n_2)=(2,1)$, $(3,1)$ or $(4,1)$ and the Dynkin diagram of $\supp\alpha$ with the numbers $m_{j,\nu}$ is one of the diagrams: \begin{align*} &\begin{xy}
\ar@{-} , (10,0) *++!D{3m} *{\cdot}*\cir<4pt>{}="F"; \ar@{-} "F"; (20,0) *++!D{4m} *\cir<4pt>{}="A", \ar@{-} "A"; (30,0) *++!D{5m} *\cir<4pt>{}="B", \ar@{-} "B"; (40,0) *+!L+!D{6m}*\cir<4pt>{}="C", \ar@{-} "C"; (50,0) *++!D{4m} *\cir<4pt>{}="D", \ar@{-} "D"; (60,0) *++!D{2m} *\cir<4pt>{}="E", \ar@{-} "C"; (40,10) *++!L{3m} *\cir<4pt>{}
\end{xy}&&|(\alpha|\alpha)|\ge 3m\allowdisplaybreaks\\ &\begin{xy}
(10,8) *{0<k<m}; \ar@{-} (0,0);(10,0) *{\cdot}*++!D{k} *\cir<4pt>{}="H"; \ar@{-} "H";
(20,0) *++!D{m} *{\cdot}*\cir<4pt>{}="F"; \ar@{-} "F"; (30,0) *++!D{2m} *\cir<4pt>{}="A", \ar@{-} "A"; (40,0) *++!D{3m} *\cir<4pt>{}="B", \ar@{-} "B"; (50,0) *+!L+!D{4m}*\cir<4pt>{}="C", \ar@{-} "C"; (60,0) *++!D{3m} *\cir<4pt>{}="D", \ar@{-} "D"; (70,0) *++!D{2m} *\cir<4pt>{}="E", \ar@{-} "E"; (80,0) *++!D{m} *\cir<4pt>{}, \ar@{-} "C"; (50,10) *++!L{2m} *\cir<4pt>{}
\end{xy}&&|(\alpha|\alpha)|\ge 2k(m-k)\allowdisplaybreaks\\ &\begin{xy}
\ar@{-} , (10,0) *++!D{m} *{\cdot}*\cir<4pt>{}="F"; \ar@{-} "F"; (20,0) *++!D{4m} *\cir<4pt>{}="A", \ar@{-} "A"; (30,0) *++!D{7m} *\cir<4pt>{}="B", \ar@{-} "B"; (40,0) *+!L+!D{\!10m}*\cir<4pt>{}="C", \ar@{-} "C"; (50,0) *++!D{\ 8m} *\cir<4pt>{}="D", \ar@{-} "D"; (60,0) *++!D{6m} *\cir<4pt>{}="E", \ar@{-} "E"; (70,0) *++!D{4m} *\cir<4pt>{}="G", \ar@{-} "G"; (80,0) *++!D{2m} *\cir<4pt>{}, \ar@{-} "C"; (40,10) *++!L{5m} *\cir<4pt>{}\ar@{-} \end{xy}
&&|(\alpha|\alpha)|\ge 2m^2 \end{align*} For example, when $(n_1,n_2)=(3,1)$, then $k:=m_{0,4}\ge 1$
because $(\alpha|\alpha_{0,3})\ne 0$ and therefore
$0<k<m$ and $|(\alpha|\alpha)|\ge k(m-2k)+m(2m+k-2m)
=2k(m-k)\ge 2m-2$ and $3|(\alpha|\alpha)|+6 - 4m\ge 3(2m-2)+6-4m > 0$. Hence \eqref{eq:beq} doesn't hold.
Other cases don't happen because of the inequalities $3\cdot 3m +6 - 6m > 0$ and $3\cdot 2m^2 + 6 - 10m > 0$.
Lastly suppose $k_0>3$. Then $(k_0,n_1,n_2)=(4,2,1)$ or $(5,2,1)$. \begin{align*} &\begin{xy}
(10,8) *{m<k<2m}; \ar@{-} (0,0); (10,0) *++!D{k} *{\cdot}*\cir<4pt>{}="F"; \ar@{-} "F"; (20,0) *++!D{2m} *{\cdot}*\cir<4pt>{}="G", \ar@{-} "G"; (30,0) *++!D{3m} *\cir<4pt>{}="H", \ar@{-} "H"; (40,0) *++!D{4m} *\cir<4pt>{}="A", \ar@{-} "A"; (50,0) *++!D{5m} *\cir<4pt>{}="B", \ar@{-} "B"; (60,0) *+!L+!D{6m}*\cir<4pt>{}="C", \ar@{-} "C"; (70,0) *++!D{4m} *\cir<4pt>{}="D", \ar@{-} "D"; (80,0) *++!D{2m} *\cir<4pt>{}, \ar@{-} "C"; (60,10) *++!L{3m} *\cir<4pt>{}, \end{xy}&&
\!|(\alpha|\alpha)|\ge 2m\allowdisplaybreaks\\ &\begin{xy}
(10,8) *{0<k<m}; \ar@{-} (0,0);(10,0) *++!D{k} *{\cdot}*\cir<4pt>{}="F"; \ar@{-} "F"; (20,0) *++!D{m} *{\cdot}*\cir<4pt>{}="G", \ar@{-} "G"; (30,0) *++!D{2m} *\cir<4pt>{}="H", \ar@{-} "H"; (40,0) *++!D{3m} *\cir<4pt>{}="I", \ar@{-} "I"; (50,0) *++!D{4m} *\cir<4pt>{}="A", \ar@{-} "A"; (60,0) *++!D{5m} *\cir<4pt>{}="B", \ar@{-} "B"; (70,0) *+!L+!D{6m}*\cir<4pt>{}="C", \ar@{-} "C"; (80,0) *++!D{4m} *\cir<4pt>{}="D", \ar@{-} "D"; (90,0) *++!D{2m} *\cir<4pt>{}, \ar@{-} "C"; (70,10) *++!L{3m} *\cir<4pt>{} \end{xy}
&&\!|(\alpha|\alpha)|\ge 2(m-1) \end{align*}
In the above first case we have $(\alpha|\alpha)|\ge 2m$, which contradicts
to \eqref{eq:beq}. Note that $(|\alpha|\alpha)|\ge k\cdot(m-2k)+m\cdot k=2k(m-k)\ge 2(m-1)$ in the above last case, which also contradicts to \eqref{eq:beq} because $3\cdot 2(m-1)+ 6 = 6m$.
Thus we have proved \eqref{eq:bineq}.
Assume $\mathbf m\notin\mathcal P_3$
to prove a different inequality \eqref{eq:b4ineq}. In this case, we may assume $(\alpha|\alpha_0)=0$, $|(\alpha|\alpha)|\ge 2$ and $n>4$. Note that \begin{equation}\label{eq:P40}
2n=n_{0,1}+n_{1,1}+\cdots+n_{p,1}\text{ \ with \ }
p\ge 3\text{ and }n_{j,1}\ge 1\text{ for }j=0,\dots,p. \end{equation} If there exists $j$ with $1\le n_{j,1}\le\frac n2-1$, \eqref{eq:b4ineq} follows from \eqref{eq:DynNorm} and $
|(\alpha|\alpha_{j,1})|=n_{j,1}(n+n_{j,2}-2 n_{j,1})\ge
2n_{j,1}(\tfrac n2- n_{j,1})\ge n-2 $.
Hence we may assume $n_{j,1}\ge \frac{n-1}2$ for $j=0,\dots,p$. Suppose there exists $j$ with $n_{j,1}=\frac{n-1}2$. Then $n$ is odd and \eqref{eq:P40} means that there also exists $j'$ with $j\ne j'$ and $n_{j',1}=\frac{n-1}2$. In this case we have \eqref{eq:b4ineq} since \[
|(\alpha|\alpha_{j,1})|+|(\alpha|\alpha_{j',1})|=
n_{j,1}(n+n_{j,2}-2 n_{j,1})+n_{j',1}(n+n_{j',2}-2 n_{j,1})
\ge \tfrac{n-1}2+\tfrac{n-1}2. \]
Now we may assume $n_{j,1}\ge \frac n2$ for $j=0,\dots,p$. Then \eqref{eq:P40} implies that $p=3$ and $n_{j,1}=\frac n2$
for $j=0,\dots,3$. Since $(\alpha|\alpha)<0$, there exists $j$ with $n_{j,2}\ge1$ and \[ \begin{split}
|(\alpha|\alpha_{j,1})|+|(\alpha|\alpha_{j,2})|&=n_{j,1}(n+n_{j,2}-2n_{j,1})
+ n_{j,2}(n_{j,1}+n_{j,3}-2n_{j,2})\\
&=
\tfrac n2 n_{j,2}+n_{j,2}(\tfrac n2+n_{j,3}-2n_{j,2})\\
&
\begin{cases}
\ge n&(n_{j,2}\ge1),\\
= n-2&(n_{j,2}=1\text{ and }n_{j,3}=0).
\end{cases} \end{split} \] Thus we have completed the proof of \eqref{eq:b4ineq}.
There are 4 basic tuples with the index of the rigidity $0$ and 13 basic tuples with the index of the rigidity $-2$, which are given in \eqref{eq:basic0} and Proposition~\ref{prop:bas2}. They satisfy \eqref{eq:pineq}.
Suppose that \eqref{eq:pineq} is not valid. We may assume that $p$ is minimal under this assumption. Then $\idx\mathbf m<-2$, $p\ge 5$ and $n=\ord\mathbf m>2$. We may assume $n>n_{0,1}\ge n_{1,1}\ge\cdots\ge n_{p,1}>0$. Since $(\alpha|\alpha_0)\le 0$, we have \begin{equation}\label{eq:basp}
n_{0,1}+n_{1,1}+\cdots+n_{p,1}\ge 2n>n_{0,1}+\cdots+n_{p-1,1}. \end{equation} In fact, if $n_{0,1}+\cdots+n_{p-1,1}\ge 2n$, the tuple $\mathbf m'=(\mathbf m_0,\dots,\mathbf m_{p-1})$
is also basic and $|(\alpha|\alpha)|-
|(\alpha_{\mathbf m'},\alpha_{\mathbf m'})|=n^2-\sum_{\nu\ge1}n_{p,\nu}^2\ge 2$, which contradicts to the minimality.
Thus we have $2n_{j,1}<n$ for $j=3,\dots,p$. If $n$ is even, $|\idx\mathbf m|\ge\sum_{j=3}^p|(\alpha|\alpha_{j,1})| =\sum_{j=3}^p(n+n_{j,2}-2n_{j,1})\ge 2(p-2)$, which contradicts to the assumption. If $n=3$, \eqref{eq:basp} assures $p=5$ and $n_{0,1}=\cdots=n_{5,0}=1$ and therefore $\idx\mathbf m=-4$, which also contradicts to the assumption. Thus $n=2m+1$ with $m\ge 2$.
Choose $k$ so that $n_{k-1,1}\ge m>n_{k,1}$. Then $|\idx\mathbf m|\ge\sum_{j=k}^p(\alpha|\alpha_{j,1})|=\sum_{j=k}^p (n+n_{j,2}-2n_{j,1})\ge 3(p-k+1)$. Owing to \eqref{eq:basp}, we have $2(2m+1)>km+(p-k)$ and $k< \frac{4m+2-p}{m-1}\le \frac{4m-3}{m-1}\le 5$,
which means $k\le 4$, $|\idx\mathbf m|\ge 3(p-3)\ge 2p-4$ and a contradiction to the assumption. \end{proof}
\section{Expression of local solutions}\label{sec:exp} Fix $\mathbf m=\bigl(m_{j,\nu}\bigr)_{\substack{j=0,\dots,p\\1\le\nu\le n_j}} \in\mathcal P_{p+1}$. Suppose $\mathbf m$ is monotone and irreducibly realizable. Let $P_{\mathbf m}$ be the universal operator with the Riemann scheme \eqref{eq:GRS}, which is given in Theorem~\ref{thm:univmodel}. Suppose $c_1=0$ and $m_{1,n_1}=1$. We give expressions of the local solution of $P_{\mathbf m}u=0$ at $x=0$ corresponding to the characteristic exponent $\lambda_{1,n_1}$.
\begin{thm}\label{thm:expsol} Retain the notation above and in Definition~\ref{def:redGRS}. Suppose $\lambda_{j,\nu}$ are generic. Let \begin{equation}
v(x) = \sum_{\nu=0}^\infty C_\nu x^{\lambda(K)_{1,n_1}+\nu} \end{equation} be the local solution of $\bigl(\partial_{\max}^KP_{\mathbf m}\bigr)v=0$ at $x=0$ with the condition $C_0=1$. Put \begin{equation}
\lambda(k)_{j,max}=\lambda(k)_{j,\ell(k)_j}. \end{equation} Note that if\/ $\mathbf m$ is rigid, then \begin{equation}
v(x) = x^{\lambda(K)_{1,n_1}}\prod_{j=2}^p
\Bigl(1-\frac x{c_j}\Bigr)^{\lambda(K)_{j,max}}. \end{equation} The function \begin{equation}\label{eq:intexp}
\begin{split}
u(x)&:=\prod_{k=0}^{K-1}
\frac{\Gamma\bigl(\lambda(k)_{1,n_1}-\lambda(k)_{1,max}+1\bigr)}
{\Gamma\bigl(\lambda(k)_{1,n_1}-\lambda(k)_{1,max}+\mu(k)+1\bigr)
\Gamma\bigl(-\mu(k)\bigr)} \\
&\int_0^{s_0}\cdots \int_0^{s_{K-1}}
\prod_{k=0}^{K-1}(s_k-s_{k+1})^{-\mu(k)-1}\\
&\quad\cdot\prod_{k=0}^{K-1}\biggl(
\Bigl(\frac{s_k}{s_{k+1}}\Bigr)^{\lambda(k)_{1,max}}
\prod_{j=2}^p\Bigl(\frac{1-c_j^{-1}s_k}{1-c_j^{-1}s_{k+1}}\Bigr)
^{\lambda(k)_{j,{max}}}
\biggr)\\
&\quad\cdot v(s_K)ds_K\cdots ds_1\Bigl|_{s_0=x}
\end{split} \end{equation} is the solution of $P_{\mathbf m}u=0$ so normalized that $u(x)\equiv x^{\lambda_{1,n_1}}\mod x^{\lambda_{1,n_1}+1}\mathcal O_0$.
Here we note that \begin{equation}\label{eq:solrepsub}
\begin{split}
&\prod_{k=0}^{K-1}\biggl(
\Bigl(\frac{s_k}{s_{k+1}}\Bigr)^{\lambda(k)_{1,max}}
\prod_{j=2}^p\Bigl(\frac{1-c_j^{-1}s_k}{1-c_j^{-1}s_{k+1}}\Bigr)
^{\lambda(k)_{j,max}}
\biggr)\\
&\quad=\frac{s_0^{\lambda(0)_{1,max}}}
{s_K^{\lambda(K-1)_{1,max}}}
\prod_{j=1}^p
\frac{(1-c_j^{-1}s_0)^{\lambda(0)_{j,{max}}}}
{(1-c_j^{-1}s_K)^{\lambda(K-1)_{j,{max}}}}\\
&\qquad\cdot\prod_{k=1}^{K-1}
\Bigl(s_k^{\lambda(k)_{1,{max}}-\lambda(k-1)_{1,{max}}
}
\prod_{j=2}^{p}
(1-c_j^{-1}s_k)
^{\lambda(k)_{j,{max}}-\lambda(k-1)_{j,{max}}}\Bigr).
\end{split} \end{equation} When\/ $\mathbf m$ is rigid, \begin{equation}\label{eq:serexpr} \begin{split}
u(x)&= x^{\lambda_{1,n_1}}
\biggl(\prod_{j=2}^p\Bigl(1-\frac x{c_j}\Bigr)^{\lambda(0)_{j,{max}}}\biggr)
\sum_{\bigl(\nu_{j,k}\bigr)
_{\substack{2\le j\le p\\ 1\le k\le K}}\in\mathbb Z_{\ge 0}^{(p-1)K}}
\\&\quad
\prod_{i=0}^{K-1}
\frac
{\bigl(\lambda(i)_{1,n_1}-\lambda(i)_{1,{max}}+1\bigr)
_{\sum_{s=2}^p\sum_{t=i+1}^{K}\nu_{s,t}}}
{\bigl(\lambda(i)_{1,n_1}-\lambda(i)_{1,{max}}+\mu(i)+1\bigr)
_{\sum_{s=2}^p\sum_{t=i+1}^{K}\nu_{s,t}}}
\\&
\quad
\cdot\prod_{i=1}^K\prod_{s=2}^p\frac{
\bigl(\lambda(i-1)_{s,{max}}-\lambda(i)_{s,{max}}\bigr)_{\nu_{s,i}}}
{\nu_{s,i}!}\cdot
\prod_{s=2}^p\Bigl(\frac{x}{c_s}\Bigr)^{\sum_{i=1}^K\nu_{s,i}}. \end{split} \end{equation} When\/ $\mathbf m$ is not rigid \begin{equation}\label{eq:serexp} \begin{split}
u(x)&= x^{\lambda_{1,n_1}}
\biggl(\prod_{j=2}^p\Bigl(1-\frac x{c_j}\Bigr)^{\lambda(0)_{j,{max}}}\biggr)
\sum_{\nu_0=0}^\infty\sum_{\bigl(\nu_{j,k}\bigr)
_{\substack{2\le j\le p\\ 1\le k\le K}}\in\mathbb Z_{\ge 0}^{(p-1)K}}
\\&\quad
\prod_{i=0}^{K-1}
\frac
{\bigl(\lambda(i)_{1,n_1}-\lambda(i)_{1,{max}}+1\bigr)
_{\nu_0+\sum_{s=2}^p\sum_{t=i+1}^{K}\nu_{s,t}}}
{\bigl(\lambda(i)_{1,n_1}-\lambda(i)_{1,{max}}+\mu(i)+1\bigr)
_{\nu_0+\sum_{s=2}^p\sum_{t=i+1}^{K}\nu_{s,t}}}\\
&
\quad\cdot
\prod_{s=2}^p\frac{
\bigl(\lambda(K-1)_{s,{max}}\bigr)_{\nu_{s,K}}
}{\nu_{s,K}!}\cdot
\prod_{i=1}^{K-1}\prod_{s=2}^p\frac{
\bigl(\lambda(i-1)_{s,{max}}
-\lambda(i)_{s,{max}}\bigr)_{\nu_{s,i}}
}{\nu_{s,i}!}
\\&\quad\cdot
C_{\nu_0}x^{\nu_0}\prod_{s=2}^p\Bigl(\frac{x}{c_s}\Bigr)^{\sum_{i=1}^K\nu_{s,i}}. \end{split} \end{equation}
Fix $j$ and $k$ and suppose \begin{equation}
\begin{cases}
\ell(k-1)_j=\ell(k)_\nu&\text{when }\mathbf m\text{ is rigid or }k<K,\\
\ell(k-1)_j=0&\text{when }\mathbf m\text{ is not rigid and }k=K.
\end{cases} \end{equation} Then the terms satisfying $\nu_{j,k}>0$ vanish because $(0)_{\nu_{j,k}}= \delta_{0,\nu_{j,k}}$ for $\nu_{j,k}=0,1,2,\ldots$. \end{thm} \begin{proof} The theorem follows from \eqref{eq:opred}, \eqref{eq:pellP2}, \eqref{eq:defpell}, \eqref{eq:seriesP} and \eqref{eq:IcP} by the induction on $K$. Note that the integral representation of the normalized solution of $\bigr(\partial_{max}P\bigr)v=0$ corresponding to the exponent $\lambda(1)_{n_1}$ equals \[
\begin{split}
v(x)&:=\prod_{k=1}^{K-1}
\frac{\Gamma\bigl(\lambda(k)_{1,n_1}-\lambda(k)_{1,max}+1\bigr)}
{\Gamma\bigl(\lambda(k)_{1,n_1}-\lambda(k)_{1,max}+\mu(k)+1\bigr)
\Gamma\bigl(-\mu(k)\bigr)}\\
&\quad\cdot\int_0^{s_1}\cdots \int_0^{s_{K-1}}
\prod_{k=0}^{K-1}(s_k-s_{k+1})^{-\mu(k)-1}\\
&\quad\cdot\prod_{k=0}^{K-1}\biggl(
\Bigl(\frac{s_k}{s_{k+1}}\Bigr)^{\lambda(k)_{1,max}}
\prod_{j=2}^p\Bigl(\frac{1-c_j^{-1}s_k}{1-c_j^{-1}s_{k+1}}\Bigr)
^{\lambda(k)_{j,{max}}}
\biggr)\\
&\quad\cdot v(s_K)ds_K\cdots ds_1\Bigl|_{s_1=x}\\
&\equiv x^{\lambda(1)_{1,n_1}}\mod x^{\lambda(1)_{1,n_1}+1}\mathcal O_0
\end{split} \] by the induction hypothesis and the normalized solution of $Pu=0$ corresponding to the exponent $\lambda_{1,n_1}$ equals \[
\begin{split}
&\frac{\Gamma\bigl(\lambda(0)_{1,n_1}-\lambda(0)_{1,max}+1\bigr)}
{\Gamma\bigl(\lambda(0)_{1,n_1}-\lambda(0)_{1,max}+\mu(0)+1\bigr)
\Gamma\bigl(-\mu(0)\bigr)}\\
&\quad{}\cdot\int_0^x(x-s_0)^{-\mu(0)-1}\frac{x^{-\lambda(0)_{1,\max}}}
{s_0^{-\lambda(0)_{1,\max}}}\prod_{j=2}^p
\Bigl(\frac{1-c_j^{-1}x}{1-c_j^{-1}s_0}\Bigr)^{-\lambda(0)_{j,\max}}
v(s_0)ds_0
\end{split} \] and hence we have \eqref{eq:intexp}. Then the integral expression \eqref{eq:intexp} with \eqref{eq:solrepsub}, \eqref{eq:seriesP} and \eqref{eq:IcP} inductively proves \eqref{eq:serexpr} and \eqref{eq:serexp}. \end{proof}
\begin{exmp}[Gauss hypergeometric equation]\label{ex:localGauss} \index{hypergeometric equation/function!Gauss} The reduction \eqref{eq:redGG} shows \begin{align*}
\lambda(0)_{j,\nu}&=\lambda_{j,\nu},\
m(0)_{j,\nu} = 1 \quad(0\le j\le 2,\ 1\le\nu\le 2),\
\mu(0) =-\lambda_{0,2}-\lambda_{1,2}-\lambda_{2,2},\\
m(1)_{j,1}&=0,\ m(1)_{j,2}=1\quad(j=0,1,2),\\
\lambda(1)_{0,1} &=\lambda_{0,1}+2\lambda_{0,2}+2\lambda_{1,2}+2\lambda_{2,2},\
\lambda(1)_{1,1}=\lambda_{1,1},\ \lambda(1)_{2,1}=\lambda_{2,1},\\
\lambda(1)_{0,2} &=2\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,2},\
\lambda(1)_{1,2} =-\lambda_{0,2}-\lambda_{2,2},\
\lambda(1)_{2,2} =-\lambda_{0,2}-\lambda_{1,2} \end{align*} and therefore \begin{align*}
\lambda(0)_{1,n_1}-\lambda(0)_{1,max}+\mu(0)+1
&=\lambda_{1,2}-\lambda_{1,1}-(\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,2})+1\\
&=\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,1},
\allowdisplaybreaks\\
\lambda(0)_{2,max}-\lambda(1)_{2,max}&=\lambda(0)_{2,1}-\lambda(1)_{2,2}
=\lambda_{2,1}+\lambda_{0,2}+\lambda_{1,2}. \end{align*} Hence \eqref{eq:intexp} says that the normalized local solution corresponding to the characteristic exponent $\lambda_{1,2}$ with $c_1=0$ and $c_2=1$ equals \begin{equation}\label{eq:gaussIrep} \begin{split}
u(x)
&=\frac{\Gamma\bigl(\lambda_{1,2}-\lambda_{1,1}+1\bigr)x^{\lambda_{1,1}}
(1-x)^{\lambda_{2,1}}}
{\Gamma\bigl(\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,1}\bigr)
\Gamma\bigl(\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,2}\bigr)}\\
&\quad\int_0^x(x-s)^{\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,2}-1}
s^{-\lambda_{0,2}-\lambda_{1,1}-\lambda_{2,2}}
(1-s)^{-\lambda_{0,2}-\lambda_{1,2} - \lambda_{2,1}}ds \end{split}\end{equation} and moreover \eqref{eq:serexpr} says \begin{equation}\begin{split}
u(x)&=x^{\lambda_{1,2}}(1-x)^{\lambda_{2,1}}
\sum_{\nu=0}^\infty\frac{(\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,1})_\nu
(\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,1})_\nu}
{(\lambda_{1,2}-\lambda_{1,1}+1)_\nu\nu!}x^\nu. \end{split}\end{equation} Note that $u(x)=F(a,b,c;x)$ when \begin{equation}\label{eq:RSgauss}
\begin{Bmatrix}
x=\infty & 0 & 1\\
\lambda_{0,1} & \lambda_{1,1} & \lambda_{2,1}\\
\lambda_{0,2} & \lambda_{1,2} & \lambda_{2,2}
\end{Bmatrix}=
\begin{Bmatrix}
x=\infty & 0 & 1\\
a & 1-c & 0\\
b & 0 & c-a-b
\end{Bmatrix}. \end{equation} The integral expression \eqref{eq:gaussIrep} is based on the minimal expression $w=s_{0,1}s_{1,1}s_{1,2}s_0$ satisfying $w\alpha_{\mathbf m}=\alpha_0$. Here $\alpha_{\mathbf m}=2\alpha_0+\sum_{j=0}^2\alpha_{j,1}$. When we replace $w$ and its minimal expression by $w'=s_{0,1}s_{1,1}s_{1,2}s_0s_{0,1}$ or $w''=s_{0,1}s_{1,1}s_{1,2}s_0s_{2,1}$, we get the different integral expressions \begin{equation}\begin{split}
u(x)
&=\frac{\Gamma\bigl(\lambda_{1,2}-\lambda_{1,1}+1\bigr)x^{\lambda_{1,1}}
(1-x)^{\lambda_{2,1}}}
{\Gamma\bigl(\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,1}\bigr)
\Gamma\bigl(\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,2}\bigr)}\\
&\quad\int_0^x(x-s)^{\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,2}-1}
s^{-\lambda_{0,1}-\lambda_{1,1}-\lambda_{2,2}}
(1-s)^{-\lambda_{0,1}-\lambda_{1,2} - \lambda_{2,1}}ds \allowdisplaybreaks\\
&=\frac{\Gamma\bigl(\lambda_{1,2}-\lambda_{1,1}+1\bigr)x^{\lambda_{1,1}}
(1-x)^{\lambda_{2,2}}}
{\Gamma\bigl(\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,2}\bigr)
\Gamma\bigl(\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,1}\bigr)}\\
&\quad\int_0^x(x-s)^{\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,1}-1}
s^{-\lambda_{0,2}-\lambda_{1,1}-\lambda_{2,1}}
(1-s)^{-\lambda_{0,2}-\lambda_{1,2} - \lambda_{2,2}}ds. \end{split}\end{equation} \index{hypergeometric equation/function!Gauss!integral expression} These give different integral expressions of $F(a,b,c;x)$ under \eqref{eq:RSgauss}.
Since $s_{\alpha_0+\alpha_{0,1}+\alpha_{0,2}}\alpha_{\mathbf m} =\alpha_{\mathbf m}$, we have \begin{align*} &\begin{Bmatrix}
x=\infty & 0 & 1\\
a & 1-c & 0\\
b & 0 & c-a-b \end{Bmatrix}
\xrightarrow{x^{c-1}} \begin{Bmatrix}
x=\infty & 0 & 1\\
a-c+1 & 0 &0\\
b-c+1 & c-1 &c-a-b \end{Bmatrix}\\ &\quad \xrightarrow{\partial^{c-d}} \begin{Bmatrix}
x=\infty & 0 & 1\\
a-d+1 & 0 &0\\
b-d+1 & d-1 &d-a-b \end{Bmatrix}
\xrightarrow{x^{1-d}} \begin{Bmatrix}
x=\infty & 0 & 1\\
a & 1-d & 0\\
b & 0 & d-a-b \end{Bmatrix} \end{align*} and hence (cf.~\eqref{eq:IcP}) \index{hypergeometric equation/function!Gauss!Euler transformation} \begin{equation} F(a,b,d;x)=\frac{\Gamma(d)x^{1-d}}{\Gamma(c)\Gamma(d-c)}
\int_0^x (x-s)^{d-c-1}s^{c-1}F(a,b,c;s)ds. \end{equation} \end{exmp}
\begin{rem}\label{rem:Irep} The integral expression of the local solution $u(x)$ as is given in Theorem~\ref{thm:expsol} is obtained from the expression of the element $w$ of $W_{\!\infty}$ satisfying $w\alpha_{\mathbf m}\in B\cup\{\alpha_0\}$ as a product of simple reflections and therefore the integral expression depends on such element $w$ and the expression of $w$ as such product. The dependence on $w$ seems non-trivial as in the preceding example but the dependence on the expression of $w$ as a product of simple reflections is understood as follows.
First note that the integral expression doesn't depend on the coordinate transformations $x\mapsto ax$ and $x\mapsto x+b$ with $a\in\mathbb C^\times$ and $b\in\mathbb C$. Since \begin{align*}
\int_c^x(x-t)^{\mu-1}\phi(t)dt
&=-\int_{\frac1c}^{\frac1x}(x-\tfrac1s)^{\mu-1}\phi(\tfrac1s)s^{-2}ds\\
&=-(-1)^{\mu-1}x^{\mu-1}\int_{\frac1c}^{\frac1x}(\tfrac1x-s)^{\mu-1}
(\tfrac1s)^{\mu+1}\phi(\tfrac1s)ds, \end{align*} we have \begin{equation}\label{eq:Iinv}
I_c^\mu(\phi)=-(-1)^{\mu-1}x^{\mu-1}\left.
\left(I_{\frac1c}^x\left.
\bigl(x^{\mu+1}\phi(x)\bigr)\right|_{x\mapsto \frac1x}\right)
\right|_{x\to\frac1x}, \end{equation} which corresponds to \eqref{eq:redcoord}. Here the value $(-1)^{\mu-1}$ depends on the branch of the value of $(x-\frac1s)^{\mu-1}$ and that of $x^{\mu-1}x^{1-\mu}(\frac1x -s)^{\mu-1}$.
Hence the argument as in the proof of Theorem~\ref{thm:KatzKac} shows that the dependence on the expression of $w$ by a product of simple reflections can be understood by the identities \eqref{eq:Iinv} and $I_c^{\mu_1}I_c^{\mu_2}=I_c^{\mu_1+\mu_2}$ (cf.~\eqref{eq:Icprod}) etc. \end{rem}
\section{Monodromy} \label{sec:monodromy} The transformation of monodromy generators for irreducible Fuchsian systems of Schlesinger canonical form under the middle convolution or the addition is studied by \cite{Kz} and \cite{DR, DR2} etc. A non-zero homomorphism of an irreducible single Fuchsian differential equation to an irreducible system of Schlesinger canonical form induces the isomorphism of their monodromies of the solutions (cf.~Remark~\ref{rem:SCFmc}). In particular since any rigid local system is realized by a single Fuchsian differential equation, their monodromies naturally coincide with each other through the correspondence of their monodromy generators. The correspondence between the local monodromies and the global monodromies is described by \cite{DR2}, which we will review.
\subsection{Middle convolution of monodromies}\label{sec:MM} For given matrices $A_j\in M(n,\mathbb C)$ for $j=1,\dots,p$ the Fuchsian system \begin{equation}\label{eq:MSCF}
\frac{dv}{dx}=\sum_{j=1}^p\frac{A_j}{x-c_j}v \end{equation} of Schlesinger canonical form (SCF) is defined. Put $A_0=-A_1-\dots-A_p$ and $\mathbf A=(A_0,A_1,\dots,A_p)$ which is an element of \begin{equation}\label{eq:Msub0}
M(n,\mathbb C)^{p+1}_0:=\{(C_0,\dots,C_p)\in
M(n,\mathbb C)^{p+1}\,;\,C_0+\cdots+C_p=0\}, \end{equation} The Riemann scheme of \eqref{eq:MSCF} is defined by \begin{equation} \begin{Bmatrix}
x = c_0=\infty & c_1&\cdots&c_p\\
[\lambda_{0,1}]_{m_{0,1}} & [\lambda_{1,1}]_{m_{1,1}} &\cdots
&[\lambda_{p,1}]_{m_{p,1}}\\
\vdots & \vdots & \vdots & \vdots\\
[\lambda_{0,n_0}]_{m_{0,n_0}} & [\lambda_{1,n_1}]_{m_{1,n_1}} &\cdots
&[\lambda_{p,n_p}]_{m_{p,1}} \end{Bmatrix},\quad [\lambda]_k:=\begin{pmatrix}
\lambda\\
\vdots\\
\lambda
\end{pmatrix}\in M(1,k,\mathbb C) \end{equation} if \[
A_j \sim L(m_{j,1},\ldots,m_{j,n_j};\lambda_{j,1},\dots,\lambda_{j,n_j})
\quad(j=0,\dots,p) \] under the notation \eqref{eq:OSNF}. Here the Fuchs relation equals \begin{equation}
\sum _{j=0}^p\sum_{\nu=1}^{n_j} m_{j,\nu}\lambda_{j,\nu}=0. \end{equation}
We define that $\mathbf A$ is \textsl{irreducible} if a subspace $V$ of $\mathbb C^n$ satisfies $A_jV\subset A_j$ for $j=0,\dots,p$, then $V=\{0\}$ or $V=\mathbb C^n$. In general, $\mathbf A=(A_0,\dots,A_p)$, $\mathbf A'=(A'_0,\dots,A'_p)\in M(n,\mathbb C)^{p+1}$, we denote by $\mathbf A\sim\mathbf A'$ if there exists $U\in GL(n,\mathbb C)$ such that $A'_j=U A_jU^{-1}$ for $j=0,\dots,p$. \index{monodromy!irreducible}
For $(\mu_0,\dots,\mu_p)\in \mathbb C^{p+1}$ with $\mu_0+\cdots+\mu_p=0$, the addition $\mathbf A'=(A'_0,\dots,A'_p)\in M(n,\mathbb C)^{p+1}_0$ of $\mathbf A$ with respect to $(\mu_0,\dots,\mu_p)$ is defined by $A'_j=A_j+\mu_j$ for $j=0,\dots,p$.
For a complex number $\mu$ the middle convolution $\bar{\mathbf A}:=mc_\mu(\mathbf A)$ of $\mathbf A$ is defined by $\bar A_j=\bar A_j(\mu)$ for $j=1,\dots,p$ and $\bar A_0=-\bar A_1-\dots-\bar A_p$ under the notation in \S\ref{sec:DR}. Then we have the following theorem.
\begin{thm}[\cite{DR, DR2}]\label{thm:schmid} Suppose that $\mathbf A$ satisfies the conditions \begin{align}
\bigcap_{\substack{1\le j\le p\\ j\ne i}}\ker A_j
\cap \ker(A_0-\tau)&=\{0\}&(i=1,\dots,p,\ \forall\tau\in\mathbb C),
\label{eq:star}\\
\bigcap_{\substack{1\le j\le p\\ j\ne i}}\ker {}^t\!A_j
\cap \ker({}^t\!A_0-\tau)&=\{0\}
&(i=1,\dots,p,\ \forall\tau\in\mathbb C).
\label{eq:starstar} \end{align}
{\rm i)} The tuple $mc_\mu(\mathbf A)=(\bar A_0,\dots,\bar A_p)$ also satisfies the same conditions as above with replacing $A_\nu$ by $\bar A_\nu$ for $\nu=0,\dots,p$, respectively. Moreover we have \begin{align}
mc_\mu(\mathbf A)&\sim mc_\mu(\mathbf A')\text{ \ if \ }\mathbf A\sim\mathbf A',\\
mc_{\mu'}\circ mc_{\mu}(\mathbf A)&\sim mc_{\mu+\mu'}(\mathbf A),\\
mc_0(\mathbf A)&\sim \mathbf A \end{align} and $\mathbf A$ is irreducible if and only if $\mathbf A'$ is irreducible.
{\rm ii) (cf.~\cite[Theorem~5.2]{O3})} Assume \begin{equation}
\mu=\lambda_{0,1}\ne0\text{ \ and \ } \lambda_{j,1}=0\text{ \ for \ }j=1,\dots,p \end{equation} and \begin{align}
\lambda_{j,\nu}=\lambda_{j,1}\text{ \ implies \ }m_{j,\nu}\le m_{j,1} \end{align} for $j=0,\dots,p$ and $\nu=2,\dots,n_j$. Then the Riemann scheme of $mc_{\mu}(\mathbf A)$ equals \begin{gather} \begin{Bmatrix} x = \infty & c_1&\cdots&c_p\\
[-\mu]_{m_{0,1}-d} & [0]_{m_{1,1}-d} &\cdots
&[0]_{m_{p,1}-d}\\
[\lambda_{0,2}-\mu]_{m_{0,2}} & [\lambda_{1,2}+\mu]_{m_{1,2}} &\cdots
&[\lambda_{p,2}+\mu]_{m_{p,2}}\\
\vdots & \vdots & \vdots & \vdots\\
[\lambda_{0,n_0}-\mu]_{m_{0,n_0}} & [\lambda_{1,n_1}+\mu]_{m_{1,n_1}} &\cdots
&[\lambda_{p,n_p}+\mu]_{m_{p,1}} \end{Bmatrix} \intertext{with} d:=m_{0,1}+\cdots+m_{p,1}-(p-1)\ord\mathbf m.\label{eq:defd2} \end{gather} \end{thm}
\begin{exmp}\label{ex:univSch} The addition of \[
mc_{-\lambda_{0,1}-\lambda_{1,2}-\lambda_{2,2}}(\{\lambda_{0,2}-\lambda_{0,1},
\lambda_{0,1}+\lambda_{1,1}+\lambda_{2,2},
\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,1}\}) \] with respect to $(-\lambda_{1,2}-\lambda_{2,2},\lambda_{1,2},\lambda_{2,2})$ give the Fuchsian system of Schlesinger canonical form \[
\begin{gathered}
\frac{du}{dx}=\frac{A_1}{x}u+\frac{A_2}{x-1}u,\\
A_1=\begin{pmatrix}
\lambda_{1,1} & \lambda_{0,1}+\lambda_{1,2}+\lambda_{2,1}\\
& \lambda_{1,2}
\end{pmatrix}\text{ \ and \ }
A_2=\begin{pmatrix}
\lambda_{2,2} & \\
\lambda_{0,1}+\lambda_{1,1}+\lambda_{2,2} & \lambda_{2,1}\\
\end{pmatrix}.
\end{gathered} \] with the Riemann scheme \[
\begin{Bmatrix}
x=\infty & 0 & 1\\
\lambda_{0,1} & \lambda_{1,1} & \lambda_{2,1}\\
\lambda_{0,2} & \lambda_{1,2} & \lambda_{2,2}\\
\end{Bmatrix} \qquad
(\lambda_{0,1}+\lambda_{0,2}+\lambda_{1,1}+\lambda_{1,2}
+\lambda_{2,1}+\lambda_{2,2}=0). \] The system is invariant as $W(x;\lambda_{j,\nu})$-modules under the transformation $\lambda_{j,\nu}\mapsto\lambda_{j,3-\nu}$ for $j=0,1,2$ and $\nu=1,2$.
Suppose $\lambda_{j,\nu}$ are generic complex numbers under the condition $
\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,1}
=\lambda_{0,2}+\lambda_{1,1}+\lambda_{2,2}=0. $ Then $A_1$ and $A_2$ have a unique simultaneous eigenspace. In fact, $A_1\binom 01=\lambda_{1,2}\binom 01$ and $A_2\binom 01=\lambda_{2,1}\binom 01$. Hence the system is not invariant as $W(x)$-modules under the transformation above and $\mathbf A$ is not irreducible in this case. \end{exmp}
To describe the monodromies, we review the multiplicative version of these operations.
Let $\mathbf M=(M_0,\dots,M_p)$ be an element of \begin{equation}\label{eq:Gsub1}
GL(n,\mathbb C)^{p+1}_1:=\{(G_0,\dots,G_p)\in GL(n,\mathbb C)^{p+1}\,;\,
G_p\cdots G_0=I_n\}. \end{equation} For $(\rho_0,\dots,\rho_p)\in\mathbb C^{p+1}$ satisfying $\rho_0\cdots \rho_p=1$, the \textsl{multiplication\/} of $\mathbf M$ with respect to $\rho$ is defined by $(\rho_0M_0,\dots,\rho_pM_p)$.
For a given $\rho\in\mathbb C^\times$, we define $\tilde M_j=\bigl(M_{j,\nu,\nu'}\bigr)_{
{\substack{1\le\nu\le n\\1\le \nu'\le p}}}\in GL(pn,\mathbb C)$ by \begin{align*}
\tilde M_{j,\nu,\nu'} =
\begin{cases}
\delta_{\nu,\nu'}I_n &(\nu\ne j),\\
M_{\nu'}-1 &(\nu=j,\ 1\le\nu' \le j-1),\\
\rho M_j&(\nu=\nu'=j),\\
\rho(M_{\nu'}-1)&(\nu=j,\ j+1\le\nu'\le p).
\end{cases} \end{align*}
Let $\bar M_j$ denote the quotient $\tilde M_j|_{\mathbb C^{pn}/V}$ of \begin{equation}\label{eq:conM}
\tilde M_j=\begin{pmatrix}
I_n\\
&\ddots\\
M_1-1 & \cdots &\rho M_j & \cdots &\rho(M_p-1)\\
& & & \ddots \\
& & & & I_n
\end{pmatrix}\in GL(pn,\mathbb C) \end{equation} for $j=1,\dots,p$ and $M_0=(M_p\dots M_1)^{-1}$. The tuple $\MC_\rho(\mathbf M)=(\bar M_0,\dots,\bar M_p)$ is called (the multiplicative version of) the middle convolution of $\mathbf M$ with respect to $\rho$. Here $V:= \ker(\tilde M-1)+\bigcap_{j=1}^p \ker(\tilde M_j-1)$ with \[
\tilde M := \begin{pmatrix}
M_1\\
&\ddots\\
& & M_p
\end{pmatrix}. \] Then we have the following theorem. \begin{thm}[\cite{DR,DR2}]\label{thm:Mmid} Let $\mathbf M=(M_0,\dots,M_p)\in GL(n,\mathbb C)^{p+1}_1$. Suppose \begin{align}
\bigcap_{\substack{1\le\nu\le p\\ \nu\le i}}\ker (M_\nu-1)\cap\ker(M_i-\tau)
&=\{0\} &&(1\le i\le p,\ \forall\tau\in\mathbb C^\times),\label{eq:mulstar}\\
\bigcap_{\substack{1\le\nu\le p\\ \nu\le i}}\ker ({}^t\!M_\nu-1)\cap\ker({}^t\!M_i-\tau)
&=\{0\}
&&(1\le i\le p,\ \forall\tau\in\mathbb C^\times).\label{eq:mulss} \end{align}
{\rm i)} The tuple $\MC_\rho(\mathbf M)=(\bar M_0,\dots,\bar M_p)$ also satisfies the same conditions as above with replacing $M_\nu$ by $\bar M_\nu$ for $\nu=0,\dots,p$, respectively. Moreover we have \begin{align}
\MC_\rho(\mathbf M)&\sim \MC_\rho(\mathbf M')
\text{ \ if \ }\mathbf M\sim\mathbf M',\\
\MC_{\rho'}\circ \MC_{\rho}(\mathbf M)&\sim \MC_{\rho\rho'}(\mathbf M),\\
\MC_1(\mathbf M)&\sim \mathbf M \end{align} and $\MC_\rho(\mathbf M)$ is irreducible if and only if $\mathbf M$ is irreducible.
{\rm ii)} Assume \begin{align}
M_j&\sim L(m_{j,1},\dots,m_{j,n_j};\rho_{j,1},\dots,\rho_{j,n_j})
\text{ \ for \ }j=0,\dots,p,\label{eq:Lisom}\\
\rho&=\rho_{0,1}\ne1\text{ \ and \ } \rho_{j,1}=1\text{ \ for \ }j=1,\dots,p \end{align} and \begin{align}
\rho_{j,\nu}=\rho_{j,1}\text{ \ implies \ }m_{j,\nu}\le m_{j,1} \end{align} for $j=0,\dots,p$ and $\nu=2,\dots,n_j$. In this case, we say that $\mathbf M$ has a spectral type $\mathbf m:=(\mathbf m_0,\dots,\mathbf m_p)$ with $\mathbf m_j=(m_{j,1},\ldots,m_{j,n_j})$.
Putting $(\bar M_0,\dots,\bar M_p)=\MC_\rho(M_0,\dots,M_p)$, we have \begin{equation}
\bar M_j\sim
\begin{cases}
L(m_{0,1}-d,m_{0,2},\dots,m_{0,n_0};\rho^{-1},\rho^{-1}\rho_{0,2},\dots
\rho^{-1}\rho_{0,n_0}) & (j=0),\\
L(m_{j,1}-d,m_{j,2},\dots,m_{j,n_j};1,\rho\rho_{j,2},\dots
\rho\rho_{j,n_j}) & (j=1,\dots,p).
\end{cases} \end{equation} Here $d$ is given by \eqref{eq:defd2}. \end{thm}
\begin{rem}\label{rem:monred} i) \ We note that some $m_{j,1}$ may be zero in Theorem~\ref{thm:schmid} and Theorem~\ref{thm:Mmid}.
ii) It follows from Theorem~\ref{thm:schmid} (resp.~Theorem~\ref{thm:Mmid}) and Scott's lemma that any irreducible tuple $\mathbf A\in M(n,\mathbb C)^{p+1}_0$ (resp.~$\mathbf M\in GL(n,\mathbb C)^{p+1}_1$) can be connected by successive applications of middle convolutions and additions (resp.~ multiplications) to an tuple whose spectral type is fundamental (cf.~Definition~\ref{def:fund}). In particular, the spectral type of $\mathbf M$ is an irreducibly realizable tuple if $\mathbf M$ is irreducible. \end{rem}
\begin{defn} Let $\mathbf M=(M_0,\dots,M_p)\in GL(n,\mathbb C)^{p+1}_1$. Suppose \eqref{eq:Lisom}. Fix $\ell=(\ell_0,\dots,\ell_p)\in\mathbb Z_{\ge 1}^{p+1}$ and define $\partial_\ell\mathbf M$ as follows. \begin{align*}
\rho_j&:=\begin{cases}
\rho_{j,\ell_j} &(0\le j\le p,\ 1\le \ell_j\le n_j),\\
\text{any complex number} &(0\le j\le p,\ n_j<\ell_j),
\end{cases}\\
\rho&:=\rho_0\rho_1\dots\rho_p,\\
(M_0',\dots,M_p')&
:=\MC_\rho(\rho_1\cdots\rho_pM_0,\rho_1^{-1}M_1,\rho_2^{-1}M_2,\dots,\rho_p^{-1}M_p),\\
\partial_\ell\mathbf M&:=(\rho_1^{-1}\cdots\rho_p^{-1}M_0',\rho_1M_1',\rho_2M_2,'\dots,\rho_pM_p'). \end{align*} Here we note that if $\ell=(1,\dots,1)$ and $\rho_{j,1}=1$ for $j=2,\dots,p$, $\partial_\ell\mathbf M=\MC_\rho(\mathbf M)$. \end{defn}
Let $u(1),\dots,u(n)$ be independent solutions of \eqref{eq:MSCF} at a generic point $q$. \index{monodromy} Let $\gamma_j$ be a closed path around $c_j$ as in the following figure. Denoting the result of the analytic continuation of $\tilde u:=(u(1),\dots,u(n))$ along $\gamma_j$ by $\gamma_j(\tilde u)$, we have a \textsl{monodromy generator} $M_j\in GL(n,\mathbb C)$ such that $\gamma_j(\tilde u)=\tilde u M_j$. \index{monodromy!generator} We call the tuple $\mathbf M=(M_0,\ldots,M_p)$ the \textsl{monodromy} of \eqref{eq:MSCF} with respect to $\tilde u$ and $\gamma_0,\dots,\gamma_p$. The connecting path first going along $\gamma_i$ and then going along $\gamma_j$ is denoted by $\gamma_i\circ\gamma_j$.
\quad \begin{equation}\label{fig:mon}
\begin{xy}
(90,20) *{\gamma_i\circ\gamma_j(\tilde u)=\gamma_j(\tilde u M_i)};
(90,14) *{\phantom{\gamma_i\circ\gamma_j(\tilde u)}=\gamma_j(\tilde u) M_i};
(90,8) *{\phantom{\gamma_i\circ\gamma_j(\tilde u)}=\tilde u M_jM_i,};
(90,2) *{M_pM_{p-1}\cdots M_1M_0=I_n.};
(15,0) *+[Fo]{{\tiny \phantom{a}\times\!c_0}}="A";
(35,0) *+[Fo]{{\tiny c_1\!\times\phantom{c}}}="B";
(55,0) *+[Fo]{{\tiny \phantom{a_1}\!\times\phantom{c}}}="C";
(33,25) *+{q};
(55,2.3) *+{\tiny c_2};
\ar@/_2pt/ @{->} (9.3,3);(9.3,-3)
\ar@/_2pt/ @{->} (20.7,-3);(20.7,3)
\ar@/_2pt/ @{->} (29.3,3);(29.3,-3)
\ar@/_2pt/ @{->} (40.7,-3);(40.7,3)
\ar@/_2pt/ @{->} (49.3,3);(49.3,-3)
\ar@/_2pt/ @{->} (60.7,-3);(60.7,3)
\ar@{{*}-}_{\gamma_0} (30,25);(17.4,4,3)
\ar@{->} (19.1,10.2);(16.5,6)
\ar@{<-} (22.6,10.2);(20,6)
\ar@{-}^{\gamma_1} (30,25);(33.8,5)
\ar@{-}^{\gamma_2} (30,25);(53,3.9)
\ar@{-}^{\gamma_3} (30,25);(53,15) \end{xy} \end{equation}
The following theorem says that the monodromy of solutions of the system obtained by a middle convolution of the system \eqref{eq:MSCF} is a multiplicative middle convolution of that of the original system \eqref{eq:MSCF}.
\begin{thm}[\cite{DR2}]\label{thm:mcMC} Let\/ $\Mon(\mathbf A)$ denote the monodromy of the equation \eqref{eq:MSCF}. Put\/ $\mathbf M=\Mon(\mathbf A)$. Suppose\/ $\mathbf M$ satisfies \eqref{eq:mulstar} and \eqref{eq:mulss} and \begin{align}
\rank(A_0-\mu) &= \rank(M_0 - e^{2\pi\sqrt{-1}\mu}),\\
\rank(A_j) &= \rank(M_j-1) \end{align} for $j=1,\dots,p$, then \begin{equation}
\Mon\bigl(mc_\mu(\mathbf A)\bigr) \sim
\MC_{e^{2\pi\sqrt{-1}\mu}}\bigl(\Mon(\mathbf A)\bigr). \end{equation} \end{thm}
Let $\mathcal F$ be a space of (multi-valued) holomorphic functions on $\mathbb C\setminus\{c_1,\dots,c_p\}$ valued in $\mathbb C^n$ such that $\mathcal F$ satisfies \eqref{eq:Fmap}, \eqref{eq:F0} and \eqref{eq:F1}. For example the solutions of the equation \eqref{eq:MSCF} defines $\mathcal F$. Fixing a base $u=\bigl(u(1),\dots,u(n)\bigr)$ of $\mathcal F(U)$ with $U\ni q$, we can define monodromy generators $(M_0,\dots,M_p)$. Fix $\mu\in\mathbb C$ and put $\rho = e^{2\pi\sqrt{-1}\mu}$ and \begin{align*}
v_j(x) =
\begin{pmatrix}
\int^{(x+,c_j+,x-,c_j-)}\frac{u(t)(x-t)^{\mu-1}}{t-c_1}dt
\\
\vdots\\
\int^{(x+,c_j+,x-,c_j-)}\frac{u(t)(x-t)^{\mu-1}}{t-c_p}dt
\end{pmatrix}\text{ \ and \ }
v(x)=\bigl(v_1(x),\dots,v_p(x)\bigr). \end{align*} Then $v(x)$ is a holomorphic function valued in $M(pn,\mathbb C)$ and the $pn$ column vectors of $v(x)$ define a \textsl{convolution} $\tilde{\mathcal F}$ of $\mathcal F$ and the following facts are shown by \cite{DR2}.
The monodromy generators of $\tilde{\mathcal F}$ with respect to the base $v(x)$ equals the \textsl{convolution} $\tilde{\mathbf M}= (\tilde M_0,\dots,\tilde M_1)$ of $\mathbf M$ given by \eqref{eq:conM} and if $\mathcal F$ corresponds to the space of solutions of \eqref{eq:SCF}, $\tilde{\mathcal F}$ corresponds to that of the system of Schlesinger canonical form defined by $\bigl(\tilde A_0(\mu),\dots,\tilde A_p(\mu)\bigr)$ in \eqref{eq:conSch}, which we denote by $\mathcal M_{\tilde{\mathbf A}}$.
The middle convolution $\MC_\rho(\mathbf M)$ of $\mathbf M$ is the induced monodromy generators on the quotient space of $\mathbb C^{pn}/V$ where $V$ is the maximal invariant subspace such the restriction of $\tilde{\mathbf M}$ on $V$ is a direct sum of finite copies of $1$-dimensional spaces with the actions $(\rho^{-1},1,\dots,1,\overset{\underset{\smallsmile}j}\rho,1,\dots,1) \in GL(1,\mathbb C)^{p+1}_1$ $(j=1,\dots,p)$ and $(1,1,\dots,1)$. The system defined by the middle convolution $mc_\mu(\mathbf A)$ is the quotient of the system $\mathcal M_{\tilde{\mathbf A}}$ by the maximal submodule such that the submodule is a direct sum of finite copies of the equations $(x-c_j)\frac{dw}{dx}=\mu w$ $(j=1,\dots,p)$ and $\frac{dw}{dx}=0$.
Suppose $\mathbf M$ and $\MC_\rho(\mathbf M)$ are irreducible and $\rho\ne 1$. Assume $\phi(x)$ is a function belonging to $\mathcal F$ such that it is defined around $x=c_j$ and corresponds to the eigenvector of the monodromy matrix $M_j$ with the eigenvalue different from 1. Then the holomorphic continuation of $\Phi(x)=\int^{(x+,c_j+,x-,c_j-)}\frac{\phi(t)(t-x)^\mu}{t-c_j}dt$ defines the monodromy isomorphic to $\MC_\rho(\mathbf M)$.
\begin{rem}\label{rem:Mon} We can define the monodromy $\mathbf M=(M_0,\dots,M_p)$ of the universal model $P_{\mathbf m}u=0$ (cf.~Theorem~\ref{thm:univmodel}) so that $\mathbf M$ is entire holomorphic with respect to the spectral parameters $\lambda_{j,\nu}$ and the accessory parameters $ g_i$ under the normalization $u(j)^{(\nu-1)}(q)= \delta_{j,\nu}$ for $j,\ \nu=1,\dots,n$ and $q\in\mathbb C\setminus\{c_1,\dots,c_p\}$. Here $u(1),\dots,u(n)$ are solutions of $P_{\mathbf m}u=0$. \end{rem}
\begin{defn}\label{def:locnondeg}\index{locally non-degenerate} Let $P$ be a Fuchsian differential operator with the Riemann scheme \eqref{eq:GRS} and the spectral type $\mathbf m=\bigl(m_{j,\nu}\bigr)_{\substack{0\le j\le p\\1\le\nu\le n_j}}$. We define that $P$ is \textsl{locally non-degenerate} \index{Fuchsian differential equation/operator!locally non-degenerate} if the tuple of the monodromy generators $\mathbf M:=(M_0,\dots,M_p)$ satisfies \begin{equation}\label{eq:nondeg1}
M_j\sim L(m_{j,1},\dots,m_{j,n_j};e^{2\pi\sqrt{-1}\lambda_{j,1}},\dots,
e^{2\pi\sqrt{-1}\lambda_{j,n_j}})\quad(j=0,\dots,p), \end{equation} which is equivalent to the condition that \index{00Z@$Z(A)$, $Z(\mathbf M)$} \begin{equation}\label{eq:nondeg0} \dim Z(M_j) = m_{j,1}^2+\cdots+m_{j,n_j}^2\quad(j=0,\dots,p). \end{equation} Suppose $\mathbf m$ is irreducibly realizable. Let $P_{\mathbf m}$ be the universal operator with the Riemann scheme \eqref{eq:GRS}. We say that the parameters $\lambda_{j,\nu}$ and $g_i$ are \textsl{locally non-degenerate} if the corresponding operator is locally non-degenerate. \end{defn} Note that the parameters are locally non-degenerate if \[
\lambda_{j,\nu}-\lambda_{j,\nu'}\notin\mathbb Z
\quad(j=0,\dots,p,\ \nu=1,\dots,n_j,\ \nu'=1,\dots,n_j). \]
Define $P_t$ as in Remark~\ref{rem:GCexp} iv). Then we can define monodromy generator $M_t$ of $P_t$ at $x=c_j$ so that $M_t$ holomorphically depend on $t$ (cf.~Remark~\ref{rem:Mon}). Then Remark~\ref{rm:1} v) proves that \eqref{eq:nondeg0} implies \eqref{eq:nondeg1} for every $j$.
The following proposition gives a sufficient condition such that an operator is locally non-degenerate. \begin{prop}\label{prop:nondeg} Let $P$ be a Fuchsian differential operator with the Riemann scheme \eqref{eq:GRS} and let $M_j$ be the monodromy generator at $x=c_j$. Fix an integer $j$ with $0\le j\le p$. Then the condition \begin{equation}\label{eq:nondegA}
\begin{split}
&\lambda_{j,\nu}-\lambda_{j,\nu'}\notin\mathbb Z\text{ \ or \ }
(\lambda_{j,\nu}-\lambda_{j,\nu'})
(\lambda_{j,\nu}+m_{j,\nu}-\lambda_{j,\nu'}-m_{j,\nu'})\le 0\\
&\qquad\text{ \ for \ }1\le \nu\le n_j\text{ \ and \ }1\le \nu'\le n_j
\end{split} \end{equation} implies $\dim Z(M_j)=m_{j,1}^2+\cdots+m_{j,n_j}^2$. In particular, $P$ is locally non-degenerate if \eqref{eq:nondegA} is valid for $j=0,\dots,p$.
Here we remark that the following condition implies \eqref{eq:nondegA}. \begin{equation}\label{eq:nondegB}
\lambda_{j,\nu}-\lambda_{j,\nu'}\notin\mathbb Z\setminus\{0\}
\quad\text{for \ }1\le \nu\le n_j\text{ \ and \ }1\le \nu'\le n_j. \end{equation} \end{prop} \begin{proof} For $\mu\in\mathbb C$ we put \[
N_\mu=\bigl\{\nu\,;\,1\le\nu\le n_j,\ \mu\in\{\lambda_{j,\nu},\lambda_{j,\nu}+1,
\dots,\lambda_{j,\nu}+m_{j,\nu}-1\}\bigr\}. \] If $N_\mu > 0$, we have a local solution $u_{\mu,\nu}(x)$ of the equation $Pu=0$ such that \begin{equation}\label{eq:nondegs1}
u_{\mu,\nu}(x)=(x-c_j)^{\mu}\log^{\nu}(x-c_j)+\mathcal O_{c_j}(\mu+1,L_\nu)
\text{ \ for \ }\nu=0,\dots,N_\mu-1. \end{equation} Here $L_\nu$ are positive integers and if $j=0$, then $x$ and $x-c_j$ should be replaced by $y=\frac1x$ and $y$, respectively.
Suppose \eqref{eq:nondegA}. Put $\rho=e^{2\pi \mu i}$, $\mathbf m'_\rho=\{m_{j,\nu}\,;\, \lambda_{j,\nu}-\mu\in\mathbb Z\}$ and $\mathbf m'_\rho=\{m'_{\rho,1},\dots, m'_{\rho,n_\rho}\}$ with $m'_{\rho,1}\ge m'_{\rho,2}\ge\cdots\ge m'_{\rho,n_\rho} \ge 1$. Then \eqref{eq:nondegA} implies \begin{equation}\label{eq:nondegs2}
n-\rank(M_j-\rho)^k \le
\begin{cases}
m'_{\rho,1}+\cdots+m'_{\rho,k}&(1\le k\le n_\rho),\\
m'_{\rho,1}+\cdots+m'_{\rho,n_\rho}&(n_\rho< k).
\end{cases} \end{equation} The above argument proving \eqref{eq:nondeg1} under the condition \eqref{eq:nondeg0} shows that the left hand side of \eqref{eq:nondegs2} is not smaller than the right hand side of \eqref{eq:nondegs2}. Hence we have the equality in \eqref{eq:nondegs2}. Thus we have \eqref{eq:nondeg0} and we can assume that $L_\nu=\nu$ in \eqref{eq:nondegs1}. \end{proof}
Theorem~\ref{thm:Mmid}, Theorem~\ref{thm:mcMC} and Proposition~\ref{prop:RAdIc} show the following corollary. One can also prove it by the same way as in the proof of \cite[Theorem~4.7]{DR2}. \begin{cor}\label{cor:irredred} Let\/ $P$ be a Fuchsian differential operator with the Riemann scheme \eqref{eq:GRS}. Let\/ $\Mon(P)$ denote the monodromy of the equation $Pu=0$. Put\/ $\Mon(P)=(M_0,\dots,M_p)$. Suppose \begin{equation}
M_j\sim L(m_{j,1},\dots,m_{j,n_j};e^{2\pi\sqrt{-1}\lambda_{j,1}},\dots,
e^{2\pi\sqrt{-1}\lambda_{j,n_j}}) \text{ \ for \ }j=0,\dots,p. \end{equation} In this case, $P$ is said to be locally non-degenerate. Under the notation in Definition~\ref{def:pell}, we fix $\ell\in\mathbb Z_{\ge 1}^{p+1}$ and suppose \eqref{eq:redok}. Assume moreover \begin{gather}
\mu_\ell\notin\mathbb Z,\\
m_{j,\nu}\le m_{j,\ell_j}\text{ \ or \ }
\lambda_{j,\ell_j}-\lambda_{j,\nu}\notin\mathbb Z\quad
(j=0,\dots,p,\ \nu=1,\dots,n_j). \end{gather} Then we have \begin{equation}
\Mon(\partial_\ell P)\sim\partial_\ell\Mon(P). \end{equation} In particular, $\Mon(P)$ is irreducible if and only if $\Mon(\partial_\ell P)$ is irreducible. \end{cor}
\subsection{Scott's lemma and Katz's rigidity}\label{sec:Scott} The results in this subsection are known but we will review them with their proof for the completeness of this paper. \begin{lem}[Scott \cite{Sc}]\index{Scott's lemma} \label{lem:Scott} Let\/ $\mathbf M\in GL(n,\mathbb C)^{p+1}_1$ and\/ $\mathbf A\in M(n,\mathbb C)^{p+1}_0$ under the notation \eqref{eq:Msub0} and \eqref{eq:Gsub1}. Then \begin{align}
\sum_{j=0}^p \codim\ker(M_j-1)&\ge
\codim\bigcap_{j=0}^p\ker(M_j-1)+\codim\bigcap_{j=0}^p\ker({}^t\!M_j-1),\\
\sum_{j=0}^p \codim\ker A_j&\ge
\codim\bigcap_{j=0}^p\ker A_j +\codim\bigcap_{j=0}^p\ker{}^t\!A_j. \end{align} In particular, if\/ $\mathbf M$ and $\mathbf A$ are irreducible, then \begin{align}
\sum_{j=0}^p\dim\ker(M_j-1)&\le (p-1)n,\\
\sum_{j=0}^p\dim\ker A_j&\le (p-1)n. \end{align} \end{lem} \begin{proof} Consider the following linear maps: \begin{align*}
V&=\IM(M_0-1)\times\cdots\times \IM(M_p-1)\subset\mathbb C^{n(p+1)},\\
\beta&:\ \mathbb C^n\to V,\quad v\mapsto ((M_0-1)v,\dots,(M_p-1)v),\\
\delta&:\ V\to \mathbb C^n,\quad (v_0,\dots,v_p)
\mapsto M_p\cdots M_1v_0+M_p\cdots M_2v_1+\cdots+M_pv_{p-1}+v_p. \end{align*} Since $M_p\cdots M_1(M_0-1)+\cdots+M_p(M_{p-1}-1)+(M_p-1) =M_p\cdots M_1M_0-1=0$, we have $\delta\circ\beta=0$. Moreover we have \begin{align*}
&\sum_{j=0}^pM_p\cdots M_{j+1}(M_j-1)v_j
=\sum_{j=0}^p\Bigl(1+\sum_{\nu=j+1}^{p}(M_\nu-1)M_{\nu-1}
\cdots M_{j+1}\Bigr)(M_j-1)v_j\\
&\quad=\sum_{j=0}^p(M_j-1)v_j+\sum_{\nu=1}^{p}\sum_{i=0}^{\nu-1}
(M_\nu-1)M_{\nu-1}\cdots M_{i+1}(M_i-1)v_i\\
&\quad=\sum_{j=0}^p(M_j-1)\Bigl(v_j +
\sum_{i=0}^{j-1} M_{j+1}\cdots M_{i-1}(M_i-1)v_i\Bigr) \end{align*} and therefore $\IM\delta=\sum_{j=0}^p\IM(M_j-1)$. Hence \begin{align*}
\dim\IM\delta&=\rank (M_0-1,\dots,M_p-1)
=\rank \begin{pmatrix}{}^t\!M_0-1\\\vdots\\ {}^t\!M_p-1\end{pmatrix} \\[-.3cm] \intertext{and}
\sum_{j=0}^p\codim\ker(M_j-1)&=\dim V=\dim\ker\delta+\dim\IM\delta\\
&\ge\dim\IM\beta+\dim\IM\delta\\
&=\codim\bigcap_{j=0}^p\ker(M_j-1)
+\codim\bigcap_{j=0}^p\ker({}^t\!M_j-1). \end{align*}
Putting \begin{align*}
V&=\IM A_0\times\cdots\times \IM A_p\subset\mathbb C^{n(p+1)}, \allowdisplaybreaks\\
\beta&:\ \mathbb C^n\to V,\quad v\mapsto (A_0 v,\dots, A_p v), \allowdisplaybreaks\\
\delta&:\ V\to \mathbb C^n,\quad (v_0,\dots,v_p)
\mapsto v_0+v_1+\cdots+v_p, \end{align*} we have the claims for $\mathbf A\in M(n,\mathbb C)^{p+1}$ in the same way as in the proof for $\mathbf M\in GL(n,\mathbb C)^{p+1}_1$. \end{proof}
\begin{cor}[Katz \cite{Kz} and \cite{SV}]\label{cor:katz} \index{Katz's rigidity} Let $\mathbf M\in GL(n,\mathbb C)^{p+1}_1$. The dimensions of the manifolds \begin{align}
V_1 &:=\{\mathbf H\in GL(n,\mathbb C)^{p+1}_1\,;\,\mathbf H\sim\mathbf M\} \intertext{and}
V_2 &:=\{\mathbf H\in GL(n,\mathbb C)^{p+1}_1\,;\,
H_j\sim M_j\quad(j=0,\dots,p)\} \end{align} are give by \index{00Z@$Z(A)$, $Z(\mathbf M)$} \begin{align}
\dim V_1 &= \codim Z(\mathbf M),\\
\dim V_2 &=\sum_{j=0}^p\codim Z(M_j)-\codim Z(\mathbf M). \end{align} Here $Z(\mathbf M):=\bigcap_{j=0}^p Z(M_j)$ and $Z(M_i)=\{X\in M(n,\mathbb C)\,;\,XM_j=M_jX\}$.
Suppose\/ $\mathbf M$ is irreducible. Then\/ $\codim Z(\mathbf M)=n^2-1$ and \begin{align}\label{eq:iridx}
\sum_{j=0}^p\codim Z(M_j)\ge 2n^2-2. \end{align} Moreover\/ $\mathbf M$ is \textsl{rigid}, namely, $V_1=V_2$ if and only if\/ $\displaystyle\sum_{j=0}^p\codim Z(M_j)=2n^2-2$. \end{cor} \begin{proof} The group $GL(n,\mathbb C)$ transitively acts on $V_1$ as simultaneous conjugations and the isotropy group with respect to $\mathbf M$ equals $Z(\mathbf M)$ and hence $\dim V_1=\codim Z(\mathbf M)$.
The group $GL(n,\mathbb C)^{p+1}$ naturally acts on $GL(n,\mathbb C)^{p+1}$ by conjugations. Putting $L=\{(g_j)\in GL(n,\mathbb C)^{p+1}\,;\,g_pM_pg_p^{-1} \cdots g_0M_0g_0^{-1}=M_p\cdots M_0\}$, $V_2$ is identified with $L/Z(M_0)\times\cdots\times Z(M_p)$. Denoting $g_j=\exp(t X_j)$ with $X_j\in M(n,\mathbb C)$ and $t\in\mathbb R$
with $|t|\ll1$ and defining $A_j\in\End\bigl(M(n,\mathbb C)\bigr)$ by $A_j X = M_jXM_j^{-1}$, we can prove that the dimension of $L$ equals the dimension of the kernel of the map \[
\gamma: M(n,\mathbb C)^{p+1}\ni
(X_0,\dots,X_p) \mapsto
\sum_{j=0}^p A_p\cdots A_{j+1}(A_j-1)X_j \] by looking at the tangent space of $L$ at the identity element because \begin{align*}
&\exp(t X_p)M_p\exp(-t X_p)\cdots \exp(tX_0)M_0(-t X_0) - M_p\cdots M_0\\
&\quad= t\Bigl(\sum_{j=0}^p A_p\cdots A_{j+1}(A_j-1)X_j\Bigr) M_p\cdots M_0+ o(t). \end{align*} We have obtained in the proof of Lemma~\ref{lem:Scott} that $\codim\ker\gamma=\dim\IM\gamma=\dim\sum_{j=0}^p\IM(A_j-1) =\codim\bigcap_{j=0}^p\ker({}^t\!A_j-1)$. We will see that $\bigcap_{j=0}^p\ker({}^t\!A_j-1)$ is identified with $Z(\mathbf M)$ and hence $\codim\ker\gamma=\codim Z(\mathbf M)$ and \[\dim V_2=\dim \ker\gamma-\sum_{j=0}^p\dim Z(M_j) =\sum_{j=0}^p\codim Z(M_j)-\codim Z(\mathbf M). \]
In general, fix $\mathbf H\in V_1$ and define $A_j\in \End\bigl(M(n,\mathbb C)\bigr)$ by $X\mapsto M_jXH_j^{-1}$ for $j=0,\dots,p$. Note that $A_pA_{p-1}\cdots A_0$ is the identity map. If we identify $M(n,\mathbb C)$ with its dual by the inner product $\trace XY$ for $X$, $Y\in M(n,\mathbb C)$, ${}^t\!A_j$ are identified with the map $Y\mapsto H_j^{-1}YM_j$, respectively.
Fix $P_j\in GL(n,\mathbb C)$ such that $H_j=P_jM_jP_j^{-1}$. Then \[
\begin{split} A_j(X)=X \Leftrightarrow&\ M_jXH_j^{-1}=X
\Leftrightarrow M_jX=XP_jM_jP_j^{-1}
\Leftrightarrow M_jXP_j=XP_jM_j,\\
{}^t\!A_j(X)=X \Leftrightarrow&\ H_j^{-1}XM_j=X
\Leftrightarrow XM_j=P_jM_jP_j^{-1}X
\Leftrightarrow P_j^{-1}XM_j=M_jP_j^{-1}X \end{split} \] and $\codim\ker(A_j-1)=\codim Z(M_j)$ and $\bigcap_{j=0}^p\ker({}^t\!A_j-1)\simeq Z(\mathbf M)$.
Suppose $\mathbf M$ is irreducible. Then $\codim Z(\mathbf M)=n^2-1$ and the inequality \eqref{eq:iridx} follows from $V_1\subset V_2$. Moreover suppose $\sum_{j=0}^p\codim Z(M_i)=2n^2-2$. Then Scott's lemma proves \[ \begin{split}
2n^2-2&=\sum_{j=0}^p\codim\ker(A_j-1)\\
&\ge n^2- \dim \bigcap_{j=0}^p
\{X\in M(n,\mathbb C)\,;\,M_jX=XH_j\}\\
&\quad+
n^2- \dim \bigcap_{j=0}^p
\{X\in M(n,\mathbb C)\,;\,H_jX=XM_j\}.
\end{split} \] Hence there exists a non-zero matrix $X$ such that $M_jX=XH_j$ ($j=0,\dots,p$) or $H_jX=XM_j$ ($j=0,\dots,p$). If $M_jX=XH_j$ (resp.~$H_jX=XM_j$) for $j=0,\dots,p$, $\ker X$ (resp.~$\IM X$) is $M_j$-stable for $j=0,\dots,p$ and hence $X\in GL(n,\mathbb C)$ because $\mathbf M$ is irreducible, Thus we have $V_1=V_2$ and we get all the claims in the corollary. \end{proof}
\section{Reducibility} \subsection{Direct decompositions}\label{sec:reddirect} For a realizable $(p+1)$-tuple $\mathbf m\in\mathcal P^{(n)}_{p+1}$, Theorem~\ref{thm:univmodel} gives the universal Fuchsian differential operator $P_{\mathbf m}(\lambda_{j,\nu},g_i)$ with the Riemann scheme \eqref{eq:GRS}. Here $g_1,\dots,g_N$ are accessory parameters and $N=\Ridx\mathbf m$.
First suppose $\mathbf m$ is basic. Choose positive numbers $n'$, $n''$, $m'_{j,1}$ and $m''_{j,1}$ such that \begin{equation}
\begin{gathered}
n= n'+n'',\quad 0<m'_{j,1}\le n',\quad 0<m''_{j,1}\le n'',\\
m'_{0,1}+\cdots+m'_{p,1}\le (p-1)n',\quad
m''_{0,1}+\cdots+m''_{p,1}\le (p-1)n''.
\end{gathered} \end{equation} We choose other positive integers $m'_{j,\nu}$ and $m''_{j,\nu}$ so that $\mathbf m'=\bigl(m'_{j,\nu}\bigr)$ and $\mathbf m''=\bigl(m''_{j,\nu}\bigr)$ are monotone tuples of partitions of $n'$ and $n''$, respectively, and moreover \begin{equation}\label{eq:tsum}
\mathbf m=\mathbf m'+\mathbf m''. \end{equation} Theorem~\ref{thm:ExF} shows that $\mathbf m'$ and $\mathbf m''$ are realizable. If $\{\lambda_{j,\nu}\}$ satisfies the Fuchs relation \begin{equation}\label{eq:FuchsProd}
\sum_{j=0}^p\sum_{\nu=1}^{n_j} m'_{j,\nu}\lambda_{j,\nu}
= n' - \frac{\idx\mathbf m'}2 \end{equation} for the Riemann scheme $\bigl\{[\lambda_{j,\nu}]_{(m'_{j,\nu})}\bigr\}$, Theorem~\ref{thm:prod} shows that the operators \begin{equation}\label{eq:2prod}
P_{\mathbf m''}(\lambda_{j,\nu}+m'_{j,\nu}-\delta_{j,0}(p-1)n',g''_i)
\cdot
P_{\mathbf m'}(\lambda_{j,\nu},g'_i) \end{equation} has the Riemann scheme $\{[\lambda_{j,\nu}]_{(m_{j,\nu})}\}$. This shows that the equation $P_{\mathbf m}(\lambda_{j,\nu},g_i)u=0$ is not irreducible when the parameters take the values corresponding to \eqref{eq:2prod}.
In this subsection, we study the condition \begin{equation}\label{eq:Rsum}
\Ridx\mathbf m=\Ridx\mathbf m'+\Ridx\mathbf m'' \end{equation} for realizable tuples $\mathbf m'$ and $\mathbf m''$ with $\mathbf m=\mathbf m'+\mathbf m''$. Under this condition the Fuchs relation \eqref{eq:FuchsProd} assures that the universal operator is reducible for any values of accessory parameters. \begin{defn}[direct decomposition] \index{direct decomposition} If realizable tuples $\mathbf m$, $\mathbf m'$ and $\mathbf m''$ satisfy \eqref{eq:tsum} and \eqref{eq:Rsum}, we define that $\mathbf m$ is the \textsl{direct sum} of $\mathbf m'$ and $\mathbf m''$ and call $\mathbf m=\mathbf m'+\mathbf m''$ a \textsl{direct decomposition} of $\mathbf m$ and express it as follows. \index{tuple of partitions!direct decomposition} \begin{equation}\label{eq:dsum}
\mathbf m=\mathbf m'\oplus\mathbf m''. \end{equation} \end{defn} \begin{thm}\label{thm:ddecmp} Let \eqref{eq:dsum} be a direct decomposition of a realizable tuple $\mathbf m$.
{\rm i) }
Suppose $\mathbf m$ is irreducibly realizable and\/ $\idx \mathbf m''>0$. Put\/ ${\overline{\mathbf m}'}=\gcd(\mathbf m')^{-1}\mathbf m'$. If\/ $\mathbf m'$ is indivisible or\/ $\idx\mathbf m\le 0$, then \begin{align}\label{eq:mref}
\alpha_{\mathbf m}&=\alpha_{\mathbf m'}-2
\frac{(\alpha_{{\overline{\mathbf m}}''}|\alpha_{\mathbf m'})}
{(\alpha_{{\overline{\mathbf m}''}}|\alpha_{{\overline{\mathbf m}}''})}
\alpha_{{\overline{\mathbf m}}''} \end{align} or $\mathbf m=\mathbf m'\oplus\mathbf m''$ is isomorphic to one of the decompositions \begin{equation}
\begin{aligned}
32,32,32,221&=22,22,22,220\oplus10,10,10,10,001\\
322,322,2221&=222,222,2220\oplus100,100,0001\\
54,3222,22221&=44,2222,22220\oplus10,1000,00001\\
76,544,2222221&=66,444,2222220\oplus10,100,0000001\\
\end{aligned} \end{equation} under the action of $\widetilde W_{\!\infty}$.
{\rm ii) } Suppose $\idx\mathbf m \le 0$ and $\idx\mathbf m'\le 0$ and $\idx\mathbf m''\le 0$. Then $\mathbf m=\mathbf m'\oplus\mathbf m''$ or $\mathbf m=\mathbf m''\oplus\mathbf m'$ is transformed into one of the decompositions \begin{equation} \begin{aligned}
\Sigma=11,11,11,11\ \ \ &111,111,111\ \ \ 22,1^4,1^4\ \ \ 33,222,1^6\\ m\Sigma&=k\Sigma\oplus \ell\Sigma\\ mm,mm,mm,m(m-1)1&=
kk,kk,kk,k(k-1)1\oplus\ell\ell,\ell\ell,\ell\ell,\ell\ell0\\ mmm,mmm,mm(m-1)1&=kkk,kkk,kkk,kk(k-1)1
\oplus\ell\ell\ell,\ell\ell\ell,\ell\ell\ell0\\ (2m)^2,m^4,mmm(m-1)1&=(2k)^2,k^4,k^4,kkk(k-1)1
\oplus(2\ell)^2,\ell^4,\ell^40\\
(3m)^2,(2m)^3,m^5(m-1)1&=(3k)^2,(2k)^3,k^5(k-1)1
\oplus (3\ell)^2,(2\ell)^3,\ell^60 \end{aligned} \end{equation} under the action of\/ $\widetilde W_{\!\infty}$. Here $m$, $k$ and $\ell$ are positive integers satisfying $m=k+\ell$. \index{00D4z@$D_4^{(m)},\ E_6^{(m)},\ E_7^{(m)},\ E_8^{(m)}$} These are expressed by \begin{equation}\begin{aligned}
m\tilde D_4&=k\tilde D_4\oplus\ell\tilde D_4,&
m\tilde E_j&=k\tilde E_j\oplus\ell\tilde E_j\quad(j=6,7,8),\\
D_4^{(m)}&=D_4^{(k)}\oplus\ell\tilde D_4,&
E_j^{(m)}&=E_j^{(k)}\oplus\ell\tilde E_j\quad(j=6,7,8). \end{aligned}\end{equation} \end{thm} \begin{proof} Put $\mathbf m'=k{\overline{\mathbf m}}'$ and $\mathbf m''
=\ell{\overline{\mathbf m}}''$ with indivisible ${\overline{\mathbf m}}'$ and ${\overline{\mathbf m}}''$. First note that \begin{equation}\label{eq:squaresum}
(\alpha_{\mathbf m}|\alpha_{\mathbf m})=
(\alpha_{\mathbf m'}|\alpha_{\mathbf m'})
+2(\alpha_{\mathbf m'}|\alpha_{\mathbf m''})
+(\alpha_{\mathbf m''}|\alpha_{\mathbf m''}). \end{equation}
{\rm ii) } Using Lemma~\ref{lem:sumlem}, we will prove the theorem. If $\idx\mathbf m=0$, then \eqref{eq:squaresum} and \eqref{eq:imneg} show
$0=(\alpha_{\mathbf m'}|\alpha_{\mathbf m''})=k\ell
(\alpha_{{\overline{\mathbf m}}'}|\alpha_{{\overline{\mathbf m}}''})$, Lemma~\ref{lem:sumlem} proves $\idx\mathbf m'=0$ and ${\overline{\mathbf m}}'={\overline{\mathbf m}}''$ and we have the theorem.
Suppose $\idx\mathbf m<0$.
If $\idx\mathbf m'<0$ and $\idx\mathbf m''<0$, we have $\Pidx\mathbf m=\Pidx\mathbf m'+\Pidx\mathbf m''$, which implies
$(\alpha_{\mathbf m'}|\alpha_{\mathbf m''})=-1$ and contradicts to Lemma~\ref{lem:sumlem}.
Hence we may assume $\idx\mathbf m''=0$.
\underline{Case: $\idx\mathbf m'<0$}. It follows from \eqref{eq:squaresum} that $2-2\Ridx\mathbf m=2-2\Ridx\mathbf m'+2\ell(\mathbf m,{\overline{\mathbf m}})$. Since $\Ridx\mathbf m=\Ridx\mathbf m'+\ell$,
we have $(\alpha_{\mathbf m}|\alpha_{\overline{\mathbf m}'})=-1$ and the theorem follows from Lemma~\ref{lem:sumlem}.
\underline{Case: $\idx\mathbf m'=0$}. It follows from \eqref{eq:squaresum} that $2-2\Ridx\mathbf m=2k\ell
(\alpha_{{\overline{\mathbf m}}'}|\alpha_{{\overline{\mathbf m}}''})$. Since the condition $\Ridx\mathbf m=k+\ell$ shows
$(\alpha_{{\overline{\mathbf m}}'}|\alpha_{{\overline{\mathbf m}}''}) =\frac1{k\ell}-\frac1k-\frac1\ell$ and we have
$(\alpha_{{\overline{\mathbf m}}'}|\alpha_{{\overline{\mathbf m}}''})=-1$. Hence the theorem also follows from Lemma~\ref{lem:sumlem}.
{\rm i) } First suppose $\idx\mathbf m'\ne0$. Note that $\mathbf m$ and $\mathbf m'$ are rigid if $\idx\mathbf m'>0$. We have $\idx\mathbf m=\idx\mathbf m'$ and $\idx\mathbf m
=(\alpha_{\mathbf m'}+\ell\alpha_{{\overline{\mathbf m}}''}|
\alpha_{\mathbf m'}+\ell\alpha_{{\overline{\mathbf m}}''})
= \idx\mathbf m'+2\ell(\alpha_{\mathbf m}|\alpha_{{\overline{\mathbf m}}''})
+2\ell^2$, which implies \eqref{eq:mref}.
Thus we may assume $\idx\mathbf m<0$ and $\idx\mathbf m'=0$. If $k=1$, $\idx\mathbf m=\idx\mathbf m'=0$ and we have \eqref{eq:mref} as above. Hence we may moreover assume $k\ge 2$. Then \eqref{eq:squaresum} and the assumption imply
$2-2k=2k\ell(\alpha_{{\overline{\mathbf m}}'}|
\alpha_{{\overline{\mathbf m}}''})+2\ell^2$, which means \[
-(\alpha_{{\overline{\mathbf m}}'}|
\alpha_{{\overline{\mathbf m}}''})=\frac{k-1+\ell^2}{k\ell}. \] Here $k$ and $\ell$ are mutually prime and hence there exists a positive integer $m$ with $k=m\ell+1$ and \[
-(\alpha_{{\overline{\mathbf m}}'}|
\alpha_{{\overline{\mathbf m}}''}) = \frac{m+\ell}{m\ell+1}
= \frac{1}{\ell+\frac1m}+\frac{1}{m+\frac1\ell}<2. \] Thus we have $m=\ell=1$, $k=2$ and
$(\alpha_{{\overline{\mathbf m}}'}|
\alpha_{{\overline{\mathbf m}}''})=-1$. By the transformation of an element of $\widetilde W_{\!\infty}$, we may assume $\overline{\mathbf m}'\in\mathcal P_{p+1}$ is a tuple in \eqref{eq:aftmp}. Since
$(\alpha_{\overline{\mathbf m}'}|\alpha_{\overline{\mathbf m}''})=-1$ and $\alpha_{\overline{\mathbf m}''}$ is a positive real root, we have the theorem by a similar argument as in the proof of Lemma~\ref{lem:sumlem}. Namely, $m'_{p,n'_p}=2$ and $m'_{p,n'_p+1}=0$ and we may assume $m''_{j,n'_j+1}=0$ for $j=0,\dots,p-1$ and $m''_{p,n'_p+1}+m''_{p,n'_p+2}+\cdots=1$, which proves the theorem in view of $\alpha_{\mathbf m''}\in\Delta^{re}_+$. \end{proof} \begin{lem}\label{lem:sumlem} Suppose $\mathbf m$ and $\mathbf m'$ are realizable and $\idx\mathbf m\le 0$ and $\idx\mathbf m'\le0$. Then \begin{align}\label{eq:imneg}
(\alpha_{\mathbf m}|\alpha_{\mathbf m'})\le0. \end{align}
If\/ $\mathbf m$ and\/ $\mathbf m'$ are basic and monotone, \begin{equation}\label{eq:maxidx}
(\alpha_{\mathbf m}|w\alpha_{\mathbf m'})\le
(\alpha_{\mathbf m}|\alpha_{\mathbf m'})
\qquad(\forall w\in W_{\!\infty}). \end{equation}
If\/ $(\alpha_{\mathbf m}|\alpha_{\mathbf m'})=0$ and $\mathbf m$ and $\mathbf m'$ are indivisible, then\/ $\idx\mathbf m=0$ and $\mathbf m=\mathbf m'$.
If\/ $(\alpha_{\mathbf m}|\alpha_{\mathbf m'})=-1$, then the pair is isomorphic to one of the pairs \begin{equation}\label{eq:idx-1} \begin{aligned}
(D_4^{(k)},\tilde D_4):&\ \bigl((kk,kk,kk,k(k-1)1),&&(11,11,11,110)\bigr)\\
(E_6^{(k)},\tilde E_6):&\ \bigl((kkk,kkk,k(k-1)1), &&(111,111,1110)\bigr)\\
(E_7^{(k)},\tilde E_7):&\ \bigl(((2k)^2,kkkk,kkk(k-1)1), &&(22,1111,11110)\bigr)\\
(E_8^{(k)},\tilde E_8):&\ \bigl(((3k)^2,(2k)^3,kkkkk(k-1)1),&&(33,222,1111110)\bigr) \end{aligned} \end{equation} under the action of\/ $\widetilde W_{\!\infty}$. \end{lem} \begin{proof} We may assume that $\mathbf m$ and $\mathbf m'$ are indivisible. Under the transformation of the Weyl group, we may assume that $\mathbf m$ is a basic monotone tuple in $\mathcal P_{p+1}$,
namely, $(\alpha_{\mathbf m}|\alpha_0)\le 0$ and
$(\alpha_{\mathbf m}|\alpha_{j,\nu})\le0$.
If $\mathbf m'$ is basic and monotone, $w\alpha_{\mathbf m'}-\alpha_{\mathbf m'}$ is a sum of positive real roots, which proves \eqref{eq:maxidx}.
Put $\alpha_{\mathbf m}=n\alpha_0+\sum n_{j,\nu}\alpha_{j,\nu}$ and $\mathbf m'=n'_0\alpha_0+\sum n'_{j,\nu}\alpha_{j,\nu}$. Then \begin{equation} \begin{split}
(\alpha_{\mathbf m}|\alpha_{\mathbf m'})&=
n'_0(\alpha_{\mathbf m}|\alpha_0)+
\sum n'_{j,\nu}(\alpha_{\mathbf m}|\alpha_{j,\nu}),\\
(\alpha_{\mathbf m}|\alpha)&\le 0\quad(\forall\alpha\in\supp\alpha_{\mathbf m}). \end{split} \end{equation} Let $k_j$ be the maximal positive integer satisfying $m_{j,k_j}=m_{j,1}$ and put $\Pi_0=\{\alpha_0,\alpha_{j,\nu}\,;\,1\le\nu<k_j,\ j=0,\dots,p\}$. Note that $\Pi_0$ defines a classical root system if $\idx\mathbf m<0$ (cf.~Remark~\ref{rem:classinbas}).
Suppose $(\alpha_{\mathbf m}|\alpha_{\mathbf m'})=0$ and $\mathbf m\in\mathcal P_{p+1}$. Then $m_{0,1}+\cdots+m_{p,1}=(p-1)\ord\mathbf m$ and $\supp\alpha_{\mathbf m'}\subset\Pi_0$
because $(\alpha_{\mathbf m}|\alpha)=0$ for $\alpha\in\supp\alpha_{\mathbf m'}$. Hence it follows from $\idx\mathbf m'\le0$ that $\idx\mathbf m=0$ and we may assume that $\mathbf m$ is one of the tuples \eqref{eq:aftmp}. Since $\supp\alpha_{\mathbf m'}\subset\supp\alpha_{\mathbf m}$ and $\idx\mathbf m'\le 0$, we conclude that $\mathbf m'=\mathbf m$.
Lastly suppose $(\alpha_{\mathbf m}|\alpha_{\mathbf m'})=-1$.
\underline{Case: $\idx\mathbf m=\idx\mathbf m'=0$}. If $\mathbf m'$ is basic and monotone and $\mathbf m'\ne\mathbf m$, then it is easy to see that $(\alpha_{\mathbf m}|\alpha_{\mathbf m'})<-1$ (cf.~Remark~\ref{rem:Kac}). Hence \eqref{eq:maxidx} assures $\mathbf m'=w\mathbf m$ with a certain $w\in W_{\!\infty}$ and therefore $\supp\mathbf m\subsetneq\supp\mathbf m'$. Moreover there exists $j_0$ and $L\ge k_{j_0}$ such that $\supp m'=\supp m\cup\{\alpha_{j_0,k_{j_0}},\alpha_{j_0,k_j+1},\dots, \alpha_{j_0,L}\}$ and $m_{j_0,k_{j_0}}=1$ and $m'_{j_0,k_{j_0}+1}=1$. Then by a transformation of an element of the Weyl group, we may assume $L=k_{j_0}$ and $\mathbf m'=r_{i_N}\cdots r_{i_1} r_{(j_0,k_{j_0})}\mathbf m$ with suitable $i_\nu$ satisfying $\alpha_{i_\nu}\in\supp\mathbf m$ for $\nu=1,\dots,N$. Applying $r_{i_1}\cdots r_{i_N}$ to the pair $(\mathbf m,\mathbf m')$, we may assume $\mathbf m'=r_{(j_0,k_{j_0})}\mathbf m$. Hence the pair $(\mathbf m,\mathbf m')$ is isomorphic to one of the pairs in the list \eqref{eq:idx-1} with $k=1$.
\underline{Case: $\idx\mathbf m<0$ and $\idx\mathbf m'\le 0$}.
There exists $j_0$ such that $\supp\alpha_{\mathbf m'}\ni\alpha_{j_0,k_j}$. Then the fact $\idx(\mathbf m,\mathbf m')=-1$ implies $n'_{j_0,k_0}=1$ and $n'_{j,k_j}=0$ for $j\ne j_0$.
Let $L$ be the maximal positive integer with $n'_{j_0,L}\ne0$. Since $(\alpha_{\mathbf m}|\alpha_{j_0,\nu})=0$ for $k_0+1\le \nu\le L$, we may assume $L=k_0$ by the transformation $r_{(j_0,k_0+1)}\circ \cdots\circ r_{(j_0,L)}$ if $L>k_0$. Since the Dynkin diagram corresponding to $\Pi_0\cup\{\alpha_{j_0,k_{j_0}}\}$ is classical or affine and $\supp\mathbf m'$ is contained in this set, $\idx\mathbf m'=0$ and $\mathbf m'$ is basic and we may assume that $\mathbf m'$ is one of the tuples \begin{equation}\label{eq:aftmp}
11,11,11,11\quad 111,111,111 \quad 22,1111,1111 \quad 33,222,111111 \end{equation}
and $j_0=p$. In particular $m'_{p,1}=\cdots=m'_{p,k_p}=1$ and $m'_{p,k_p+1}=0$. It follows from $(\alpha_{\mathbf m}|\alpha_{p,k_p})=-1$ that there exists an integer $L'\ge k_p+1$ satisfying $\supp\mathbf m=\supp\mathbf m'\cup\{\alpha_{p,\nu}\,;\,k_p\le \nu< L'\}$ and $m_{p,k_p}=m_{p,k_p-1}-1$. In particular, $m_{j,\nu}=m_{j,1}$ for $\nu=1,\dots,k_j-\delta_{j,p}$ and $j=0,\dots,p$. Since $\sum_{j=0}^pm_{j,1}=(p-1)\ord\mathbf m$, there exists a positive integer $k$ such that \[
m_{j,\nu}=
\begin{cases}
km'_{j,1}&(j=0,\dots,p,\ \nu=1,\dots,k_j-\delta_{j,p}),\\
km'_{p,1}-1&(j=p,\ \nu=k_p).
\end{cases} \] Hence $m_{p,k_p+1}=1$ and $L'=k_p+1$ and the pair $(\mathbf m,\mathbf m')$ is one of the pairs in the list \eqref{eq:idx-1} with $k>1$. \end{proof}
\begin{rem} Let $k$ be an integer with $k\ge2$ and let $P$ be a differential operator with the spectral type $D_4^{(k)}$, $E_6^{(k)}$, $E_7^{(k)}$ or $E_8^{(k)}$. It follows from Theorem~\ref{thm:prod} and Theorem~\ref{thm:univmodel} that $P$ is reducible for any values of accessory parameters when the characteristic exponents satisfy Fuchs relation with respect to the subtuple given in \eqref{eq:idx-1}. For example, the Fuchsian differential operator $P$ with the Riemann scheme \begin{align*}
&\begin{Bmatrix}
[\lambda_{0,1}]_{(k)} & [\lambda_{1,1}]_{(k)} & [\lambda_{2,1}]_{(k)}
& [\lambda_{3,1}]_{(k)}\\
[\lambda_{0,2}]_{(k)} & [\lambda_{1,2}]_{(k)} & [\lambda_{2,2}]_{(k)}
& [\lambda_{3,2}]_{(k-1)}\\
&&& \lambda_{3,2}+2k-2
\end{Bmatrix} \end{align*} is reducible. \end{rem} \begin{exmp}\label{ex:JPH} i) \ (generalized Jordan-Pochhammer)\index{Jordan-Pochhammer!generalized} If $\mathbf m=k\mathbf m'\oplus\ell\mathbf m''$ with a rigid tuples $\mathbf m$, $\mathbf m'$ and $\mathbf m''$ and positive integers $k$ and $\ell$ satisfying $1\le k\le \ell$, we have \begin{equation}
(\alpha_{\mathbf m'}|\alpha_{\mathbf m''})=-\frac{k^2+\ell^2-1}{k\ell}
\in\mathbb Z. \end{equation} For positive integers $k$ and $\ell$ satisfying $1\le k\le \ell$ and \begin{equation}\label{eq:pkl}
p:=\frac{k^2+\ell^2-1}{k\ell}+1\in\mathbb Z, \end{equation} we have an example of direct decompositions \begin{equation}\label{eq:kl} \begin{split}
\overbrace{\ell k,\ell k,\ldots,\ell k}^{p+1\text{ partitions}}
&= 0k,0k,\ldots,0k\oplus\ell0,\ell0,\ldots,\ell0\\
&=((p-1)k-\ell)k,((p-1)k-\ell)k,\ldots,((p-1)k-\ell)k\\
&\quad
\oplus(2\ell-(p-1)k)0,(2\ell-(p-1)k)0,\ldots,(2\ell-(p-1)k)0. \end{split}\end{equation} Here $p=3+\frac{(k-\ell)^2-1}{k\ell}\ge 2$ and the condition $p=2$ implies $k=\ell=1$ and the condition $p=3$ implies
$\ell=k+1$. If $k=1$, then $(\alpha_{\mathbf m'}|\alpha_{\mathbf m''})=-\ell$ and we have an example corresponding to Jordan-Pochhammer equation: \index{Jordan-Pochhammer} \begin{equation}
\overbrace{\ell1,\cdots,\ell1}^{\ell+2\text{ partitions}}
=01,\cdots,01\oplus \ell0,\cdots,\ell0.\\ \end{equation} When $\ell=k+1$, we have
$(\alpha_{\mathbf m'}|\alpha_{\mathbf m''})=-2k$ and an example \begin{equation}\begin{split}
&(k+1)k,(k+1)k,(k+1)k,(k+1)k\\
&\quad=0k,0k,0k,0k\oplus(k+1)0,(k+1)0,(k+1)0,(k+1)0\\
&\quad=(k-1)k,(k-1)k,(k-1)k,(k-1)k\oplus 20,20,20,20. \end{split}\end{equation} We have another example \begin{equation}\begin{split}
83,83,83,83,83&=03,03,03,03,03\oplus 80,80,80,80,80\\
&=13,13,13,13,13\oplus 70,70,70,70,70 \end{split}\end{equation} in the case $(k,\ell)=(3,8)$, which is a special case where $\ell=k^2-1$, $p=k+1$ and
$(\alpha_{\mathbf m'}|\alpha_{\mathbf m''})=-k$.
When $p$ is odd, the equation \eqref{eq:pkl} is equal to the Pell equation \begin{equation}
y^2 - (m^2-1)x^2=1 \end{equation} by putting $p-1=2m$, $x=\ell$ and $y=m\ell-k$ and hence the reduction of the tuple of partition \eqref{eq:kl} by $\partial_{\max}$ and its inverse give all the integer solutions of this Pell equation. \index{Pell equation}
The tuple of partitions $\ell k,\ell k,\ldots,\ell k\in\mathcal P_{p+1}^{(\ell+k)}$ with \eqref{eq:pkl} is called a \textsl{generalized Jordan-Pochhammer} tuple and denoted by $P_{p+1,\ell+k}$. In particular, $P_{n+1,n}$ is simply denoted by $P_n$. \index{00P@$P_{p+1,n},\ P_n$}
ii) We give an example of direct decompositions of a rigid tuple: \begin{align*} 3322,532,532&=0022,202,202\oplus 3300,330,330:1\\
&=1122,312,312\oplus 2200,220,220:1\\
&=0322,232,232\oplus 3000,300,300:2\\
&=3302,332,332\oplus 0020,200,200:2\\
&=1212,321,321\oplus 2110,211,211:4\\
&=2211,321,312\oplus 1111,211,220:2\\
&=2212,421,322\oplus 1110,111,210:4\\
&=2222,431,422\oplus 1100,101,110:2\\
&=2312,422,422\oplus 1010,110,110:4\\
&=2322,522,432\oplus 1000,010,100:4. \end{align*} They are all the direct decompositions of the tuple $3322,532,532$ modulo obvious symmetries. Here we indicate the number of the decompositions of the same type.
\end{exmp} \begin{cor} Let\/ $\mathbf m\in\mathcal P$ be realizable. Put\/ $\mathbf m=\gcd(\mathbf m)\overline{\mathbf m}$. Then\/ $\mathbf m$ has no direct decomposition\/ \eqref{eq:dsum} if and only if \begin{align} &\ord\mathbf m=1\\ \intertext{or} &\idx\mathbf m=0\text{ and basic}\\ \intertext{or} \begin{split} &\idx\mathbf m<0\text{ and\/ $\overline{\mathbf m}$ is basic and\/ $\mathbf m$ is not isomorphic to any one of tuples}\\ &\text{in\/ {\rm Example~\ref{ex:special}} with\/ $m>1$}. \end{split} \end{align} \end{cor} Moreover we have the following result. \begin{prop}\label{prop:rigdecomp} The direct decomposition\/ $\mathbf m=\mathbf m'\oplus\mathbf m''$ is called \textsl{rigid decomposition} \index{tuple of partitions!rigid decomposition} if\/ $\mathbf m$, $\mathbf m'$ and $\mathbf m''$ are rigid. If\/ $\mathbf m\in\mathcal P$ is rigid and\/ $\ord\mathbf m>1$, there exists a rigid decomposition. \end{prop} \begin{proof} We may assume that $\mathbf m$ is monotone and there exist a non-negative integer $p$ such that $m_{j,2}\ne 0$ if and only if $0\le j<p+1$. If $\ord\partial\mathbf m=1$, then we may assume $\mathbf m=(p-1)1,(p-1)1,\dots,(p-1)1\in\mathcal P^{(p)}_{p+1}$ and there exists a decomposition \[
(p-1)1,(p-1)1,\dots,(p-1)1=01,10,\dots,10\oplus
(p-1)0,(p-2)1,\dots,(p-2)1. \] Suppose $\ord{\partial \mathbf m}>1$. Put $d=\idx(\mathbf m,\mathbf 1)=m_{0,1}+\cdots+m_{p,1} -(p-1)\cdot\ord\mathbf m>0$.
The induction hypothesis assures the existence of a decomposition $\partial \mathbf m=\bar{\mathbf m}'\oplus\bar{\mathbf m}''$ such that $\bar{\mathbf m}'$ and $\bar{\mathbf m}''$ are rigid. If $\partial\bar{\mathbf m}'$ and $\partial\bar{\mathbf m}''$ are well-defined, we have the decomposition $\mathbf m=\partial^2\mathbf m=\partial\bar{\mathbf m}'\oplus \partial\bar{\mathbf m}''$ and the proposition.
If $\ord \bar{\mathbf m}'>1$, $\partial\bar{\mathbf m}'$ is well-defined. Suppose $\bar{\mathbf m}' =\bigl(\delta_{\nu,\ell_j}\bigr)_{\substack{j=0,\dots,p\\ \nu=1,2,\dots}}$. Then \begin{align*}
\idx(\partial \mathbf m,\mathbf 1)-
\idx(\partial \mathbf m,\bar{\mathbf m}')
&=\sum_{j=0}^p\bigl((m_{j,1}-d -(m_{j,\ell_j}-d\delta_{\ell_j,1})\bigr)\\
&\ge-d\#\{j\,;\,\ell_j>1,\ 0\le j\le p\}. \end{align*} Since $\idx(\partial \mathbf m,\mathbf 1)=-d$ and $\idx(\partial \mathbf m,\bar{\mathbf m}')=1$, we have $d\#\{j\,;\,\ell_j>1,\ 0\le j\le p\}\ge d+1$ and therefore $\#\{j\,;\,\ell_j>1,\ 0\le j\le p\}\ge 2$. Hence $\partial\bar{\mathbf m}'$ is well-defined. \end{proof}
\begin{rem} The author's original construction of a differential operator with a given rigid Riemann scheme doesn't use the middle convolutions and additions but uses Proposition~\ref{prop:rigdecomp}. \end{rem}
\begin{exmp} We give direct decompositions of a rigid tuple: \begin{equation}
\begin{split}
721,3331,22222
&=200,2000,20000\oplus521,1331,02222:15\\
&=210,1110,11100\oplus511,2221,11122:10\\
&=310,1111,11110\oplus411,2220,11112:\phantom{1}5
\end{split} \end{equation}
The following irreducibly realizable tuple has only two direct decompositions: \begin{equation} \begin{split}
44,311111,311111
&=20,200000,200000\oplus 24,111111,111111\\
&=02,200000,200000\oplus 42,111111,111111 \end{split} \end{equation} But it cannot be a direct sum of two irreducibly realizable tuples.
\end{exmp} \index{direct decomposition!example} \subsection{Reduction of reducibility}\label{sec:redred} We give a necessary and sufficient condition so that a Fuchsian differential equation is irreducible, which follows from \cite{Kz} and \cite{DR,DR2}. Note that a Fuchsian differential equation is irreducible if and only if its monodromy is irreducible.
\begin{thm}\label{thm:irred} Retain the notation in\/ {\rm\S\ref{sec:reddirect}}. Suppose\/ $\mathbf m$ is monotone, realizable and $\partial_{max}\mathbf m$ is well-defined and \begin{align}
d &:= m_{0,1}+\cdots+m_{p,1} - (p-1)\ord\mathbf m\ge0.\label{eq:mc1}\\ \intertext{Put $P=P_{\mathbf m}$ {\rm(cf.~\eqref{eq:uinvPm})} and}
\mu&:=\lambda_{0,1}+\lambda_{1,1}+\cdots+\lambda_{p,1}-1,\\
Q&:=\partial_{max}P,\\
P^o&:=P|_{\lambda_{j,\nu}=\lambda_{j,\nu}^o,\ g_i=g_i^o},\quad
Q^o:=Q|_{\lambda_{j,\nu}=\lambda_{j,\nu}^o,\ g_i=g_i^o} \end{align}
with some complex numbers $\lambda_{j,\nu}^o$ and $g_i^o$ satisfying the Fuchs relation $|\{\lambda^o_{\mathbf m}\}|=0$.
{\rm i)} The Riemann scheme $\{\tilde\lambda_{\tilde{\mathbf m}}\}$ of Q is given by \begin{equation}
\begin{cases}
\tilde m_{j,\nu}=m_{j,\nu}-d\delta_{\nu,1},\\
\tilde\lambda_{j,\nu}=
\lambda_{j,\nu}+\bigl((-1)^{\delta_{j,0}}-\delta_{\nu,1}\bigr)\mu.
\end{cases} \label{eq:mc2} \end{equation}
{\rm ii)} Assume that the equation $P^ou=0$ is irreducible. If $d>0$, then $\mu\notin\mathbb Z$. If the parameters given by $\lambda_{j,\nu}^o$ and $g_i^o$ are locally non-degenerate, the equation $Q^ov=0$ is irreducible and the parameters are locally non-degenerate.
{\rm iii)} Assume that the equation $Q^ov=0$ is irreducible and the parameters given by $\lambda_{j,\nu}^o$ and $g_i^o$ are locally non-degenerate. Then the equation $P^ov=0$ is irreducible if and only if \begin{align}\label{eq:irrmax}
\sum_{j=0}^p\lambda^o_{j,1+\delta_{j,j_o}(\nu_o-1)}\notin\mathbb Z
\text{ \ for any \ }(j_o,\nu_o)\text{ \ satisfying \ }
m_{j_o,\nu_o}> m_{j_o,1}-d. \end{align} If the equation $P^ov=0$ is irreducible, the parameters are locally non-degenerate.
{\rm iv)} Put\/ $\mathbf m(k):=\partial_{max}^k\mathbf m$ and $P(k)=\partial_{max}^kP$. Let $K$ be a non-negative integer such that $\ord \mathbf m(0)>\ord\mathbf m(1)>\cdots>\ord\mathbf m(K)$ and\/ $\mathbf m(K)$ is fundamental. The operator $P(k)$ is essentially the universal operator of type\/
$\mathbf m(k)$ but parametrized by\/ $\lambda_{j,\nu}$ and $g_i$. Put $P(k)^o=P(k)|_{\lambda_{j,\nu}=\lambda_{j,\nu}^o}$.
If the equation\ $P^ou=0$ is irreducible and the parameters are locally non-degenerate, so are $P(k)^ou=0$ for $k=1,\dots,K$.
If the equation $P^ou=0$ is irreducible and locally non-degenerate, so is the equation $P(K)^ou=0$.
Suppose the equation $P(K)^ou=0$ is irreducible and locally non-degenerate, which is always valid when $\mathbf m$ is rigid. Then the equation $P^ou=0$ is irreducible if and only if the equation $P(k)^ou=0$ satisfy the condition \eqref{eq:irrmax} for $k=0,\dots,K-1$. If the equation $P^ou=0$ is irreducible, it is locally non-degenerate. \end{thm} \begin{proof} The claim i) follows from Theorem~\ref{thm:GRSmid} and the claims ii) and iii) follow from Lemma~\ref{lem:irrred} and Corollary~\ref{cor:irredred}, which implies the claim iv). \end{proof}
\begin{rem}\label{rem:irredrigid} {\rm i)} In the preceding theorem the equation $P^ou=0$ may not be locally non-degenerate even if it is irreducible. For example the equation satisfied by ${}_3F_2$ is contained in the universal operator of type $111,111,111$.
{\rm ii)} It is also proved as follows that the irreducible differential equation with a rigid spectral type is locally non-degenerate.
The monodromy generators $M_j$ of the equation with the Riemann scheme at $x=c_j$ satisfy \[
\rank(M'_j-e^{2\pi\sqrt{-1}\lambda_{j,1}})
\cdots(M'_j-e^{2\pi\sqrt{-1}\lambda_{j,k}})
\le m_{j,k+1}+\cdots+m_{j,n_j}\quad(k=1,\dots,n_j) \] for $j=0,\dots,p$. The equality in the above is clear when $\lambda_{j,\nu}-\lambda_{j,\nu'}\notin\mathbb Z$ for $1\le\nu<\nu'\le n_j$ and hence the above is proved by the continuity for general $\lambda_{j,\nu}$. The rigidity index of $\mathbf M$ is calculated by the dimension of the centralizer of $M_j$ and it should be 2 if $\mathbf M$ is irreducible and rigid, the equality in the above is valid (cf.~\cite{Kz}, \cite{O3}), which means the equation is locally non-degenerate.
{\rm iii)} \ The same results as in Theorem~\ref{thm:irred} are also valid in the case of the Fuchsian system of Schlesinger canonical form \eqref{eq:MSCF} since the same proof works. A similar result is given by a different proof (cf.~\cite{CB}).
{\rm iv)} \ Let $(M_0,\dots,M_p)$ be a tuple of matrices in $GL(n,\mathbb C)$ with $M_pM_{p-1}\cdots M_0=I_n$. Then $(M_0,\dots,M_p)$ is called \textsl{rigid} if for any $g_0,\dots,g_p\in GL(n,\mathbb C)$ satisfying $g_pM_pg_p^{-1}\cdot g_{p-1}M_{p-1}g_{p-1}^{-1}\cdots g_0M_0g_0^{-1}=I_n$, there exists $g\in GL(n,\mathbb C)$ such that $g_iM_ig_i^{-1}=gM_ig^{-1}$ for $i=0,\dots,p$. The tuple $(M_0,\dots,M_p)$ is called \textsl{irreducible} if no subspace $V$ of $\mathbb C^n$ satisfies $\{0\}\subsetneqq V \subsetneqq \mathbb C^n$ and $M_iV\subset V$ for $i=0,\dots,p$. Choose $\mathbf m\in\mathcal P^{(n)}_{p+1}$ and $\{\mu_{j,\nu}\}$ such that $L(\mathbf m;\mu_{j,1},\dots,\mu_{j,n_j})$ are in the conjugacy classes containing $M_j$, respectively. Suppose $(M_0,\dots,M_p)$ is irreducible and rigid. Then Katz \cite{Kz} shows that $\mathbf m$ is rigid and gives a construction of irreducible and rigid $(M_0,\dots,M_p)$ for any rigid $\mathbf m$ (cf.~Remark \ref{rem:monred} ii)). It is an open problem given by Katz \cite{Kz} whether the monodromy generators $M_j$ are realized by solutions of a single Fuchsian differential equations without an apparent singularity, whose affirmative answer is given by the following corollary. \end{rem}
\begin{cor}\label{cor:irred} Let\/ $\mathbf m=\bigl(m_{j,\nu}\bigr)_{\substack{0\le j\le p\\1\le\nu\le n_j}}$ be a rigid monotone $(p+1)$-tuple of partitions with\/ $\ord\mathbf m>1$. Retain the notation in Definition~\ref{def:redGRS}.
{\rm i)} Fix complex numbers $\lambda_{j,\nu}$ for $0\le j\le p$ and $1\le \nu_j$ such that it satisfies the Fuchs relation \begin{equation}\label{eq:Fuchsirr}
\sum_{j=0}^p\sum_{\nu=1}^{n_j}m_{j,\nu}\lambda_{j,\nu}=\ord\mathbf m-1 \end{equation} The universal operator $P_{\mathbf m}(\lambda)u=0$ with the Riemann scheme \eqref{eq:IGRS} is irreducible if and only if the condition \begin{multline}\label{Cond:irr}
\sum_{j=0}^p\lambda(k)_{j,\ell(k)_j+\delta_{j,j_o}(\nu_o-\ell(k)_j)}
\notin\mathbb Z
\\\text{ \ for any \ }(j_o,\nu_o)\text{ \ satisfying \ }
m(k)_{j_o,\nu_o}> m(k)_{j_o,\ell(k)_{j_o}}-d(k) \end{multline} is satisfied for $k=0,\dots,K-1$.
{\rm ii)} Define $\tilde \mu(k)$ and $\mu(k)_{j,\nu}$ for $k=0,\dots,K$ by \begin{align}
\mu(0)_{j,\nu}&=\mu_{j,\nu}\quad(j=0,\dots,p,\ \nu=1,\dots,n_j),\\
\tilde\mu(k) &= \prod_{j=0}^p\mu(k)_{j,\ell(k)_j},\\
\mu(k+1)_{j,\nu}
&=\mu(k)_{j,\nu}\cdot
\tilde\mu(k)^{(-1)^{\delta_{j,0}}-\delta_{\nu,1}}. \end{align} Then there exists an irreducible tuple $(M_0,\dots,M_p)$ of matrices satisfying \begin{equation}\label{eq:monDS} \begin{gathered}
M_p\cdots M_0=I_n,\\
M_j\sim L(m_{j,1},\dots,m_{j,n_j};\mu_{j,1},\dots,\mu_{j,n_j})\quad(j=0,\dots,p) \end{gathered} \end{equation} under the notation \eqref{eq:OSNF} if and only if \begin{equation} \prod_{j=0}^p\prod_{\nu=1}^{n_j}\mu_{j,\nu}^{m_{j,\nu}}=1 \end{equation} and the condition \begin{multline}
\prod_{j=0}^p\mu(k)_{j,\ell(k)_j+\delta_{j,j_o}(\nu_o-\ell(k)_j)}
\ne1
\\\text{ \ for any \ }(j_o,\nu_o)\text{ \ satisfying \ }
m(k)_{j_o,\nu_o}> m(k)_{j_o,\ell(k)_{j_o}}-d(k) \end{multline} is satisfied for $k=0,\dots,K-1$.
{\rm iii)} Let $(M_0,\dots,M_p)$ be an irreducible tuple of matrices satisfying \eqref{eq:monDS}. Then there uniquely exists a Fuchsian differential equation $Pu=0$ with $p+1$ singular points $c_0,\dots,c_p$ and its local independent solutions $u_1,\dots,u_{\ord \mathbf m}$ in a neighborhood of a non-singular point $q$ such that the monodromy generators around the points $c_j$ with respect to the solutions equal $M_j$, respectively, for $j=0,\dots,p$ {\rm (cf.~\eqref{fig:mon})}. \end{cor} \begin{proof} The clam i) is a direct consequence of Theorem~\ref{thm:irred} and the claim ii) is proved by Theorem~\ref{thm:Mmid} and Lemma~\ref{lem:Scott} as in the case of the proof of Theorem~\ref{thm:irred} (cf.~Remark~\ref{rem:monred} ii)).
iii) Since $\gcd\mathbf m=1$, we can choose $\lambda_{j,\nu}\in\mathbb C$ such that $e^{2\pi\sqrt{-1}\lambda_{j,\nu}}=\mu_{j,\nu}$ and $\sum_{j,\nu}m_{j,\nu}\lambda_{j,\nu}=\ord\mathbf m-1$. Then we have a universal operator $P_{\mathbf m}(\lambda_{j,\nu})u=0$ with the Riemann scheme \eqref{eq:IGRS}. The irreducibility of $(M_p,\dots,M_0)$ and Theorem~\ref{thm:mcMC} assure the claim. \end{proof}
Now we state the condition \eqref{Cond:irr} using the terminology of the Kac-Moody root system. Suppose $\mathbf m\in\mathcal P$ is monotone and irreducibly realizable. Let $\{\lambda_{\mathbf m}\}$ be the Riemann scheme of the universal operator $P_{\mathbf m}$. According to Remark~\ref{rem:idxFuchs} iii) we may relax the definition of $\ell_{max}(\mathbf m)$ as is given by \eqref{eq:lmaxe} and then we may assume \begin{equation}
v_k s_0\cdots v_1s_0\Lambda(\lambda)
\in W'_{\infty}\Lambda\bigl(\lambda(k)\bigr)\qquad(k=1,\dots,K) \end{equation} under the notation in Definition~\ref{def:redGRS} and \eqref{eq:vmrp}. Then we have the following theorem.
\begin{thm}\label{thm:irrKac} Let\/ $\mathbf m=\bigl(m_{j,\nu}\bigr)_{\substack{0\le j\le p\\1\le \nu\le n_j}}$ be an irreducibly realizable monotone tuple of partition in $\mathcal P$. Under the notation in\/ {\rm Corollary~\ref{cor:irred}} and\/ {\rm \S\ref{sec:KM}}, there uniquely exists a bijection \index{000pix@$\varpi$} \begin{equation}
\begin{split}
\varpi:\Delta(\mathbf m) \xrightarrow\sim\ &
\bigl\{(k,j_0,\nu_0)\,;\,
0\le k<K,\ 0\le j_0\le p,\ 1\le\nu_0\le n_{j_0},
\\
&\qquad \nu_0\ne\ell(k)_{j_0}\text{ \ and \ }
m(k)_{j_0,\nu_0}>m(k)_{j_0,\ell(k)_{j_0}}-d(k)
\bigr\}\\
&\quad\cup\bigl\{(k,0,\ell(k)_0)\,;\,0\le k<K\bigr\}
\end{split} \end{equation} such that \begin{equation}
(\Lambda(\lambda)|\alpha)
=\sum_{j=0}^p\lambda(k)_{j,\ell(k)_j+\delta_{j,j_o}(\nu_o-\ell(k)_j)}
\text{ \ when \ }\varpi(\alpha)=(k,j_0,\nu_0). \end{equation} Moreover we have \begin{equation}
\begin{split}
(\alpha|\alpha_{\mathbf m})&=
m(k)_{j_0,\nu_0}-m(k)_{j_0,\ell(k)_{j_0}}+d(k)\\
&\qquad\qquad
(\alpha\in\Delta(\mathbf m),\ (k,j_0,\nu_0)=\varpi(\alpha))
\end{split} \end{equation} and if the universal equation $P_{\mathbf m}(\lambda)u=0$ is irreducible, we have \begin{equation}\label{eq:airred}
(\Lambda(\lambda)|\alpha)\notin\mathbb Z
\quad\text{ for any \ }\alpha\in\Delta(\mathbf m). \end{equation} In particular, if $\mathbf m$ is rigid and \eqref{eq:airred} is valid, the universal equation is irreducible. \end{thm}
\begin{proof} Assume $\ord \mathbf m>1$ and use the notation in Theorem~\ref{thm:irred}. Since $\tilde{\mathbf m}$ may not be monotone, we consider the monotone tuple $\mathbf m'=s\tilde{\mathbf m}$ in $S'_\infty\tilde{\mathbf m}$ (cf.~Definition~\ref{def:Sinfty}). First note that \[
d-m_{j,1}+m_{j,\nu}
= (\alpha_0+\alpha_{j,1}+\cdots+\alpha_{j,\nu-1}|\alpha_{\mathbf m}). \] Let $\bar \nu_j$ be the positive integers defined by \[
m_{j,\bar\nu_j+1}\le m_{j,1}-d<m_{j,\bar\nu_j} \] for $j=0,\dots,p$. Then \[
\alpha_{\mathbf m'}=v^{-1}\alpha_{\tilde{\mathbf m}}
\text{ \ with \ }
v:=\Bigl(\prod_{j=0}^p s_{j,1}\cdots s_{j,\bar\nu_j-1}
\Bigr) \] and $w(\mathbf m)=s_0vs_{\alpha_{\tilde{\mathbf m}}}$ and \[ \begin{split}
\Delta(\mathbf m)
&= \Xi\cup s_0v\Delta(\mathbf m'),\\
\Xi&:=\{\alpha_0\}\cup
\bigcup_{\substack{0\le j\le p\\\nu_j\ne 1}}
\{\alpha_0+\alpha_{j,1}+\cdots+\alpha_{j,\nu}\,;\,
\nu=1,\ldots,\bar\nu_j-1\}. \end{split} \]
Note that $\ell(0)=(1,\dots,1)$ and the condition $m_{j_0,\nu_0}>m_{j_0,1}-d(0)$ is valid if and only if $\nu_0\in\{1,\dots,\bar\nu_{j_0}\}$. Since \[
\sum_{j=0}^p\lambda(0)_{j,1+\delta_{j,j_0}(\nu_0-1)}
=(\Lambda(\lambda)|\alpha_0+\alpha_{j_0,1}+\cdots+\alpha_{j_0,\nu_0-1})+1, \] we have \[
L(0)=\bigl\{(\Lambda(\lambda)|\alpha)+1\,;\,\alpha\in\Xi\bigr\} \] by denoting \[
L(k):=\bigl\{\sum_{j=0}^p\lambda(k)_{j,\ell(k)_j+\delta_{j,j_o}(\nu_o-\ell(k)_j)}\,;\,
m(k)_{j_o,\nu_o}> m(k)_{j_o,\ell(k)_{j_o}}-d(k)\bigr\}. \] Applying $v^{-1}s_0$ to $\mathbf m$ and $\{\lambda_{\mathbf m}\}$, they changes into $\mathbf m'$ and $\{\lambda'_{\mathbf m'}\}$, respectively, such that $\Lambda(\lambda') - v^{-1}s_0\Lambda(\lambda)\in\mathbb C\Lambda_0$. Hence we obtain the corollary by the induction as in the proof of Corollary~\ref{cor:irred}. \end{proof} \begin{rem}
Let $\mathbf m$ be an irreducibly realizable monotone tuple in $\mathcal P$. Fix $\alpha\in\Delta(\mathbf m)$. We have $\alpha=\alpha_{\mathbf m'}$ with a rigid tuple $\mathbf m'\in\mathcal P$ and \begin{equation}
|\{\lambda_{\mathbf m'}\}|=(\Lambda(\lambda)|\alpha). \end{equation}
\end{rem}
\index{00indexa@$\idx_{\mathbf m}$} \begin{defn}\label{def:linidx} Define an \textsl{index} $\idx_{\mathbf m}\bigl(\ell(\lambda)\bigr)$ of the non-zero linear form $\ell(\lambda) =\sum_{j=0}^p\sum_{\nu=1}^{n_j}k_{j,\nu}\lambda_{j,\nu}$ of with $k_{j,\nu}\in\mathbb Z_{\ge 0}$ as the positive integer $d_i$ such that \begin{equation}
\Bigl\{\sum_{j=0}^p\sum_{\nu=1}^{n_j} k_{j,\nu}\epsilon_{j,\nu}\,;\,
\epsilon_{j,\nu}\in\mathbb Z\text{ and }\sum_{j=0}^p\sum_{\nu=1}^{n_j}
m_{j,\nu}\epsilon_{j,\nu}=0\Bigr\} = \mathbb Z d_i. \end{equation} \end{defn}
\begin{prop}\label{prop:subrep} For a rigid tuple $\mathbf m$ in\/ {\rm Corollary~\ref{cor:irred},} define rigid tuples $\mathbf m^{(1)},\dots,\mathbf m^{(N)}$ with a non-negative integer $N$ so that $\Delta(\mathbf m)=\{\mathbf m^{(1)},\dots,\mathbf m^{(N)}\}$ and put \begin{equation}
\ell_i(\lambda):=\sum_{j=0}^p\sum_{\nu=1}^{n_j}
m^{(i)}_{j,\nu}\lambda_{j,\nu}
\qquad(i=1,\dots,N). \end{equation} Here we note that\/ {\rm Theorem~\ref{thm:irrKac}} implies that $P_{\mathbf m}(\lambda)$ is irreducible if and only if $\ell_i(\lambda)\notin\mathbb Z$ for $i=1,\dots,n$.
Fix a function $\ell(\lambda)$ of $\lambda_{j,\nu}$ such that $\ell(\lambda)=\ell_i(\lambda)-r$ with $i\in\{1,\dots,N\}$ and $r\in\mathbb Z$. Moreover fix generic complex numbers $\lambda_{j,\nu}\in\mathbb C$ under
the condition $\ell(\lambda)=|\{\lambda_{\mathbf m}\}|=0$ and a decomposition $P_{\mathbf m}(\lambda)=P''P'$ such that $P',\,P''\in W(x)$, $0<n':=\ord P'<n$ and the differential equation $P'v=0$ is irreducible. Then there exists an irreducibly realizable subtuple\/ $\mathbf m'$ of\/ $\mathbf m$ compatible to $\ell(\lambda)$ such that the monodromy generators $M'_j$ of the equation $P'u=0$ satisfies \[
\rank(M_j-e^{2\pi\sqrt{-1}\lambda_{j,1}})
\cdots(M_j-e^{2\pi\sqrt{-1}\lambda_{j,k}})
\le m'_{j,k+1}+\cdots+m'_{j,n_j}\quad(k=1,\dots,n_j) \] for $j=0,\dots,p$. Here we define that the decomposition \begin{equation}\label{eq:subrep}
\mathbf m=\mathbf m'+\mathbf m''\quad( \mathbf m'\in\mathcal P^{(n')}_{p+1},\
\mathbf m''\in\mathcal P^{(n'')}_{p+1},\ 0<n'<n) \end{equation} is \textsl{compatible} to $\ell(\lambda)$ and that $\mathbf m'$ is a subtuple of $\mathbf m$ \textsl{compatible} to $\ell(\lambda)$ if the following conditions are valid \begin{align}
&|\{\lambda_{\mathbf m'}\}|\in\mathbb Z_{\le 0} \text{ \ and \ }
|\{\lambda_{\mathbf m''}\}|\in\mathbb Z,\\ &\text{$\mathbf m'$ is realizable if there exists $(j,\nu)$ such that $m''_{j,\nu}=m_{j,\nu}>0$,}\\ &\text{$\mathbf m''$ is realizable if there exists $(j,\nu)$ such that $m'_{j,\nu}=m_{j,\nu}>0$}. \end{align}
Here we note $|\{\lambda_{\mathbf m'}\}|+|\{\lambda_{\mathbf m''}\}|=1$ if\/ $\mathbf m'$ and\/ $\mathbf m''$ are rigid. \end{prop} \begin{proof} The equation $P_{\mathbf m}(\lambda)u=0$ is reducible since $\ell(\lambda)=0$. We may assume $\lambda_{j,\nu}-\lambda_{j,\nu'}\ne 0$ for $1\le \nu<\nu'\le n_j$ and $j=0,\dots,p$. The solutions of the equation define the map $\mathcal F$ given by \eqref{eq:Fmap} and the reducibility implies the existence of an irreducible submap $\mathcal F'$ such that $\mathcal F'(U)\subset \mathcal F(U)$ and $0<n':=\dim\mathcal F'(U)<n$. Then $\mathcal F'$ defines a irreducible Fuchsian differential equation $P'v=0$ which has regular singularities at $x=c_0=\infty,c_1,\dots,c_p$ and may have other apparent singularities $c'_1,\dots,c'_q$. Then the characteristic exponents of $P'$ at the singular points are as follows.
There exists a decomposition $\mathbf m=\mathbf m'+\mathbf m''$ such that $\mathbf m'\in\mathcal P^{(n')}$ and $\mathbf m''\in\mathcal P^{(n'')}$ with $n'':=n-n'$. The sets of characteristic exponents of $P'$ at $x=c_j$ are $\{\lambda'_{j,\nu,i}\,;\,i=1,\dots,m'_{j,\nu},\ \nu=1,\dots,n\}$ which satisfy \[
\lambda'_{j,\nu,i}-\lambda_{j,\nu}\in\{0,1,\dots,m_{j,\nu}-1\}\text{ \ and \ }
\lambda'_{j,\nu,1}<\lambda'_{j,\nu,2}<\cdots<\lambda'_{j,\nu,m'_{j,\nu}} \] for $j=0,\dots,p$. The sets of characteristic exponents at $x=c'_j$ are $\{\mu_{j,1},\dots,\mu_{j,n'}\}$, which satisfy $\mu_{j,i}\in\mathbb Z$ and $0\le \mu_{j,1}<\cdots<\mu_{j,n'}$ for $j=1,\dots,q$. Then Remark~\ref{rem:generic} ii) says that the Fuchs relation of the equation $P'v=0$ implies
$|\{\lambda_{\mathbf m'}\}|\in\mathbb Z_{\le 0}$.
Note that there exists a Fuchsian differential operator $P''\in W(x)$ such that $P=P''P'$. If there exists $j_o$ and $\nu_o$ such that $m'_{j_o,n_o}=0$, namely, $m''_{j_o,\nu_o}=m_{j_o,\nu_o}>0$, the exponents of the monodromy generators of the solution $P'v=0$ are generic and hence $\mathbf m'$ should be realizable. The same claim is also true for the tuple $\mathbf m''$. Hence we have the proposition. \end{proof}
\begin{exmp}\label{exmp:irred} {\rm i)}\index{hypergeometric equation/function!Gauss!reducibility} The reduction of the universal operator with the spectral type $11,11,11$ which is given by Theorem~\ref{thm:irred} is \begin{equation}\begin{split}
&\begin{Bmatrix}
x=\infty & 0 & 1\\
\lambda_{0,1} & \lambda_{1,1} & \lambda_{2,1}\\
\lambda_{0,2} & \lambda_{1,2} & \lambda_{2,2}
\end{Bmatrix}\quad(\sum \lambda_{j,\nu}=1)\\
&\longrightarrow
\begin{Bmatrix}
x=\infty & 0 & 1\\
2\lambda_{0,2}+\lambda_{1,1}+\lambda_{2,1}
& -\lambda_{0,2}-\lambda_{2,2} & -\lambda_{0,2}-\lambda_{1,2}
\end{Bmatrix} \end{split}\label{eq:redGG} \end{equation} because $\mu=\lambda_{0,1} + \lambda_{1,1} + \lambda_{2,1} -1
=-\lambda_{0,2} - \lambda_{1,2} - \lambda_{2,2}$. Hence the necessary and sufficient condition for the irreducibility of the universal operator given by \eqref{eq:irrmax} is \begin{equation*}
\begin{cases}
\lambda_{0,1}+\lambda_{1,1}+\lambda_{2,1}\notin\mathbb Z,\\
\lambda_{0,2}+\lambda_{1,1}+\lambda_{2,1}\notin\mathbb Z,\\
\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,1}\notin\mathbb Z,\\
\lambda_{0,1}+\lambda_{1,1}+\lambda_{2,2}\notin\mathbb Z,
\end{cases} \end{equation*} which is equivalent to \begin{equation}
\lambda_{0,i}+\lambda_{1,1}+\lambda_{2,j}\notin\mathbb Z
\quad\text{for \ }i=1,2 \text{ and }j=1,2. \end{equation} The rigid tuple $\mathbf m=11,11,11$ corresponds to the real root $\alpha_{\mathbf m}= 2\alpha_0+\alpha_{0,1}+\alpha_{1,1}+\alpha_{2,1}$ under the notation in \S\ref{sec:KM}. Then $\Delta(\mathbf m)
=\{\alpha_0,\alpha_0+\alpha_{j,1}\,;\,j=0,1,2\}$
and $(\Lambda|\alpha_0)=\lambda_{0,1}+\lambda_{1,1}+\lambda_{2,1}$
and $(\Lambda|\alpha_0+\alpha_{0,1})=\lambda_{0,2}+\lambda_{1,1}+\lambda_{2,1}$, etc. under the notation in Theorem~\ref{thm:irrKac}.
The Riemann scheme for the Gauss hypergeometric series ${}_2F_1(a,b,c;z)$ is given by $\begin{Bmatrix}
x=\infty & 0 & 1\\
a & 0 & 0\\
b & 1-c & c-a-b \end{Bmatrix}$ and therefore the condition for the irreducibility is \begin{equation}
a\notin\mathbb Z,\ b\notin\mathbb Z,\ c-b\notin\mathbb Z\text{ \ and \ } c-a\notin\mathbb Z. \end{equation}
{\rm ii)} The reduction of the Riemann scheme for the equation corresponding to ${}_3F_2(\alpha_1,\alpha_2,\alpha_3,\beta_1,\beta_2;x)$ is \begin{equation}\begin{split}
&\begin{Bmatrix}
x = \infty & 0 & 1\\
\alpha_1 & 0 & [0]_{(2)} \\
\alpha_2 & 1-\beta_1 & -\beta_3\\
\alpha_3 & 1-\beta_2
\end{Bmatrix}\qquad(\sum_{i=1}^3\alpha_i=\sum_{i=1}^3\beta_i)\\ &\longrightarrow
\begin{Bmatrix}
x = \infty & 0 & 1\\
\alpha_2-\alpha_1+1 & \alpha_1-\beta_1 & 0\\
\alpha_3-\alpha_1+1 & \alpha_1-\beta_2 & \alpha_1-\beta_3-1
\end{Bmatrix} \end{split}\end{equation} with $\mu=\alpha_1-1$. Hence Theorem~\ref{thm:irred} says that the condition for the irreducibility equals \begin{align*}
&\begin{cases}
\alpha_i\notin\mathbb Z&(i=1,2,3),\\
\alpha_1-\beta_j\notin\mathbb Z&(j=1,2)
\end{cases} \intertext{together with}
&\alpha_i-\beta_j\notin\mathbb Z\qquad(i=2,3,\ j=1,2). \end{align*} Here the second condition follows from {\rm i)}. Hence the condition for the irreducibility is \begin{equation}
\alpha_i\notin\mathbb Z\text{ \ and \ }
\alpha_i-\beta_j\notin\mathbb Z\quad(i=1,2,3,\ j=1,2). \end{equation}
{\rm iii)} The reduction of the even family is as follows: \begin{align*}
\begin{Bmatrix}
x = \infty & 0 & 1\\
\alpha_1 & [0]_{(2)} & [0]_{(2)} \\
\alpha_2 & 1-\beta_1 & [-\beta_3]_{(2)}\\
\alpha_3 & 1-\beta_2\\
\alpha_4
\end{Bmatrix}
\longrightarrow&
\begin{Bmatrix}
x = \infty & 0 & 1\\
\alpha_2-\alpha_1+1 & 0 & 0 \\
\alpha_3-\alpha_1+1 & \alpha_1-\beta_1 & [\alpha_1-\beta_3-1]_{(2)}\\
\alpha_4-\alpha_1+1 & \alpha_1-\beta_2
\end{Bmatrix}\\
\xrightarrow{(x-1)^{-\alpha_1+\beta_3+1}}&
\begin{Bmatrix}
x = \infty & 0 & 1\\
\alpha_2-\beta_3 & 0 & -\alpha_1+\beta_3+1 \\
\alpha_3-\beta_3 & \alpha_1-\beta_1 & [0]_{(2)}\\
\alpha_4-\beta_3 & \alpha_1-\beta_2
\end{Bmatrix}. \end{align*} Hence the condition for the irreducibility is \begin{align*}
&\begin{cases}
\alpha_i\notin\mathbb Z&(i=1,2,3,4),\\
\alpha_1-\beta_3\notin\mathbb Z
\end{cases} \intertext{together with}
&\begin{cases}
\alpha_i-\beta_3\notin\mathbb Z&(i=2,3,4).\\
\alpha_1+\alpha_i-\beta_j-\beta_3\notin\mathbb Z&(i=2,3,4,\ j=1,2)
\end{cases} \end{align*} by the result in ii). Thus the condition is \begin{equation}
\begin{split}
&\alpha_i\notin\mathbb Z,\ \alpha_i-\beta_3\notin\mathbb Z\text{ and }
\alpha_1+\alpha_k-\beta_j-\beta_3\notin\mathbb Z\\
&\qquad\qquad(i=1,2,3,4,\ j=1,2,\ k=2,3,4).
\end{split} \end{equation} Hence the condition for the irreducibility for the equation with the Riemann scheme \begin{equation}
\begin{Bmatrix}
\lambda_{0,1} & [\lambda_{1,1}]_{(2)} & [\lambda_{2,1}]_{(2)}\\
\lambda_{0,2} & \lambda_{1,2} & [\lambda_{2,2}]_{(2)}\\
\lambda_{0,3} & \lambda_{1,3}\\
\lambda_{0,4}
\end{Bmatrix} \end{equation}
of type $1111,211,22$ is \begin{equation}\label{eq:e4irred}
\begin{cases}
\lambda_{0,\nu} + \lambda_{1,1}+\lambda_{2,k}\notin\mathbb Z
&(\nu=1,2,3,4,\ k=1,2)\\
\lambda_{0,\nu}+\lambda_{0,\nu'} + \lambda_{1,1}+ \lambda_{1,2} + \lambda_{2,1}
+\lambda_{2,2}\notin\mathbb Z&(1\le\nu<\nu'\le 4).
\end{cases} \end{equation} This condition corresponds to the rigid decompositions \begin{equation}\label{eq:e4ddec}
1^4,21^2,2^2 = 1,10,1\oplus1^3,11^2,21 = 1^2,11,1^2\oplus1^2,11,1^2, \end{equation} which are also important in the connection formula.
{\rm iv)} (generalized Jordan-Pochhammer)\index{Jordan-Pochhammer!generalized} The reduction of the universal operator of the rigid spectral type $32,32,32,32$ is as follows: \begin{align*}
&\begin{Bmatrix}
[\lambda_{0,1}]_{(3)} & [\lambda_{1,1}]_{(3)}
&[\lambda_{2,1}]_{(3)} & [\lambda_{3,1}]_{(3)}\\
[\lambda_{0,2}]_{(2)} & [\lambda_{1,2}]_{(2)} & [\lambda_{2,2}]_{(2)}
& [\lambda_{3,2}]_{(2)}
\end{Bmatrix}
\quad(3\sum_{j=0}^3\lambda_{j,1}+2\sum_{j=0}^3\lambda_{j,2}=4)\\
&\longrightarrow
\begin{Bmatrix}
\lambda_{0,1}-2\mu & \lambda_{1,1} &\lambda_{2,1} & \lambda_{3,1}\\
[\lambda_{0,2}-\mu]_{(2)}& [\lambda_{1,2}+\mu]_{(2)} & [\lambda_{2,2}+\mu]_{(2)}
& [\lambda_{3,2}+\mu]_{(2)}
\end{Bmatrix} \end{align*} with $\mu=\lambda_{0,1}+\lambda_{1,1}+\lambda_{2,1}+\lambda_{3,1}-1$. Hence the condition for the irreducibility is \begin{equation}\label{eq:red32}
\begin{cases}
\sum_{j=0}^3\lambda_{j,1+\delta_{j,k}}\notin\mathbb Z&(k=0,1,2,3,4),\\
\sum_{j=0}^3(1+\delta_{j,k})\lambda_{j,1}
+\sum_{j=0}^3(1-\delta_{j,k})\lambda_{j,2}\notin\mathbb Z &(k=0,1,2,3,4).
\end{cases} \end{equation} Note that under the notation defined by Definition~\ref{def:linidx} we have \begin{equation}\label{eq:32idx}
\idx_{\mathbf m}
\bigl(\lambda_{0,1}+\lambda_{1,1}+\lambda_{2,1}+\lambda_{3,1}\bigr)=2 \end{equation} and the index of any other linear form in \eqref{eq:red32} is 1.
In general, the universal operator with the Riemann scheme \begin{align}
\begin{split}
&\begin{Bmatrix}
[\lambda_{0,1}]_{(k)} & [\lambda_{1,1}]_{(k)}
&[\lambda_{2,1}]_{(k)} & [\lambda_{3,1}]_{(k)}\\
[\lambda_{0,2}]_{(k-1)} & [\lambda_{1,2}]_{(k-1)} & [\lambda_{2,2}]_{(k-1)}
& [\lambda_{3,2}]_{(k-1)}
\end{Bmatrix}\\
&\qquad(k\sum_{j=0}^3\lambda_{j,1}+(k-1)\sum_{j=0}^3\lambda_{j,2}=2k) \end{split} \end{align} is irreducible if and only if \begin{equation}\label{eq:redgen}
\begin{cases}
\sum_{j=0}^3(\nu-\delta_{j,k})\lambda_{j,1}
+\sum_{j=0}^3(\nu-1+\delta_{j,k})\lambda_{j,1} \notin\mathbb Z&(k=0,1,2,3,4),\\
\sum_{j=0}^3(\nu'+\delta_{j,k})\lambda_{j,1}
+\sum_{j=0}^3(\nu'-\delta_{j,k})\lambda_{j,2}\notin\mathbb Z &(k=0,1,2,3,4),
\end{cases} \end{equation} for any integers $\nu$ and $\nu'$ satisfying $1\le 2\nu\le k$ and $1\le 2\nu'\le k-1$.
The rigid decomposition \begin{equation}
65,65,65,65=12,21,21,21\oplus 53,44,44,44 \end{equation} gives an example of the decomposition $\mathbf m=\mathbf m'\oplus\mathbf m''$ with $\supp\alpha_{\mathbf m}=\supp\alpha_{\mathbf m'}=\supp\alpha_{\mathbf m''}$.
{\rm v)} The rigid Fuchsian differential equation with the Riemann scheme \index{tuple of partitions!rigid!831,93,93,93} \[
\begin{Bmatrix}
x=0 & 1& c_3& c_4&\infty\\ [0]_{(9)} & [0]_{(9)} & [0]_{(9)} & [0]_{(9)} & [e_0]_{(8)}\\ [a]_{(3)} & [b]_{(3)} & [c]_{(3)} & [d]_{(3)} & [e_1]_{(3)}\\
& & & & e_2 \end{Bmatrix} \] is reducible when \[
a+b+c+d+3e_0+e_1\in\mathbb Z, \] which is equivalent to $\frac13(e_0-e_2-1)\in\mathbb Z$ under the Fuchs relation. At the generic point of this reducible condition, the spectral types of the decomposition in the Grothendieck group of the monodromy is \[
93,93,93,93,831=31,31,31,31,211+31,31,31,31,310+31,31,31,31,310. \] Note that the following reduction of the spectral types \[ \begin{matrix}
93,93,93,93,831&\to&13,13,13,13,031&\to&10,10,10,10,001\\
31,31,31,31,211&\to&11,11,11,11,011\\
31,31,31,31,310&\to&01,01,01,01,010 \end{matrix} \] and $\idx(31,31,31,31,211)=-2$. \end{exmp}
\section{Shift operators}\label{sec:shift} In this section we study an integer shift of spectral parameters $\lambda_{j,\nu}$ of the Fuchsian equation $P_{\mathbf m}(\lambda)u=0$. Here $P_{\mathbf m}(\lambda)$ is the universal operator (cf.~Theorem~\ref{thm:univmodel}) corresponding to the spectral type $\mathbf m =(m_{j,\nu}\bigr)_{\substack{j=0,\dots,p\\\nu=1,\dots,n_j}}$. For simplicity, we assume that $\mathbf m$ is rigid in this section unless otherwise stated.
\subsection{Construction of shift operators and recurrence relations} \label{sec:shift1} First we construct shift operators for general shifts. \begin{defn} For $\mathbf m=\bigl(m_{j,\nu}\bigr)_{\substack{j=0,\dots,p\\\nu=1,\dots,n_j}} \in\mathcal P^{(n)}_{p+1}$, a set of integers $\bigl(\epsilon_{j,\nu}\bigr)_{\substack{j=0,\dots,p\\\nu=1,\dots,n_j}}$ parametrized by $j$ and $\nu$ is called a \textsl{shift compatible to} $\mathbf m$ if \index{characteristic exponent!(compatible) shift} \begin{equation}
\sum_{j=0}^p\sum_{\nu=1}^{n_j}\epsilon_{j,\nu}m_{j,\nu}=0. \end{equation} \end{defn}
\index{shift operator} \index{00Rme@$R_{\mathbf m}(\epsilon,\lambda)$} \begin{thm}[shift operator]\label{thm:irredrigid} Fix a shift\/ $(\epsilon_{j,\nu})$ compatible to $\mathbf m\in\mathcal P^{(n)}_{p+1}$. Then there is a \textsl{shift operator}\index{shift operator} $R_{\mathbf m}(\epsilon,\lambda)\in W[x] \otimes\mathbb C[\lambda_{j,\nu}]$ which gives a homomorphism of the equation $P_{\mathbf m}(\lambda')v=0$ to $P_{\mathbf m}(\lambda)u=0$ defined by $v=R_{\mathbf m}(\epsilon,\lambda)u$. Here the Riemann scheme of $P_{\mathbf m}(\lambda)$ is $\{\lambda_{\mathbf m}\}=\bigl\{[\lambda_{j,\nu}]_{(m_{j,\nu})}\bigr\} _{\substack{j=0,\dots,p\\\nu=1,\dots,n_j}}$ and that of $P_{\mathbf m}(\lambda')$ is $\{\lambda'_{\mathbf m}\}$ defined by $\lambda'_{j,\nu}=\lambda_{j,\nu}+\epsilon_{j,\nu}$. Moreover we may assume $\ord R_{\mathbf m}(\epsilon,\lambda) < \ord\mathbf m$ and $R_{\mathbf m}(\epsilon,\lambda)$ never vanishes as a function of $\lambda$ and then $R_{\mathbf m}(\epsilon,\lambda)$ is uniquely determined up to a constant multiple.
Putting \begin{equation}\label{eq:sfttau}
\tau=\bigl(\tau_{j,\nu}\bigr)
_{\substack{0\le j\le p\\1\le\nu\le n_j}} \text{ \ with \ }
\tau_{j,\nu}:=\bigl(2+(p-1)n\bigr)\delta_{j,0}-m_{j,\nu} \end{equation} and $d=\ord R_{\mathbf m}(\epsilon,\lambda)$, we have \begin{equation}\label{eq:Sid}
P_{\mathbf m}(\lambda+\epsilon)R_{\mathbf m}(\epsilon,\lambda)
=(-1)^dR_{\mathbf m}(\epsilon,\tau-\lambda-\epsilon)^*
P_{\mathbf m}(\lambda) \end{equation} under the notation in\/ {\rm Theorem~\ref{thm:prod} ii).} \end{thm} \begin{proof} We will prove the theorem by the induction on $\ord\mathbf m$. The theorem is clear if $\ord\mathbf m=1$.
We may assume that $\mathbf m$ is monotone. Then the reduction $\{\tilde\lambda_{\tilde{\mathbf m}}\}$ of the Riemann scheme is defined by \eqref{eq:mc2}. Hence putting \begin{equation}
\begin{cases}
{\tilde \epsilon}_1 = \epsilon_{0,1}+\cdots+\epsilon_{p,1},\\
{\tilde \epsilon}_{j,\nu} =\epsilon_{j,\nu} +
\bigl((-1)^{\delta_{j,0}}-\delta_{\nu,1}\bigr){\tilde \epsilon}_1
&(j=0,\dots,p,\ \nu=1,\dots,n_j),
\end{cases} \end{equation} there is a shift operator $R(\tilde\epsilon, \tilde\lambda)$ of the equation $P_{\tilde{\mathbf m}}(\tilde\lambda')\tilde v=0$ to $P_{\tilde{\mathbf m}}(\tilde\lambda)\tilde u=0$ defined by $\tilde v=R(\tilde\epsilon, \tilde\lambda)\tilde u$. Note that \begin{align*}
P_{\tilde{\mathbf m}}(\tilde\lambda)
&=\partial_{max}P_{\mathbf m}(\lambda)
= \Ad\bigl(\prod_{j=1}^p(x-c_j)^{\lambda_{j,1}}\bigr)
\prod_{j=1}^p(x-c_j)^{m_{j,1}-d}\partial^{-d}\!\Ad(\partial^{-\mu})\\
&\quad
\prod_{j=1}^p(x-c_j)^{-m_{j,1}}
\Ad\bigl(\prod_{j=1}^p(x-c_j)^{-\lambda_{j,1}}\bigr)
P_{\mathbf m}(\lambda), \allowdisplaybreaks\\
P_{\tilde{\mathbf m}}(\tilde\lambda')
&=\partial_{max}P_{\mathbf m}(\lambda')
= \Ad\bigl(\prod_{j=1}^p(x-c_j)^{\lambda_{j,1}}\bigr)
\prod_{j=1}^p(x-c_j)^{m_{j,1}-d}\partial^{-d}\!\Ad(\partial^{-\mu'})\\
&\quad
\prod_{j=1}^p(x-c_j)^{-m_{j,1}}
\Ad\bigl(\prod_{j=1}^p(x-c_j)^{-\lambda'_{j,1}}\bigr)
P_{\mathbf m}(\lambda'). \end{align*}
Suppose $\lambda_{j,\nu}$ are generic. Let $u(x)$ be a local solution of $P_{\mathbf m}(\lambda)u=0$ at $x=c_1$ corresponding to a characteristic exponent different from $\lambda_{1,1}$. Then \[
\tilde u(x):=\prod_{j=1}^p(x-c_j)^{\lambda_{j,1}}
\partial^{-\mu}\prod_{j=1}^p(x-c_j)^{-\lambda_{j,1}}u(x) \] satisfies $P_{\tilde{\mathbf m}}(\tilde\lambda)\tilde u(x)=0$. Putting \[
\begin{split}
\tilde v(x)&:=R(\tilde\epsilon,\tilde\lambda)\tilde u(x),
\allowdisplaybreaks\\
v(x)&:=\prod_{j=1}^p(x-c_j)^{\lambda'_{j,1}}
\partial^{\mu'}\prod_{j=1}^p(x-c_j)^{\lambda'_{j,1}}\tilde v(x),
\allowdisplaybreaks\\
\tilde R(\tilde\epsilon,\tilde\lambda)&
:=\Ad(\prod_{j=1}^p(x-c_j)^{\lambda_{j,1}})R(\tilde\epsilon,\tilde\lambda)
\end{split} \] we have $P_{\tilde{\mathbf m}}(\tilde\lambda')\tilde u(x)=0$, $P_{\mathbf m}(\lambda')v(x)=0$ and \[
\prod_{j=1}^p(x-c_j)^{\epsilon_{j,1}}
\partial^{-\mu'}\prod_{j=1}^p(x-c_j)^{-\lambda_{j,1}'}
v(x)=\tilde R(\tilde\epsilon,\tilde\lambda)\partial^{-\mu}
\prod_{j=1}^p(x-c_j)^{-\lambda_{j,1}}u(x). \]
In general, if \begin{equation}\label{eq:sht1}
S_2\prod_{j=1}^p(x-c_j)^{\epsilon_{j,1}}
\partial^{-\mu'}\prod_{j=1}^p(x-c_j)^{-\lambda_{j,1}'}
v(x)=S_1\partial^{-\mu}
\prod_{j=1}^p(x-c_j)^{-\lambda_{j,1}}u(x) \end{equation} with $S_1$, $S_2\in W[x]$, we have \begin{equation}\label{eq:sht2}
R_2 v(x) = R_1 u(x) \end{equation} by putting \begin{equation}\label{eq:sht3}\begin{split}
R_1 &=
\prod_{j=1}^p(x-c_j)^{\lambda_{j,\nu}+k_{1,j}}
\partial^{\mu+\ell}\prod_{j=1}^p(x-c_j)^{k_{2,j}} S_1\prod_{j=1}^{\epsilon_{j,1}}\partial^{-\mu}
\prod_{j=1}^p(x-c_j)^{-\lambda_{j,\nu}},\\
R_2 &=
\prod_{j=1}^p(x-c_j)^{\lambda_{j,\nu}+k_{1,j}}
\partial^{\mu+\ell}\prod_{j=1}^p(x-c_j)^{k_{2,j}}
S_2\prod_{j=1}^{\epsilon_{j,1}}\partial^{-\mu'}
\prod_{j=1}^p(x-c_j)^{-\lambda_{j,\nu}'} \end{split}\end{equation} with suitable integers $k_{1,j}$, $k_{2,j}$ and $\ell$ so that $R_1,\ R_2\in W[x;\lambda]$.
We choose a non-zero polynomial $S_2\in\mathbb C[x]$ so that $S_1=S_2\tilde R(\tilde\epsilon,\tilde\lambda)\in W[x]$. Since $P_{\mathbf m}(\lambda')$ is irreducible in $W(x;\lambda)$ and $R_2v(x)$ is not zero, there exists $R_3 \in W(x;\xi)$ such that $R_3 R_2-1\in W(x;\lambda)P_{\mathbf m}(\lambda')$. Then $v(x)=R u(x)$ with the operator $R=R_3 R_1\in W(x;\lambda)$.
Since the equations $P_{\mathbf m}(\lambda)u=0$ and $P_{\mathbf m}(\lambda')v=0$ are irreducible $W(x;\lambda)$-modules, the correspondence $v=Ru$ gives an isomorphism between these two modules. Since any solutions of these equations are holomorphically continued along the path contained in $\mathbb C\setminus\{c_1,\dots,c_p\}$, the coefficients of the operator $R$ are holomorphic in $\mathbb C\setminus\{c_1,\dots,c_p\}$. Multiplying $R$ by a suitable element of $\mathbb C(\lambda)$, we may assume $R\in W(x)\otimes \mathbb C[\lambda]$ and $R$ does not vanish at any $\lambda_{j,\nu}\in\mathbb C$.
Put $f(x)=\prod_{j=1}^p(x-c_j)^n$. Since $R_{\mathbf m}(\epsilon,\lambda)$ is a shift operator, there exists $S_{\mathbf m}(\epsilon,\lambda)\in W(x;\lambda)$ such that \begin{equation}\label{eq:sft01}
f^{-1}P_{\mathbf m}(\lambda+\epsilon)R_{\mathbf m}(\epsilon,\lambda)
= S_{\mathbf m}(\epsilon,\lambda)f^{-1}P_{\mathbf m}(\lambda). \end{equation} Then Theorem~\ref{thm:prod} ii) shows \begin{align}
R_{\mathbf m}(\epsilon,\lambda)^*
\bigl(f^{-1}P_{\mathbf m}(\lambda+\epsilon)\bigr)^*
&=\bigl(f^{-1}P_{\mathbf m}(\lambda)\bigr)^*
S_{\mathbf m}(\epsilon,\lambda)^*,\notag\allowdisplaybreaks\\
R_{\mathbf m}(\epsilon,\lambda)^*\cdot f^{-1}
P_{\mathbf m}(\lambda+\epsilon)^\vee
&=f^{-1}P_{\mathbf m}(\lambda)^\vee\cdot
S_{\mathbf m}(\epsilon,\lambda)^*,\notag\allowdisplaybreaks\\
R_{\mathbf m}(\epsilon,\lambda)^*f^{-1}P_{\mathbf m}(\rho-\lambda-\epsilon)
&=f^{-1}P_{\mathbf m}(\rho-\lambda)S_{\mathbf m}(\epsilon,\lambda)^*,
\notag\allowdisplaybreaks\\
R_{\mathbf m}(\epsilon,\rho-\mu-\epsilon)^*f^{-1}P_{\mathbf m}(\mu)
&=f^{-1}P_{\mathbf m}(\mu+\epsilon)S_{\mathbf m}(\epsilon,\rho-\mu-\epsilon)^*.
\label{eq:sht02} \end{align} Here we use the notation \eqref{eq:dualop} and put $\rho_{j,\nu}=2(1-n)\delta_{j,0}+n-m_{j,\nu}$ and $\mu=\rho-\lambda-\epsilon$. Comparing \eqref{eq:sht02} with \eqref{eq:sft01}, we see that $S_{\mathbf m}(\epsilon,\lambda)$ is a constant multiple of the operator $R_{\mathbf m}(\epsilon,\rho-\lambda-\epsilon)^*$ and $fR_{\mathbf m}(\epsilon,\rho-\lambda-\epsilon)^*f^{-1} =\bigl(f^{-1}R_{\mathbf m}(\epsilon,\rho-\lambda-\epsilon)f\bigr)^* =R_{\mathbf m}(\epsilon,\tau-\lambda-\epsilon)^*$ and we have \eqref{eq:Sid}. \end{proof}
Note that the operator $R_{\mathbf m}(\epsilon,\lambda)$ is uniquely defined up to a constant multiple.
The following theorem gives a recurrence relation among specific local solutions with a rigid spectral type and a relation between the shift operator $R_{\mathbf m}(\epsilon,\lambda)$ and the universal operator $P_{\mathbf m}(\lambda)$. \begin{thm}\label{thm:shifm1} Retain the notation in {\rm Corollary~\ref{cor:irred}} and\/ {\rm Theorem~\ref{thm:shiftC}} with a rigid tuple $\mathbf m$. Assume $m_{j,n_j}=1$ for $j=0$, $1$ and $2$. Put $\epsilon=(\epsilon_{j,\nu})$, $\epsilon'=(\epsilon'_{j,\nu})$, \begin{equation}
\epsilon_{j,\nu} = \delta_{j,1}\delta_{\nu,n_1}-\delta_{j,2}\delta_{\nu,n_2}
\text{ \ and \ }
\epsilon'_{j,\nu} = \delta_{j,0}\delta_{\nu,n_0}-\delta_{j,2}\delta_{\nu,n_2} \end{equation} for $j=0,\dots,p$ and $\nu=1,\dots,n_j$.
{\rm i)} Define $Q_{\mathbf m}(\lambda)\in W(x;\lambda)$ so that $Q_{\mathbf m}(\lambda) P_{\mathbf m}(\lambda+\epsilon') - 1 \in W(x;\lambda)P_{\mathbf m}(\lambda+\epsilon)$. Then \begin{equation}\label{eq:shtUniv}
R_{\mathbf m}(\epsilon,\lambda) - C(\lambda)Q_{\mathbf m}(\lambda)
P_{\mathbf m}(\lambda+\epsilon') \in W(x;\lambda)P_{\mathbf m}(\lambda) \end{equation} with a rational function $C(\lambda)$ of $\lambda_{j,\nu}$.
{\rm ii)} Let $u_{\lambda}(x)$ be the local solution of $P_{\mathbf m}(\lambda)u=0$ such that $u_{\lambda}(x)\equiv (x-c_1)^{\lambda_{1,n_1}}\mod (x-c_1)^{\lambda_{1,n_1}+1}O_{c_1}$ for generic $\lambda_{j,\nu}$. Then we have the recurrence relation \begin{align}\label{eq:recF}
u_{\lambda}(x) &=
u_{\lambda+\epsilon'}(x)+(c_1-c_2)\prod_{\nu=0}^{K-1}
\frac{\lambda(\nu+1)_{1,n_1}-\lambda(\nu)_{1,\ell(\nu)_1}+1}
{\lambda(\nu)_{1,n_1}-\lambda(\nu)_{1,\ell(\nu)_1}+1}\cdot
u_{\lambda+\epsilon}(x). \end{align} \end{thm}
\begin{proof} Under the notation in Corollary~\ref{cor:irred}, $\ell(k)_j\ne n_j$ for $j=0,1,2$ and $k=0,\dots,K-1$ and therefore the operation $\partial_{max}^K$ on $P_{\mathbf m}(\lambda)$ is equals to $\partial_{max}^K$ on $P_{\mathbf m}(\lambda+\epsilon)$ if they are realized by the product of the operators of the form \eqref{eq:opred}. Hence by the induction on $K$, the proof of Theorem~\ref{thm:irredrigid} (cf.~\eqref{eq:sht1}, \eqref{eq:sht2} and \eqref{eq:sht3}) shows \begin{equation}\label{eq:shifm1}
P_{\mathbf m}(\lambda+\epsilon')u(x)=P_{\mathbf m}(\lambda+\epsilon')v(x) \end{equation} for suitable functions $u(x)$ and $v(x)$ satisfying $P_{\mathbf m}(\lambda)u(x)=P_{\mathbf m}(\lambda+\epsilon)v(x)=0$ and moreover \eqref{eq:recF} is calculated by \eqref{eq:IcP}. Note that the identities \begin{align*}
(c_1-c_2)\prod_{j=1}^p(x-c_j)^{\lambda_j+\epsilon'_j}
&=\prod_{j=1}^p(x-c_j)^{\lambda_j}-\prod_{j=1}^p(x-c_j)^{\lambda_j+\epsilon_j},
\allowdisplaybreaks\\
\Bigl(\partial - \sum_{j=1}^p\frac{\lambda_j+\epsilon'_j}
{x-c_j}\Bigr)\prod_{j=1}^p(x-c_j)^{\lambda_j}
&= \Bigl(\partial - \sum_{j=1}^p\frac{\lambda_j+\epsilon'_j}{x-c_j}\Bigr)
\prod_{j=1}^p(x-c_j)^{\lambda_j+\epsilon_j} \end{align*} correspond to \eqref{eq:recF} and \eqref{eq:shifm1}, respectively, when $K=0$.
Note that \eqref{eq:shifm1} may be proved by \eqref{eq:recF}. The claim i) in this theorem follows from the fact
$v(x)=Q_{\mathbf m}(\lambda)P_{\mathbf m}(\lambda+\epsilon')v(x) =Q_{\mathbf m}(\lambda)P_{\mathbf m}(\lambda+\epsilon')u(x)$. \end{proof}
In general, we have the following theorem for the recurrence relation.
\begin{thm}[recurrence relations] Let\/ $\mathbf m\in\mathcal P^{(n)}$ be a rigid tuple with\/ $m_{1,n_1}=1$ and let\/ $u_1(\lambda,x)$ be the normalized solution of the equation\/ $P_{\mathbf m}(\lambda)u=0$ with respect to the exponent\/ $\lambda_{1,n_1}$ at $x=c_1$. Let\/ $\epsilon^{(i)}$ be shifts compatible to\/ $\mathbf m$ for\/ $i=0,\dots,n$. Then there exists polynomial functions\/ $r_i(x,\lambda)\in\mathbb C[x,\lambda]$ such that $(r_0,\dots,r_n)\ne 0$ and \begin{equation}
\sum_{i=0}^{n} r_i(x,\lambda)u_1(\lambda+\epsilon^{(i)},x) = 0. \end{equation} \end{thm}
\begin{proof} There exist $R_i\in \mathbb C(\lambda)R_{\mathbf m}(\epsilon^{(i)},\lambda)$ satisfying $u_1(\lambda+\epsilon^{(i)},x)=R_iu_1(\lambda,x)$ and $\ord R_i< n$. We have $r_i(x,\lambda)$ with $\sum_{i=0}^n r_i(x,\lambda)R_i=0$ and the claim. \end{proof}
\begin{exmp}[Gauss hypergeometric equation] \index{hypergeometric equation/function!Gauss} Let $P_\lambda u=0$ and $P_{\lambda'}v=0$ be Fuchsian differential equations with the Riemann Scheme \[
\begin{Bmatrix}
x=\infty & 0 & 1\\
\lambda_{0,1} & \lambda_{1,1} & \lambda_{2,1}\\
\lambda_{0,2} & \lambda_{1,2} & \lambda_{2,2}
\end{Bmatrix}\text{ and }
\begin{Bmatrix}
x=\infty & 0 & 1\\
\lambda'_{0,1}=\lambda_{0,1} &
\lambda'_{1,1}=\lambda_{1,1} & \lambda'_{2,1}=\lambda_{2,1}\\
\lambda'_{0,2}=\lambda_{0,2} &
\lambda'_{1,2}=\lambda_{1,2}+1 & \lambda_{2,2}=\lambda_{2,2}-1
\end{Bmatrix}, \] respectively. Here the operators $P_\lambda=P_{\lambda_{0,1},\lambda_{0,2},\lambda_{1,1}, \lambda_{1,2},\lambda_{2,1},\lambda_{2,1}}$ and $P_{\lambda'}$ are given in \eqref{eq:GH2}. The normalized local solution $u_{\lambda}(x)$ of $P_\lambda u=0$ corresponding to the exponent $\lambda_{1,2}$ at $x=0$ is \begin{equation} x^{\lambda_{1,2}}(1-x)^{\lambda_{2,1}}
F(\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,1},
\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,1},
1-\lambda_{1,1}+\lambda_{1,2};x). \end{equation} By the reduction $\begin{Bmatrix}
x=\infty & 0 & 1\\
\lambda_{0,1} & \lambda_{1,1} & \lambda_{2,1}\\
\lambda_{0,2} & \lambda_{1,2} & \lambda_{2,2}
\end{Bmatrix}\to
\begin{Bmatrix}
x=\infty & 0 & 1\\
\lambda_{0,2} -\mu&
\lambda_{1,2} +\mu&
\lambda_{2,2} +\mu
\end{Bmatrix} $ with $\mu=\lambda_{0,1}+\lambda_{1,1}+\lambda_{2,1}-1$, the recurrence relation \eqref{eq:recF} means \begin{align*}
&x^{\lambda_{1,2}}(1-x)^{\lambda_{2,1}}
F(\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,1},
\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,1},
1-\lambda_{1,1}+\lambda_{1,2};x)\\
&=x^{\lambda_{1,2}}(1-x)^{\lambda_{2,1}}
F(\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,1},
\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,1}+1,
1-\lambda_{1,1}+\lambda_{1,2};x)\\
&\quad{}-\frac{\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,1}}
{1-\lambda_{1,1}+\lambda_{1,2}}
x^{\lambda_{1,2}+1}(1-x)^{\lambda_{2,1}}\\
&\qquad \cdot F(\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,1}+1,
\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,1}+1,
2-\lambda_{1,1}+\lambda_{1,2};x), \end{align*} which is equivalent to the recurrence relation \begin{equation}
F(\alpha,\beta,\gamma,x)=F(\alpha,\beta+1,\gamma;x)
-\frac{\alpha}{\gamma}x F(\alpha+1,\beta+1,\gamma+1;x). \end{equation}
Using the expression \eqref{eq:GH2}, we have \begin{align*}
P_{\lambda+\epsilon'} - P_{\lambda}&
=x^2(x-1)\partial+\lambda_{0,1}x^2-(\lambda_{0,1}+\lambda_{2,1})x,\\
P_{\lambda+\epsilon'} - P_{\lambda+\epsilon}&=x(x-1)^2\partial+\lambda_{0,1}x^2
-(\lambda_{0,1}+\lambda_{1,1})x-\lambda_{1,1},\\
(x-1)P_{\lambda+\epsilon}&= \bigl(x(x-1)\partial +(\lambda_{0,2}-2)x+\lambda_{1,2}
+1\bigr)
\bigl(P_{\lambda+\epsilon'} - P_{\lambda+\epsilon}\bigr)\\
&{}\quad-(\lambda_{0,1}+\lambda_{1,1}+\lambda_{2,1})(\lambda_{0,2}
+\lambda_{1,2}+\lambda_{2,1})x(x-1),\\
x^{-1}(x-1)^{-1}&\bigl(x(x-1)\partial +(\lambda_{0,2}-2)x+\lambda_{1,2}
+1\bigr)(P_{\lambda+\epsilon'} - P_{\lambda})-(x-1)^{-1}P_\lambda\\
&=-(\lambda_{0,1}+\lambda_{1,1}+\lambda_{2,1})
\bigl(x\partial-\lambda_{1,2}-\frac{\lambda_{2,1} x}{x-1}\bigr) \end{align*} and hence \eqref{eq:shtUniv} says \begin{equation}
R_{\mathbf m}(\epsilon,\lambda)=x\partial-\lambda_{1,2}-
\lambda_{2,1}\frac{x}{x-1}. \end{equation} In the same way we have \begin{equation}
R_{\mathbf m}(-\epsilon,\lambda+\epsilon)=
(x-1)\partial-\lambda_{2,2}+1-\lambda_{1,1}\frac{x-1}{x}. \end{equation} Then \begin{equation}
\begin{split}
R_{\mathbf m}(-\epsilon,\lambda+\epsilon)R_{\mathbf m}(\epsilon,\lambda)
&-x^{-1}(x-1)^{-1}P_\lambda\\
&=-(\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,1})
(\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,1})
\end{split} \end{equation} and since $-R_{\mathbf m}(\epsilon,\tau-\lambda-\epsilon)^* =-\bigl(x\partial+(\lambda_{1,2}+2)+(\lambda_{2,1}+1)\frac x{x-1}\bigr)^* =x\partial - \lambda_{1,2}-1-(\lambda_{2,1}+1)\frac x{x-1}$ with $\tau$ given by \eqref{eq:sfttau}, the identity \eqref{eq:Sid} means \begin{equation}
\begin{split}
P_\lambda R_{\mathbf m}(\epsilon,\lambda)
= \Bigl(x\partial-(\lambda_{1,2}+1) -(\lambda_{2,1}+1)\frac{x}{x-1}\Bigr)
P_{\lambda+\epsilon}.
\end{split} \end{equation} \end{exmp} \begin{rem} Suppose $\mathbf m$ is irreducibly realizable but it is not rigid. If the reductions of $\{\lambda_{\mathbf m}\}$ and $\{\lambda'_{\mathbf m}\}$ to Riemann schemes with a fundamental tuple of partitions are transformed into each other by suitable additions, we can construct a shift operator as in Theorem~\ref{thm:irredrigid}. If they are not so, we need a shift operator for equations whose spectral type are fundamental and such an operator is called a \textsl{Schlesinger transformation}. \end{rem}
Now we examine the condition that a universal operator defines a shift operator.
\begin{thm}[universal operator and shift operator]\label{thm:sftUniv}\index{shift operator} Let\/ $\mathbf m=\bigl(m_{j,\nu}\bigr)_{\substack{0\le j\le p\\1\le\nu\le n_j}}$ and\/ $\mathbf m'=\bigl(m'_{j,\nu}\bigr)_{\substack{0\le j\le p\\1\le\nu\le n_j}} \in \mathcal P_{p+1}$ be irreducibly realizable and monotone. They may not be rigid. Suppose\/ $\ord\mathbf m>\ord\mathbf m'$. Fix $j_0$ with $0\le j_0\le p$. Let $n'_{j_0}$ be a positive integer such that $m'_{j_0,n'_{j_0}}>m'_{j_0,n'_{j_0+1}}=0$ and let $P_{\mathbf m}(\lambda)$ be the universal operator corresponding to $\{\lambda_{\mathbf m}\}$. Putting $\lambda'_{j,\nu}=\lambda_{j,\nu}$ when $(j,\nu)\ne(j_0,n'_{j_0})$, we define the universal operator $P^{j_0}_{\mathbf m'}(\lambda):=P_{\mathbf m'}(\lambda')$ with the Riemann scheme $\{\lambda'_{\mathbf m'}\}$. Here $\lambda'_{j_0,n'_{j_0}}$ is determined by the Fuchs condition. Suppose \begin{equation}\label{eq:sftUinv}
(\alpha_{\mathbf m}|\alpha_{\mathbf m'})\Bigl(=\sum_{j=0}^p\sum_{\nu=1}^{n_j}m_{j,\nu}m'_{j,\nu}
-(p-1)\ord\mathbf m\cdot\ord\mathbf m'\Bigr)
=m_{j_0,n'_{j_0}}m'_{j_0,n'_{j_0}}. \end{equation} Then\/ $\mathbf m'$ is rigid and the universal operator $P^{j_0}_{\mathbf m'}(\lambda)$ is the shift operator $R_{\mathbf m}(\epsilon,\lambda)$: \begin{equation}
\begin{split}
&\Bigl\{
\bigl[\lambda_{j,\nu}\bigr]_{(m_{j,\nu})}
\Bigr\}_{\substack{0\le j\le p\\1\le\nu\le n_j}}
\xrightarrow{R_{\mathbf m}(\epsilon,\lambda)=P^{j_0}_{\mathbf m'}(\lambda)}
\Bigl\{
\bigl[\lambda_{j,\nu}+\epsilon_{j,\nu}\bigr]_{(m_{j,\nu})}
\Bigr\}_{\substack{0\le j\le p\\1\le\nu\le n_j}}\\
&\qquad\text{with \ }\epsilon_{j,\nu}=
\bigl(1-\delta_{j,j_0}\delta_{\nu,n'_{j_0}}\bigr)m'_{j,\nu}
-\delta_{j,0}\cdot(p-1)\ord\mathbf m'. \end{split} \end{equation} \end{thm}
\begin{proof} We may assume $\lambda$ is generic. Let $u(x)$ be the solution of the irreducible differential equation $P_{\mathbf m}(\lambda)u=0$. Then \begin{align*}
P_{\mathbf m'}(\lambda')(x-c_j)^{\lambda_{j,\nu}}\mathcal O_{c_j}&\subset
(x-c_j)^{\lambda_{j,\nu}+(1-\delta_{j,j_0}\delta_{\nu,n'_{j_0}})m'_{j,\nu}}
\mathcal O_{c_j},\\
P_{\mathbf m'}(\lambda')x^{-\lambda_{0,\nu}}\mathcal O_{\!\infty}&\subset
x^{-\lambda_{0,\nu}-(1-\delta_{0,j_0}\delta_{\nu,n'_{j_0}})
m'_{0,\nu}+(p-1)\ord\mathbf m'}
\mathcal O_{\!\infty} \end{align*} and $P_{\mathbf m'}(\lambda')u(x)$ satisfies a Fuchsian differential equation. Hence the fact $R_{\mathbf m}(\epsilon,\lambda)=P_{\mathbf m'}(\lambda')$ is clear from the characteristic exponents of the equation at each singular points. Note that the left hand side of \eqref{eq:sftUinv} is never larger than the right hand side and if they are not equal, $P_{\mathbf m'}(\lambda')u(x)$ satisfies a Fuchsian differential equation with apparent singularities for the solutions $u(x)$ of $P_{\mathbf m}(\lambda)u=0$.
It follows from Lemma~\ref{lem:sumlem} that the condition \eqref{eq:sftUinv} means that at least one of the irreducibly realizable tuples $\mathbf m$ and $\mathbf m'$ is rigid and therefore if $\mathbf m$ is rigid, so is $\mathbf m'$ because $R_{\mathbf m}(\epsilon,\lambda)$ is unique up to constant multiple. \end{proof}
If $\ord\mathbf m'=1$, the condition \eqref{eq:sftUinv} means that $\mathbf m$ is of Okubo type, which will be examined in the next subsection. It will be interesting to examine other cases. When $\mathbf m=\mathbf m'\oplus\mathbf m''$ is a rigid decomposition or $\alpha_{\mathbf m'}\in\Delta(\mathbf m)$, we easily have many examples satisfying \eqref{eq:sftUinv}.
Here we give examples of the pairs $(\mathbf m\,;\mathbf m')$ with $\ord\mathbf m'>1$: \begin{equation}
\begin{aligned}
&(1^n,1^n,n-11\,;1^{n-1},1^{n-1},n-21)&&
(221,32,32,41\,;\,110,11,11,20)\\
&(1^{2m},mm-11,m^2\,;1^2,110,1^2) &&
(1^{2m+1},m^21,m+1m\,;1^2,1^20,11)\\
&(221,221,221\,;110,110,110)\ &&
(211,221,221\,;110,110,110).
\end{aligned} \end{equation}
\subsection{Relation to reducibility} In this subsection, we will examine whether the shift operator defines a $W(x)$-isomorphism or doesn't. \begin{thm}\label{thm:shiftC} \index{00cme@$c_{\mathbf m}(\epsilon;\lambda)$} Retain the notation in\/ {\rm Theorem~\ref{thm:irredrigid}} and define a polynomial function $c_{\mathbf m}(\epsilon;\lambda)$ of $\lambda_{j,\nu}$ by \begin{equation}
R_{\mathbf m}(-\epsilon,\lambda+\epsilon)R_{\mathbf m}(\epsilon,\lambda)
-c_{\mathbf m}(\epsilon;\lambda)
\in \bigl(W[x]\otimes\mathbb C[\lambda]\bigr)P_{\mathbf m}(\lambda). \end{equation}
{\rm i)} Fix $\lambda_{j,\nu}^o\in\mathbb C$. If $c_{\mathbf m}(\epsilon;\lambda^o)\ne0$, the equation $P_{\mathbf m}(\lambda^o)u=0$ is isomorphic to the equation $P_{\mathbf m}(\lambda^o+\epsilon)v=0$. If $c_{\mathbf m}(\epsilon;\lambda^o)=0$, then the equations $P_{\mathbf m}(\lambda^o)u=0$ and $P_{\mathbf m}(\lambda^o+\epsilon)v=0$ are not irreducible.
{\rm ii)} Under the notation in\/ {\rm Proposition~\ref{prop:subrep},} there exists a set $\Lambda$ whose elements $(i,k)$ are in $\{1,\dots,N\}\times \mathbb Z$ such that \begin{equation}
c_{\mathbf m}(\epsilon;\lambda)=C\prod_{(i,k)\in\Lambda}
\bigl(\ell_i(\lambda)-k\bigr) \end{equation} with a constant $C\in\mathbb C^\times$. Here $\Lambda$ may contain some elements $(i,k)$ with multiplicities. \end{thm} \begin{proof} Since $u\mapsto R_{\mathbf m}(-\epsilon,\lambda+\epsilon)R_{\mathbf m}(\epsilon,\lambda)u$ defined an endomorphism of the irreducible equation $P_{\mathbf m}(\lambda)u=0$, the existence of $c_{\mathbf m}(\epsilon;\lambda)$ is clear.
If $c_{\mathbf m}(\epsilon;\lambda^o)=0$, the non-zero homomorphism of $P_{\mathbf m}(\lambda^o)u=0$ to $P_{\mathbf m}(\lambda^o+\epsilon)v=0$ defined by $u=R_{\mathbf m}(\epsilon;\lambda^o)v$ is not surjective nor injective. Hence the equations are not irreducible.
If $c_{\mathbf m}(\epsilon;\lambda^o)\ne0$, then the homomorphism is an isomorphism and the equations are isomorphic to each other.
The claim ii) follows from Proposition~\ref{prop:subrep}. \end{proof}
\begin{thm}\label{thm:isom} Retain the notation in\/ {\rm Theorem~\ref{thm:shiftC}} with a rigid tuple\/ $\mathbf m$. Fix a linear function $\ell(\lambda)$ of $\lambda$ such that the condition $\ell(\lambda)=0$ implies the reducibility of the universal equation $P_{\mathbf m}(\lambda)u=0$.
{\rm i) } If there is no irreducible realizable subtuple\/ $\mathbf m'$ of\/ $\mathbf m$ which is compatible to $\ell(\lambda)$ and $\ell(\lambda+\epsilon)$, $\ell(\lambda)$ is a factor of $c_{\mathbf m}(\epsilon;\lambda)$.
If there is no dual decomposition of\/ $\mathbf m$ with respect to the pair $\ell(\lambda)$ and $\ell(\lambda+\epsilon)$, $\ell(\lambda)$ is not a factor of $c_{\mathbf m}(\epsilon;\lambda)$. Here we define that the decomposition \eqref{eq:subrep} is \textsl{dual} with respect to the pair $\ell(\lambda)$ and $\ell(\lambda+\epsilon)$ if the following conditions are valid. \begin{align} &\text{$\mathbf m'$ is an irreducibly realizable subtuple of\/ $\mathbf m$ compatible to $\ell(\lambda)$},\\ &\text{$\mathbf m''$ is a subtuple of\/ $\mathbf m$ compatible to $\ell(\lambda+\epsilon)$}. \end{align}
{\rm ii)} Suppose there exists a decomposition\/ $\mathbf m=\mathbf m'\oplus\mathbf m''$ with rigid tuples\/ $\mathbf m'$ and\/ $\mathbf m''$ such that
$\ell(\lambda)=|\{\lambda_{\mathbf m}\}|+k$ with $k\in\mathbb Z$ and $\ell(\lambda+\epsilon)=\ell(\lambda)+1$. Then $\ell(\lambda)$ is a factor of $c_{\mathbf m}(\epsilon;\lambda)$ if and only if $k=0$. \end{thm}
\begin{proof} Fix generic complex numbers $\lambda_{j,\nu}\in\mathbb C$ satisfying
$\ell(\lambda)=|\{\lambda_{\mathbf m}\}|=0$. Then we may assume $\lambda_{j,\nu}-\lambda_{j,\nu'}\notin\mathbb Z$ for $1\le\nu<\nu'\le n_j$ and $j=0,\dots,p$.
i) The shift operator $R:=R_{\mathbf m}(-\epsilon,\lambda+\epsilon)$ gives a non-zero $W(x)$-homomorphism of the equation $P_{\mathbf m}(\lambda+\epsilon)v=0$ to $P_{\mathbf m}(\lambda)u=0$ by the correspondence $v=Ru$. Since the equation $P_{\mathbf m}(\lambda)u=0$ is reducible, we examine the decompositions of $\mathbf m$ described in Proposition~\ref{prop:subrep}. Note that the genericity of $\lambda_{j,\nu}\in\mathbb C$ assures that the subtuple $\mathbf m'$ of $\mathbf m$ corresponding to a decomposition $P_{\mathbf m}(\lambda)=P''P'$ is uniquely determined, namely, $\mathbf m'$ corresponds to the spectral type of the monodromy of the equation $P'u=0$.
If the shift operator $R$ is bijective, there exists a subtuple $\mathbf m'$ of $\mathbf m$ compatible to $\ell(\lambda)$ and $\ell(\lambda+\epsilon)$ because $R$ indices an isomorphism of monodromy.
Suppose $\ell(\lambda)$ is a factor of $c_{\mathbf m}(\epsilon;\lambda)$. Then $R$ is not bijective. We assume that the image of $R$ is the equation $P''\bar u=0$ and the kernel of $R$ is the equation $P'_\epsilon \bar v=0$. Then $P_{\mathbf m}(\lambda)=P''P'$ and $P_{\mathbf m}(\lambda+\epsilon)=P'_\epsilon P''_\epsilon$ with suitable Fuchsian differential operators $P'$ and $P''_\epsilon$. Note that the spectral type of the monodromy of $P'u=0$ and $P''_\epsilon v=0$ corresponds to $\mathbf m'$ and $\mathbf m''$ with $\mathbf m=\mathbf m'+\mathbf m''$. Applying Proposition~\ref{prop:subrep} to the decompositions $P_{\mathbf m}(\lambda)=P''P'$ and $P_{\mathbf m}(\lambda+\epsilon) =P'_\epsilon P''_\epsilon$, we have a dual decomposition \eqref{eq:subrep} of $\mathbf m$ with respect to the pair $\ell(\lambda)$ and $\ell(\lambda+\epsilon)$.
ii) Since $P_{\mathbf m}(\lambda)u=0$ is reducible, we have a decomposition $P_{\mathbf m}(\lambda)=P''P'$ with $0<\ord P'<\ord P_{\mathbf m}(\lambda)$. We may assume $P'u=0$ and let $\tilde{\mathbf m}'$ be the spectral type of the monodromy of the equation $P'u=0$. Then $\tilde{\mathbf m}'=\ell_1\mathbf m'+\ell_2\mathbf m''$ with integers $\ell_1$ and $\ell_2$ because
$|\{\lambda_{\tilde{\mathbf m}'}\}|\in\mathbb Z_{\le 0}$. Since $P'u=0$ is irreducible, $2\ge \idx\tilde{\mathbf m}'=2(\ell_1^2-\ell_1\ell_2+\ell_2^2)$ and therefore $(\ell_1,\ell_2)=(1,0)$ or $(0,1)$. Hence the claim follows from i) and the identity
$|\{\lambda_{\mathbf m'}\}|+|\{\lambda_{\mathbf m''}\}|=1$ \end{proof} \begin{rem} i) \ The reducibility of $P_{\mathbf m}(\lambda)$ implies that of the dual of $P_{\mathbf m}(\lambda)$.
ii) When $\mathbf m$ is simply reducible (cf.~Definition \ref{def:fund}), each linear form of $\lambda_{j,\nu}$ describing the reducibility uniquely corresponds to a rigid decomposition of $\mathbf m$ and therefore Theorem~\ref{thm:isom} gives the necessary and sufficient condition for the bijectivity of the shift operator $R_{\mathbf m}(\epsilon,\lambda)$. \end{rem}
\begin{exmp}[$EO_4$] \index{even family!$EO_4$} \index{tuple of partitions!rigid!1111,211,22} Let $P(\lambda)u=0$ and $P(\lambda')v=0$ be the Fuchsian differential equation with the Riemann schemes \[ \begin{Bmatrix}
\lambda_{0,1} & [\lambda_{1,1}]_{(2)} & [\lambda_{2,1}]_{(2)}\\
\lambda_{0,2} & \lambda_{1,2} & [\lambda_{2,2}]_{(2)}\\
\lambda_{0,3} & \lambda_{1,3}\\
\lambda_{0,4}
\end{Bmatrix} \text{ \ and \ } \begin{Bmatrix}
\lambda_{0,1} & [\lambda_{1,1}]_{(2)} & [\lambda_{2,1}]_{(2)}\\
\lambda_{0,2} & \lambda_{1,2} & [\lambda_{2,2}]_{(2)}\\
\lambda_{0,3} & \lambda_{1,3}+1\\
\lambda_{0,4}-1
\end{Bmatrix}, \] respectively. Since the condition of the reducibility of the equation corresponds to rigid decompositions \eqref{eq:e4ddec}, it easily follows from Theorem~\ref{thm:isom} that the shift operator between $P(\lambda)u=0$ and $P(\lambda')v=0$ is bijective if and only if \begin{equation*}
\begin{cases}
\lambda_{0,4}+\lambda_{1,2}+\lambda_{2,\mu}-1\ne 0
&(1\le \mu\le2),\\
\lambda_{0,\nu}+\lambda_{0,\nu'}
+\lambda_{1,1}+\lambda_{1,3}+\lambda_{2,1}+\lambda_{2,2}-1\ne 0
&(1\le \nu<\nu'\le 3).
\end{cases} \end{equation*}
In general, for a shift $\epsilon=(\epsilon_{j,\nu})$ compatible to the spectral type $1111,211,22$, the shift operator between $P(\lambda)u=0$ and $P(\lambda+\epsilon)v=0$ is bijective if and only if the values of each function in the list \begin{align}
&\lambda_{0,\nu}+\lambda_{1,1}+\lambda_{2,\mu}
&&(1\le\nu\le 4,\ 1\le\mu\le 2),\\
&\lambda_{0,\nu}+\lambda_{0,\nu'}+\lambda_{1,1}+\lambda_{1,3}+
\lambda_{2,1}+\lambda_{2,2}-1
&&(1\le\nu<\nu'\le4) \end{align} are \begin{equation}\label{eq:non-pos}
\begin{cases}
\text{\hspace{12.5pt}not integers for $\lambda$ and $\lambda+\epsilon$}\\
\text{or positive integers for $\lambda$ and $\lambda+\epsilon$}\\
\text{or non-positive integers for $\lambda$ and $\lambda+\epsilon$}.
\end{cases} \end{equation} Note that the shift operator gives a homomorphism between monodromies (cf.~\eqref{eq:isoWM}). \end{exmp} The following conjecture gives $c_{\mathbf m}(\epsilon;\lambda)$ under certain conditions. \begin{conj}\label{conj:shift} Retain the assumption that $\mathbf m=\bigl(\lambda_{j,\nu}\bigr)_{\substack{0\le j\le p\\1\le\nu\le n_j}} \in\mathcal P^{(n)}_{p+1}$ is rigid.
{\rm i) } If $\ell(\lambda)=\ell(\lambda+\epsilon)$ in Theorem~\ref{thm:isom}, then $\ell(\lambda)$ is not a factor of $c_{\mathbf m}(\epsilon;\lambda)$,
{\rm ii) } Assume $m_{1,n_1}=m_{2,n_2}=1$ and \begin{equation}
\epsilon:=\bigl(\epsilon_{j,\nu}\bigr)_{\substack{0\le j\le p\\1\le\nu\le n_j}},
\quad \epsilon_{j,\nu}=\delta_{j,1}\delta_{\nu,n_1}
-\delta_{j,2}\delta_{\nu,n_2}, \end{equation} Then we have \begin{equation}\label{eq:cef}
c_{\mathbf m}(\epsilon;\lambda)=C\prod_{\substack
{\mathbf m=\mathbf m'\oplus\mathbf m''\\m'_{1,n_1}=m''_{2,n_2}=1}}
|\{\lambda_{\mathbf m'}\}| \end{equation} with $C\in\mathbb C^\times$. \end{conj} Suppose the spectral type $\mathbf m$ is of \textsl{Okubo type}, namely, \index{Okubo type} \begin{equation}\label{eq:OkuboT}
m_{1,1}+\cdots+m_{p,1}=(p-1)\ord\mathbf m. \end{equation} Then some shift operators are easily obtained as follows. By a suitable addition we may assume that the Riemann scheme is \begin{equation}\label{eq:GRSshift}
\begin{Bmatrix}
x = \infty & x=c_1 & \cdots & x=c_p\\
[\lambda_{0,1}]_{(m_0,1)} & [0]_{(m_{1,1})} & \cdots & [0]_{(m_{p,1})}\\
[\lambda_{0,2}]_{(m_{0,2})} & [\lambda_{1,2}]_{(m_{1,2})} & \cdots
& [\lambda_{p,2}]_{(m_{p,2})}\\
\vdots & \vdots & \vdots & \vdots\\
[\lambda_{0,n_0}]_{(m_{0,n_0})} & [\lambda_{1,n_1}]_{(m_{1,n_1})} & \cdots
& [\lambda_{p,n_p}]_{(m_{p,n_p})}
\end{Bmatrix} \end{equation} and the corresponding differential equation $Pu=0$ is of the form \[
P_{\mathbf m}(\lambda)=\prod_{j=1}^p(x-c_j)^{n-m_{j,1}}\frac{d^n}{dx^n}
+ \sum_{k=0}^{n-1}
\prod_{j=1}^p
(x-c_j)^{\max\{k-m_{j,1},0\}}a_{k}(x)\frac{d^k}{dx^k}. \] Here $a_k(x)$ is a polynomial of $x$ whose degree is not larger than $k - \sum_{j=1}^n\max\{k-m_{j,1},0\}$. Moreover we have \begin{equation}
a_0(x)=\prod_{\nu=1}^{n_0}\prod_{i=0}^{m_{0,\nu}-1}(\lambda_{0,\nu}+i). \end{equation} Define the differential operators $R_1$ and $R_{\mathbf m}(\lambda)\in W[x]\otimes\mathbb C[\lambda]$ by \begin{equation}\label{eq:SftOku} R_1 = \tfrac{d}{dx} \text{ \ and \ }P_{\mathbf m}(\lambda) = -R_{\mathbf m}(\lambda)R_1+a_0(x). \end{equation} Let $P_{\mathbf m}(\lambda')v=0$ be the differential equation with the Riemann scheme \begin{equation}
\begin{Bmatrix}
x = \infty & x=c_1 & \cdots & x=c_p\\
[\lambda_{0,1}+1]_{(m_0,1)} & [0]_{(m_{1,1})} & \cdots & [0]_{(m_{p,1})}\\
[\lambda_{0,2}+1]_{(m_{0,2})} & [\lambda_{1,2}-1]_{(m_{1,2})} & \cdots
& [\lambda_{p,2}-1]_{(m_{p,2})}\\
\vdots & \vdots & \vdots & \vdots\\
[\lambda_{0,n_0}+1]_{(m_{0,n_0})}
& [\lambda_{1,n_1}-1]_{(m_{1,n_1})} & \cdots
& [\lambda_{p,n_p}-1]_{(m_{p,n_p})}
\end{Bmatrix}. \end{equation} Then the correspondences $u=R_{\mathbf m}(\lambda)v$ and $v=R_1u$ give $W(x)$-homomorphisms between the differential equations.
\begin{prop}\label{prop:shift} Let\/ $\mathbf m=\{m_{j,\nu}\}_{\substack{0\le j\le p\\1\le \nu\le n_j}}$ be a rigid tuple of partitions satisfying \eqref{eq:OkuboT}. Putting \begin{equation}\label{eq:shifte}
\epsilon_{j,\nu} =
\begin{cases}
1 & (j=0,\ 1\le \nu\le n_0),\\
\delta_{\nu,0}-1 & (1\le j\le p,\ 1\le \nu\le n_j),
\end{cases} \end{equation} we have \begin{equation} c_{\mathbf m}(\epsilon;\lambda) =
\prod_{\nu=1}^{n_0}\prod_{i=0}^{m_{0,\nu}-1}
(\lambda_{0,\nu}+\lambda_{1,1}+\cdots+\lambda_{p,1}+i). \end{equation} \end{prop} \begin{proof} By suitable additions the proposition follows from the result assuming $\lambda_{j,1}=0$ for $j=1,\dots,p$, which has been shown. \end{proof} \begin{exmp} \index{hypergeometric equation/function!generalized!shift operator} The generalized hypergeometric equations with the Riemann schemes \begin{align}
\begin{Bmatrix}
\lambda_{0,1} & \lambda_{1,1} & [\lambda_{2,1}]_{(n-1)}\\
\vdots & \vdots\\
\lambda_{0,\nu} & \lambda_{1,\nu_o} \\
\vdots & \vdots \\
\lambda_{0,n} & \lambda_{1,n} & \lambda_{2,2}
\end{Bmatrix}
\text{ \ and \ }
\begin{Bmatrix}
\lambda_{0,1} & \lambda_{1,1} & [\lambda_{2,1}]_{(n-1)}\\
\vdots & \vdots\\
\lambda_{0,\nu} & \lambda_{1,\nu_o} +1 \\
\vdots & \vdots \\
\lambda_{0,n} & \lambda_{1,n} & \lambda_{2,2} -1
\end{Bmatrix}, \end{align} respectively, whose spectral type is $\mathbf m=1^n,1^n,(n-1)1$ are isomorphic to each other by the shift operator if and only if \begin{equation}
\lambda_{0,\nu}+\lambda_{1,\nu_o}+\lambda_{2,1}\ne 0\quad(\nu=1,\dots,n). \end{equation} This statement follows from Proposition~\ref{prop:shift} with suitable additions.
Theorem~\ref{thm:isom} shows that in general $P(\lambda)u=0$ with the Riemann scheme $\{\lambda_{\mathbf m}\}$ is $W(x)$-isomorphic to $P(\lambda+\epsilon)v=0$ by the shift operator if and only if the values of the function $\lambda_{0,\nu}+\lambda_{1,\mu}+\lambda_{2,1}$ satisfy \eqref{eq:non-pos} for $1\le \nu\le n$ and $1\le\mu\le n$. Here $\epsilon$ is any shift compatible to $\mathbf m$.
The shift operator between \begin{align}
\begin{Bmatrix}
\lambda_{0,1} & \lambda_{1,1} & [\lambda_{2,1}]_{(n-1)}\\
\lambda_{0,2} & \lambda_{1,2} & \lambda_{2,2}\\
\vdots & \vdots \\
\lambda_{0,n} & \lambda_{1,n}
\end{Bmatrix}
\text{ \ and \ }
\begin{Bmatrix}
\lambda_{0,1} & \lambda_{1,1}+1 & [\lambda_{2,1}]_{(n-1)}\\
\lambda_{0,2} & \lambda_{1,2}-1 & \lambda_{2,2}\\
\vdots & \vdots \\
\lambda_{0,n} & \lambda_{1,n}
\end{Bmatrix} \end{align} is bijective if and only if \[
\lambda_{0,\nu}+\lambda_{1,1}+\lambda_{2,1}\ne 0
\text{\ \ and\ \ }
\lambda_{0,\nu}+\lambda_{1,2}+\lambda_{2,1}\ne 1
\text{ \ for \ }
\nu=1,\dots,n. \] Hence if $\lambda_{1,1}=0$ and $\lambda_{1,2}=1$ and $\lambda_{0,1}+\lambda_{2,1}=0$, the shift operator defines a non-zero endomorphism which is not bijective and therefore the monodromy of the space of the solutions are decomposed into a direct sum of the spaces of solutions of two Fuchsian differential equations. The other parameters are generic in this case, the decomposition is unique and the dimension of the smaller space equals 1. When $n=2$ and $(c_0,c_1,c_2)=(\infty,1,0)$ and $\lambda_{2,1}$ and $\lambda_{2,2}$ are generic, the space equals $\mathbb Cx^{\lambda_{2,1}}\oplus\mathbb C x^{\lambda_{2,2}}$ \end{exmp}
\subsection{Polynomial solutions}\label{sec:polyn} We characterize some polynomial solutions of a differential equation of Okubo type. \index{Okubo type!polynomial solution} \index{polynomial solution} \begin{prop}\label{prop:polynomial} Retain the notation in $\S\ref{sec:shift1}$. Let $P_{\mathbf m}(\lambda)u=0$ be the differential equation with the Riemann scheme \eqref{eq:GRSshift}. Suppose that\/ $\mathbf m$ is rigid and satisfies \eqref{eq:OkuboT}. Suppose moreover that there exists $j_o$ satisfying $m_{j_o,1}=1$ and $0\le j_o\le p$. Fix a complex number $C$. Suppose $\lambda_{0,1}=-C$ and $\lambda_{j,\nu}\notin\mathbb Z$ for $j=0,\dots,p$ and $\nu=2,\dots,n_j$. Then the equation has a polynomial solution of degree $k$ if and only if\/ $C=k$.
We denote the polynomial solution by $p_\lambda$. Then $p'_\lambda$ is a polynomial solution of $P_{\mathbf m}(\lambda+\epsilon)v=0$ under the notation \eqref{eq:shifte}. Moreover \begin{equation}\label{eq:getpol}
R_{\mathbf m}(\lambda)\circ R_{\mathbf m}(\lambda+\epsilon)\circ
\cdots\circ R_{\mathbf m}(\lambda+(k-1)\epsilon)1 \end{equation} is a non-zero constant multiple of $p_\lambda$ under the notation \eqref{eq:SftOku}. \end{prop} \begin{proof} Since $\mathbf m=\bigl(\delta_{1,\nu}\bigr)
_{\substack{0\le j\le p\\1\le\nu\le n_j}}
\oplus \bigl(m_{j,\nu}-\delta_{1,\nu}\bigr)
_{\substack{0\le j\le p\\1\le\nu\le n_j}}$ is a rigid decomposition of $\mathbf m$, we have $P_{\mathbf m}(\lambda) = P_1\partial$ with suitable $P_1\in W(x)$ when $C=0$. Note that $R_{\mathbf m}(\lambda+\ell\epsilon)$ defines an isomorphism of the equation $P_{\mathbf m}(\lambda+(\ell+1)\epsilon)u_{k+1}=0$ to the equation $P_{\mathbf m}(\lambda+\ell\epsilon)u_k=0$ by $u_k=R_{\mathbf m}(\lambda+\ell\epsilon)u_{k+1}$ if $C\ne\ell$, the function \eqref{eq:getpol} is a polynomial solution of $P_{\mathbf m}(\lambda)u=0$. The remaining part of the proposition is clear. \end{proof} \begin{rem} We have not used the assumption that $\mathbf m$ is rigid in Proposition~\ref{prop:shift} and Proposition~\ref{prop:polynomial} and hence the propositions are valid without this assumption. \end{rem}
\section{Connection problem}\label{sec:C} \subsection{Connection formula}\label{sec:C1} For a realizable tuple $\mathbf m\in\mathcal P_{p+1}$, let $P_{\mathbf m}u=0$ be a universal Fuchsian differential equation with the Riemann scheme \begin{equation}\label{eq:GRSC}
\begin{Bmatrix}
x =0 & {c_1}=1 & \cdots &{c_j} &\cdots& {c_p}=\infty\\
[\lambda_{0,1}]_{(m_{0,1})} & [\lambda_{1,1}]_{(m_{1,1})}&\cdots
&[\lambda_{j,1}]_{(m_{j,1})}&\cdots&[\lambda_{p,1}]_{(m_{p,1})}\\
\vdots & \vdots & \vdots & \vdots &\vdots &\vdots\\
[\lambda_{0,n_0}]_{(m_{0,n_0})} & [\lambda_{1,n_1}]_{(m_{1,n_1})}&\cdots
&[\lambda_{j,n_j}]_{(m_{j,n_j})}&\cdots&[\lambda_{p,n_p}]_{(m_{p,n_p})}
\end{Bmatrix}. \end{equation} The singular points of the equation are ${c_j}$ for $j=0,\dots,p$. In this subsection we always assume ${c_0}=0$, ${c_1}=1$ and ${c_p}=\infty$ and $c_j\notin[0,1]$ for $j=2,\dots,p-1$. We also assume that $\lambda_{j,\nu}$ are generic. \index{connection coefficient} \index{00c@$c(\lambda_{j,\nu}\rightsquigarrow\lambda_{j',\nu'})$}
\begin{defn}[connection coefficients]\label{def:cc} Suppose $\lambda_{j,\nu}$ are generic under the Fuchs relation. Let $u_0^{\lambda_{0,\nu_0}}$ and $u_1^{\lambda_{1,\nu_1}}$ be normalized local solutions of $P_\mathbf m=0$ at $x=0$ and $x=1$ corresponding to the exponents $\lambda_{0,\nu_0}$ and $\lambda_{1,\nu_1}$, respectively, so that $u_0^{\lambda_{0,\nu_0}}\equiv x^{\lambda_{0,\nu_0}}\mod x^{\lambda_{0,\nu_0}+1}\mathcal O_0$ and $u_1^{\lambda_{1,\nu_1}}\equiv (1-x)^{\lambda_{1,\nu_1}}\mod (1-x)^{\lambda_{1,\nu_1}+1}\mathcal O_1$. Here $1\le\nu_0\le n_0$ and $1\le\nu_1\le n_1$. If $m_{0,\nu_0}=1$, $u_0^{\lambda_{0,\nu_0}}$ is uniquely determined and then the analytic continuation of $u_0^{\lambda_{0,\nu_0}}$ to $x=1$ along $(0,1)\subset\mathbb R$ defines a \textsl{connection coefficient} with respect to $u_1^{\lambda_{1,\nu_1}}$, which is denoted by $c(0\!:\!\lambda_{0,\nu_0}\!\rightsquigarrow\!1\!:\!\lambda_{1,\nu_1})$ or simply by $c(\lambda_{0,\nu_0}\!\rightsquigarrow\!\lambda_{1,\nu_1})$. The connection coefficient $c(1\!:\!\lambda_{1,\nu_1}\!\rightsquigarrow\!0\!:\!\lambda_{0.\nu_0})$ or $c(\lambda_{1,\nu_1}\!\rightsquigarrow\!\lambda_{0.\nu_0})$ of $u_1^{\lambda_{1,\nu_1}}$ with respect to $u_0^{\lambda_{0,\nu_0}}$ are similarly defined if $m_{1,\nu_1}=1$. \index{connection coefficient}
Moreover we define $c({c_i}:\lambda_{i,\nu_i}\!\rightsquigarrow\! {c_j}:\lambda_{j,\nu_j})$ by using a suitable linear fractional transformation $T$ of $\mathbb C\cup\{\infty\}$ which transforms $\{{c_i},{c_j}\}$ to $\{0,1\}$ so that $T({c_\nu})\notin(0,1)$ for $\nu=0,\dots,p$. If $p=2$, we define the map $T$ so that $T({c_k})=\infty$ for the other singular point ${c_k}$. For example if $c_j\notin[0,1]$ for $j=2,\dots,p-1$, we put $T(x)=\frac{x}{x-1}$ to define $c(0:\lambda_{0,\nu_0}\!\rightsquigarrow\!\infty:\lambda_{p,\nu_p})$ or $c(\infty:\lambda_{p,\nu_p}\!\rightsquigarrow\!0:\lambda_{0,\nu_0})$. \end{defn}
In the definition $u_0^{\lambda_{0,\nu_0}}(x)=x^{\lambda_{0,\nu_0}} \phi(x)$ with analytic function $\phi(x)$ at $0$ which satisfies $\phi(0)=1$ and if $\RE\lambda_{1,\nu_1}<\RE\lambda_{1,\nu}$ for $\nu\ne \nu_1$, we have \begin{equation}
c(\lambda_{0,\nu_0}\!\rightsquigarrow\!\lambda_{1,\nu_1})
= \lim_{x\to 1-0}(1-x)^{-\lambda_{1,\nu_1}}u_0^{\lambda_{0,\nu_0}}(x)
\qquad(x\in[0,1)) \end{equation} by the analytic continuation. The connection coefficient $c(\lambda_{0.\nu_0}\!\rightsquigarrow\!\lambda_{1,\nu_1})$ meromorphically depends on spectral parameters $\lambda_{j,\nu}$. It also holomorphically depends on accessory parameters $g_i$ and singular points $\frac1{c_j}$ $(j=2,\dots,p-1)$ in a neighborhood of given values of parameters.
The main purpose in this subsection is to get the explicit expression of the connection coefficients in terms of gamma functions when $\mathbf m$ is rigid and $m_{0,\nu}=m_{1,\nu'}=1$.
Fist we prove the following key lemma which describes the effect of a middle convolution on connection coefficients. \begin{lem}\label{lem:conn} Using the integral transformation \eqref{eq:fracdif}, we put \begin{align} (T_{a,b}^\mu u)(x)&:=x^{-a-\mu}(1-x)^{-b-\mu}I_0^\mu x^{a}(1-x)^{b}u(x),\\ (S_{a,b}^\mu u)(x)&:=x^{-a-\mu}I_0^\mu x^{a}(1-x)^bu(x) \end{align} for a continuous function $u(x)$ on $[0,1]$. Suppose\/ $\RE a \ge 0$ and\/ $\RE \mu > 0$. Under the condition\/ $\RE b+\RE\mu<0$ or\/ $\RE b+\RE\mu>0$, $(T_{a,b}^\mu u)(x)$ or $S_{a,b}^\mu (u)(x)$ defines a continuous function on\/ $[0,1]$, respectively, and we have \begin{align} T_{a,b}^\mu(u)(0)&
=S_{a,b}^\mu(u)(0)=\frac{\Gamma(a+1)}{\Gamma(a+\mu+1)}u(0), \allowdisplaybreaks\\ \frac{T_{a,b}^\mu(u)(1)}{T_{a,b}^\mu(u)(0)}
&=\frac{u(1)}{u(0)}C_{a,b}^\mu,\quad
C_{a,b}^\mu:=\frac{\Gamma(a+\mu+1)\Gamma(-\mu-b)} {\Gamma(a+1)\Gamma(-b)},\label{eq:CR1} \allowdisplaybreaks\\ \frac{S_{a,b}^\mu(u)(1)}{S_{a,b}^\mu(u)(0)} &=\frac1{u(0)}\frac
{\Gamma(a+\mu+1)}{\Gamma(\mu)\Gamma(a+1)} \int_0^1t^a(1-t)^{b+\mu-1}u(t)dt. \end{align} \end{lem} \begin{proof} Suppose $\RE a\ge 0$ and $0<\RE\mu<-\RE b$. Then \begin{align*} \Gamma(\mu)&T_{a,b}^\mu(u)(x)\\ &={x^{-a-\mu}(1-x)^{-b-\mu}}
\int_0^x t^a(1-t)^b (x-t)^{\mu-1}u(t)dt&\hspace{-1.2cm}(t=xs_1,\ 0\le x<1)
\allowdisplaybreaks\\ &={(1-x)^{-b-\mu}}
\int_0^1 s_1^{a}(1-s_1)^{\mu-1}(1-xs_1)^{b}u(xs_1)ds_1
\allowdisplaybreaks\\ &=\int_0^1 s_1^{a}\Bigl(\frac{1-s_1}{1-x}\Bigr)^\mu
\Bigl(\frac{1-xs_1}{1-x}\Bigr)^{b}u(xs_1)\frac{ds}{1-s_1}
\allowdisplaybreaks\\ &=\int_0^1 (1-s_2)^{a}\Bigl(\frac{s_2}{1-x}\Bigr)^\mu
\Bigl(1+\frac {xs_2}{1-x}\Bigr)^b
u(x-xs_2)\frac{ds_2}{s_2}&(s_1=1-s_2)\allowdisplaybreaks\\ &=\int_0^\frac1{1-x} \bigl(1-s(1-x)\bigr)^{a}s^\mu
(1+xs)^{b}u\bigl(x-x(1-x)s\bigr)\frac{ds}{s}
&(s_2=(1-x)s). \end{align*}
Since \[
\left|s_1^{a}(1-s_1)^{\mu-1}(1-xs_1)^{b}u(xs_1)\right|
\le \max\{(1-s_1)^{\RE\mu-1},1\}3^{-\RE b}\max_{0\le t\le 1}|u(t)| \] for $0\le s_1< 1$ and $0\le x\le \frac23$, $T^\mu_{a,b}(u)(x)$ is continuous for $x\in[0,\tfrac23)$. We have \begin{align*}
\left|\bigl(1-s(1-x)\bigr)^{a}s^{\mu-1}
(1+xs)^{b}u\bigl(x-x(1-x)s)\bigr)\right|
\le s^{\RE\mu-1}(1+\tfrac s2)^{\RE b}\max_{0\le t\le 1}|u(t)| \end{align*} for $\frac12\le x\le 1$ and $0<s\le \frac1{1-x}$ and therefore $T_{a,b}^\mu(u)(x)$ is continuous for $x\in(\frac12,1]$. Hence $T_{a,b}^\mu(x)$ defines a continuous function on $[0,1]$ and \begin{align*} T_{a,b}^\mu(u)(0)&
=\frac{1}{\Gamma(\mu)}\int_0^1(1-s_2)^{a}s_2^\mu u(0)\frac{ds_2}{s_2}
=\frac{\Gamma(a+1)}{\Gamma(a+\mu+1)}u(0),\\ T_{a,b}^\mu(u)(1)&
=\frac{1}{\Gamma(\mu)}\int_0^\infty s^\mu (1+s)^{b}u(1)\frac{ds}s \intertext{$(t=\frac s{1+s}=1-\frac1{1+s},\ \frac1{1+s}=1-t,\ 1+s=\frac1{1-t},\ s=\frac1{1-t}-1=\frac{t}{1-t},\ \frac{ds}{dt}=-\frac1{(1-t)^2})$}
&=\frac{1}{\Gamma(\mu)}\int_0^1\Bigl(\frac{t}{1-t}\Bigr)^{\mu-1}(1-t)^{-b-2}u(1)dt
=\frac{\Gamma(-\mu-b)}{\Gamma(-b)}u(1). \end{align*} The claims for $S_{a,b}^\mu$ are clear from \[
\Gamma(\mu)S^\mu_{a,b}(u)(x)=
\int_0^1 s_1^a(1-s_1)^{\mu-1}(1-xs_1)^b u(xs_1)ds_1. \]\\[-1cm] \end{proof}
This lemma is useful for the middle convolution $mc_\mu$ not only when it gives a reduction but also when it doesn't change the spectral type.
\begin{exmp}\index{Jordan-Pochhammer} Applying Lemma~\ref{lem:conn} to the solution \[
u_0^{\lambda_0+\mu}(x)=\int_0^x t^{\lambda_0}(1-t)^{\lambda_1}
\biggl(\prod_{j=2}^{p-1}
\Bigl(1-\frac t{c_j}\Bigr)^{\lambda_j}\biggr)(x-t)^{\mu-1}dt \] of the Jordan-Pochhammer equation (cf.\ Example~\ref{ex:midconv} iii)) with the Riemann scheme \[
\begin{Bmatrix}
x =0 & {c_1}=1 & \cdots &{c_j} &\cdots& {c_p}=\infty\\
[0]_{(p-1)} & [0]_{(p-1)}&\cdots
&[0]_{(p-1)}&\cdots&[1-\mu]_{(p-1)}\\
\lambda_0+\mu&\lambda_1+\mu&\cdots&\lambda_j+\mu&\cdots&
-\sum_{\nu=0}^{p-1}\lambda_\nu-\mu
\end{Bmatrix}, \] we have \begin{align*} c(0\!:\!\lambda_0+\mu\!\rightsquigarrow\!1\!:\!\lambda_1+\mu)
&=\frac{\Gamma(\lambda_0+\mu+1)\Gamma(-\lambda_1-\mu)}
{\Gamma(\lambda_0+1)\Gamma(-\lambda_1)}
\prod_{j=2}^{p-1}\Bigl(1-\frac1{c_j}\Bigr)^{\lambda_j},
\allowdisplaybreaks\\ c(0\!:\!\lambda_0+\mu\!\rightsquigarrow\!1:\!0)&=
\frac{\Gamma(\lambda_0+\mu+1)}{\Gamma(\mu)\Gamma(\lambda_0+1)}\int_0^1
t^{\lambda_0}(1-t)^{\lambda_1+\mu-1}\prod_{j=1}^{p-1}
\Bigl(1-\frac t{c_j}\Bigr)^{\lambda_j}dt. \end{align*} Moreover the equation $Pu=0$ with \[
P:=\RAd(\partial^{-\mu'})\RAd(x^{\lambda'})
\RAd(\partial^{-\mu})\RAd(x^{\lambda_0}
(1-x)^{\lambda_1})\partial \] is satisfied by the generalized hypergeometric function ${}_3F_2$ with the Riemann scheme \[
\begin{Bmatrix}
x = 0 &1& \infty\\
0 &[0]_{(2)}&1-\mu'\\
\lambda'+\mu'& &1-\lambda'-\mu-\mu'\\
\lambda_0+\lambda'+\mu+\mu'&\lambda_1+\mu+\mu'&
-\lambda_0-\lambda_1-\lambda'-\mu-\mu'
\end{Bmatrix} \] corresponding to $111,21,111$ and therefore \begin{align*}
&c(\lambda_0+\lambda'+\mu+\mu'\!\rightsquigarrow\!\lambda_1+\mu+\mu')=
C_{\lambda_0,\lambda_1}^\mu\cdot
C_{\lambda_0+\lambda'+\mu,\lambda_1+\mu}^{\mu'}\\
&\quad=
\frac{\Gamma(\lambda_0+\mu+1)\Gamma(-\lambda_1-\mu)}
{\Gamma(\lambda_0+1)\Gamma(-\lambda_1)}\cdot
\frac{\Gamma(\lambda_0+\lambda'+\mu+\mu'+1)\Gamma(-\lambda_1-\mu-\mu')}
{\Gamma(\lambda_0+\lambda'+\mu+1)\Gamma(-\lambda_1-\mu)}\\
&\quad=
\frac{\Gamma(\lambda_0+\mu+1)
\Gamma(\lambda_0+\lambda'+\mu+\mu'+1)\Gamma(-\lambda_1-\mu-\mu')}
{\Gamma(\lambda_0+1)\Gamma(-\lambda_1)\Gamma(\lambda_0+\lambda'+\mu+1)}. \end{align*} \end{exmp}
We further examine the connection coefficient.
In general, putting $c_0=0$ and $c_1=1$ and $\lambda_1=\sum_{k=0}^p\lambda_{k,1}-1$, we have { \begin{align*} &\begin{Bmatrix} x={c_j}\quad(j=0,\dots,p-1)&\infty \\ [\lambda_{j,\nu}-(\delta_{j,0}+\delta_{j,1})
\lambda_{j,n_j}]_{(m_{j,\nu})}
&[\lambda_{p,\nu}+\lambda_{0,n_0}+\lambda_{1,n_1}]_{(m_{0,\nu})} \end{Bmatrix}\allowdisplaybreaks\\ &\xrightarrow{x^{\lambda_{0,n_0}}(1-x)^{\lambda_{1,n_1}}}{} \begin{Bmatrix} x= {c_j} &\infty\\
[\lambda_{j,\nu}]_{(m_{j,\nu})}
&[\lambda_{p,\nu}]_{(m_{p,\nu})} \end{Bmatrix}\allowdisplaybreaks\\ \\ &\xrightarrow{x^{-\lambda_{0,1}}\prod_{j=1}^{p-1}
(1-c_j^{-1}x)^{-\lambda_{j,1}}}{} \begin{Bmatrix}
[0]_{(m_{j,1})}
& [\lambda_{p,1}+\sum_{k=0}^{p-1} \lambda_{k,1}]_{(m_{p,1})} \\
[\lambda_{j,\nu}-\lambda_{j,1}]_{(m_{j,\nu})}
&[\lambda_{p,\nu}+\sum_{k=0}^{p-1} \lambda_{k,1}]_{(m_{p,\nu})} \end{Bmatrix}\allowdisplaybreaks\\ &\xrightarrow{\partial^{1-\sum_{k=0}^p \lambda_{k,1}}}{} \begin{Bmatrix}
[0]_{(m_{j,1}-d)}
&[\lambda_{p,1}+\sum_{k=0}^{p-1} \lambda_{k,1}-2\lambda_1]_{(m_{p,1}-d)} \\
[\lambda_{j,\nu}-\lambda_{j,1}+\lambda_1]_{(m_{j,\nu})}
& [\lambda_{p,\nu}+\sum_{k=0}^{p-1} \lambda_{k,1}-\lambda_1]_{(m_{p,\nu})} \end{Bmatrix}\\ &\qquad(d=\sum_{k=0}^pm_{k,1}-(p-1)n) \allowdisplaybreaks\\ &\xrightarrow{x^{\lambda_{0,1}}
\prod_{j=1}^{p-1} (1-c_j^{-1}x)^{\lambda_{j,1}}}{} \begin{Bmatrix} x=\frac1{c_j} & \infty\\
[\lambda_{j,1}]_{(m_{j,1}-d)}
&[\lambda_{p,1}-2\lambda_1]_{(m_{p,1}-d)} \\
[\lambda_{j,\nu}+\lambda_1]_{(m_{j,\nu})}
&[\lambda_{p,\nu}-\lambda_1]_{(m_{p,\nu})} \end{Bmatrix},\allowdisplaybreaks\\ &C_{\lambda_{0,n_1}-\lambda_{0,1},\lambda_{1,n_1}-\lambda_{1,1}}
^{\lambda_1}= \frac{\Gamma(\lambda_{0,n_0}+\lambda_1-\lambda_{0,1}+1)
\Gamma(\lambda_{1,1}-\lambda_{1,n_1}-\lambda_1)}
{\Gamma(\lambda_{0,n_0}-\lambda_{0,1}+1)
\Gamma(\lambda_{1,1}-\lambda_{1,n_1})
}. \end{align*}}
In general, the following theorem is a direct consequence of Definition~\ref{def:pell} and Lemma~\ref{lem:conn}.
\begin{thm}\label{thm:GC} Put $c_0=\infty$, $c_1=1$ and $c_j\in\mathbb C\setminus\{0\}$ for $j=3,\dots,p-1$. By the transformation \[ \begin{split}
&\RAd\Bigl(x^{\lambda_{0,1}}\!
\prod_{j=1}^{p-1}\bigl(1-\frac x{c_j}\bigr)^{\lambda_{j,1}}\Bigr)
\circ
\RAd\Bigl(\partial^{1-\sum_{k=0}^p\lambda_{k,1}}\Bigr)\circ
\RAd\Bigl(x^{-\lambda_{0,1}}\!\prod_{j=1}^{p-1}
\bigl(1-\frac x{c_j}\bigr)^{-\lambda_{j,1}}\Bigr) \end{split} \] the Riemann scheme of a Fuchsian ordinary differential equation and its connection coefficient change as follows: { \begin{align*} &\left\{\lambda_{\bf m}\right\} =\left\{[\lambda_{j,\nu}]_{(m_{j,\nu})}\right\}
_{\substack{0\le j\le p\\1\le\nu\le n_j}} =\begin{Bmatrix} x={c_j}&\infty\\ [\lambda_{j,1}]_{(m_{j,1})} & [\lambda_{p,1}]_{(m_{p,1})}\\ [\lambda_{j,\nu}]_{(m_{j,\nu})} &
[\lambda_{p,\nu}]_{(m_{p,\nu})} \end{Bmatrix} \allowdisplaybreaks\\& \ \mapsto \{\lambda'_{\mathbf m'}\} = \left\{[\lambda'_{j,\nu}]_{(m'_{j,\nu})}\right\}
_{\substack{0\le j\le p\\1\le\nu\le n_j}} \\&\qquad =\begin{Bmatrix} x={c_j} & \infty\\
[\lambda_{j,1}]_{(m_{j,1}-d)}
&[\lambda_{p,1}-2\sum_{k=0}^p\lambda_{k,1}+2]_{(m_{p,1}-d)} \\
[\lambda_{j,\nu}+\sum_{k=0}^p\lambda_{k,1}-1]_{(m_{j,\nu})}
&[\lambda_{p,\nu}-\sum_{k=0}^p\lambda_{k,1}+1]_{(m_{p,\nu})} \end{Bmatrix} \intertext{with} &\qquad d=m_{0,1}+\cdots+m_{p,1}-(p-1)\ord\mathbf m,\\ &\qquad m_{j,\nu}'=m_{j,\nu}-d\delta_{\nu,1}\quad(j=0,\dots,p,\ \nu=1,\dots,n_j),\\ &\qquad \lambda'_{j,1}=\lambda_{j,1}\quad(j=0,\dots,p-1),\
\lambda'_{p,1}=-2\lambda_{0,1}-\cdots-2\lambda_{p-1,1}-\lambda_{p,1}+2,\\ &\qquad \lambda'_{j,\nu}=\lambda_{j,\nu}+\lambda_{0,1}+\lambda_{1,1}
+\cdots+\lambda_{p,1}-1\quad(j=0,\dots,p-1,\ \nu=2,\dots,n_j),\\ &\qquad \lambda'_{p,\nu}=\lambda_{p,\nu}-\lambda_{0,1}-\cdots-\lambda_{p,1}+1 \end{align*}} and if $m_{0,n_0}=1$ and $n_0>1$ and $n_1>1$, then \begin{equation}\label{eq:cid} \frac{c'(\lambda'_{0,n_0}\!\rightsquigarrow\!\lambda'_{1,n_1})}
{\Gamma(\lambda'_{0,n_0}-\lambda'_{0,1}+1)
\Gamma(\lambda'_{1,1}-\lambda'_{1,n_1})} = \frac{c(\lambda_{0,n_0}\!\rightsquigarrow\!\lambda_{1,n_1})}
{\Gamma(\lambda_{0,n_0}-\lambda_{0,1}+1)
\Gamma(\lambda_{1,1}-\lambda_{1,n_1})}. \end{equation} \end{thm}
Applying the successive reduction by $\partial_{max}$ to the above theorem, we obtain the following theorem. \begin{thm}\label{thm:conG} Suppose that a tuple\/ $\mathbf m\in\mathcal P$ is irreducibly realizable and $m_{0,n_0}=m_{1,n_1}=1$ in the Riemann scheme \eqref{eq:GRSC}. Then the connection coefficient satisfies \begin{align*}
&\frac{c(\lambda_{0,n_0}\!\rightsquigarrow\!\lambda_{1,n_1})}
{\bar c\bigl(\lambda(K)_{0,n_0}\!\rightsquigarrow\!\lambda(K)_{1,n_1}\bigr)}\\
&\quad= \prod_{k=0}^{K-1}
\frac{\Gamma\bigl(\lambda(k)_{0,n_0}-\lambda(k)_{0,\ell(k)_0}+1\bigr)
\cdot\Gamma\bigl(\lambda(k)_{1,\ell(k)_1}-\lambda(k)_{1,n_1}\bigr)}
{\Gamma\bigl(\lambda(k+1)_{0,n_0}-\lambda(k+1)_{0,\ell(k)_0}+1\bigr)
\cdot\Gamma\bigl(\lambda(k+1)_{1,\ell(k)_1}-\lambda(k+1)_{1,n_1}\bigr)} \end{align*} under the notation in Definitions~\ref{def:redGRS}. Here $\bar c\bigl(\lambda(K)_{0,n_0}\!\rightsquigarrow\!\lambda(K)_{1,n_1}\bigr)$ is a corresponding connection coefficient for the equation $(\partial_{max}^K P_{\mathbf m})v=0$ with the fundamental spectral type $f\mathbf m$. We note that \begin{equation}\label{eq:cdeqn} \begin{split}
&\bigl(\lambda(k+1)_{0,n_0}-\lambda(k+1)_{0,\ell(k)_0}+1\bigr)
+\bigl(\lambda(k+1)_{1,\ell(k)_1}-\lambda(k+1)_{1,n_1}\bigr)\\
&\quad=\bigl(\lambda(k)_{0,n_0}-\lambda(k)_{0,\ell(k)_0}+1\bigr)+
\bigl(\lambda(k)_{1,\ell(k)_1}-\lambda(k)_{1,n_1}\bigr) \end{split} \end{equation} for $k=0,\dots,K-1$. \end{thm}
When $\mathbf m$ is rigid in the theorem above, we note that $\bar c(\lambda_{0,n_0}(K)\!\rightsquigarrow\!\lambda_{1,n_1}(K))=1$ and we have the following more explicit result.
\begin{thm}\label{thm:c} Let\/ $\mathbf m\in\mathcal P$ be a rigid tuple. Assume $m_{0,n_0}=m_{1,n_1}=1$, $n_0>1$ and $n_1>1$ in the Riemann scheme \eqref{eq:GRSC}. Then \index{000lambda@$\arrowvert$\textbraceleft$\lambda_{\mathbf m}$\textbraceright$\arrowvert$} \begin{gather}\label{eq:connection} \begin{split}
c(\lambda_{0,n_0}\!\rightsquigarrow\!\lambda_{1,n_1})
=\frac
{\displaystyle\prod_{\nu=1}^{n_0-1}
\Gamma\bigl(\lambda_{0,n_0}-\lambda_{0,\nu}+1\bigr)
\cdot\prod_{\nu=1}^{n_1-1}
\Gamma\bigl(\lambda_{1,\nu}-\lambda_{1,n_1}\bigr)
}
{\displaystyle\prod_{\substack{\mathbf m'\oplus\mathbf m''=\mathbf m\\
m'_{0,n_0}=m''_{1,n_1}=1}}
\Gamma\bigl(|\{\lambda_{\mathbf m'}\}|\bigr)
\cdot\prod_{j=2}^{p-1}\Bigl(1-\frac1{c_j}\Bigr)^{-\lambda(K)_{j,\ell(K)_j}}
},\\ \end{split}\allowdisplaybreaks\\
\sum_{\substack{\mathbf m'\oplus\mathbf m''=\mathbf m\\
m'_{0,n_0}=m''_{1,n_1}=1}}
\!\!\!\!\!\! m'_{j,\nu}
= (n_1-1)m_{j,\nu}-\delta_{j,0}(1-n_0\delta_{\nu,_{n_0}})
+\delta_{j,1}(1-n_1\delta_{\nu,_{n_1}})\label{eq:concob}\\[-.5cm]
\hspace{4.5cm}(1\le\nu\le n_j,\ 0\le j\le p)\notag \end{gather} under the notation in Definitions~\ref{def:FRLM} and \ref{def:redGRS}. \end{thm}
\begin{proof} We may assume $\mathbf m$ is monotone and $\ord\mathbf m>1$.
We will prove this theorem by the induction on $\ord\mathbf m$. Suppose \begin{equation}\label{eq:DecC} \mathbf m=\mathbf m'\oplus\mathbf m''\text{ \ with \ } m'_{0,n_0}=m''_{1,n_1}=1. \end{equation}
If $\partial_{\mathbf 1}\mathbf m'$ is not well-defined, then \begin{equation}\label{eq:DecC0}
\ord\mathbf m'=1\text{ \ and \ }
m'_{j,1}=1\text{ \ for \ }j=1,2,\dots,p \end{equation} and $1+m_{1,1}+\cdots+m_{p,1}-(p-1)\ord\mathbf m=1$ because $\idx(\mathbf m,\mathbf m')=1$ and therefore \begin{equation} d_{\mathbf 1}(\mathbf m)=m_{0,1}. \end{equation} If $\partial_{\mathbf 1}\mathbf m''$ is not well-defined, \begin{equation}\label{eq:DecC1} \begin{split}
\ord\mathbf m''&=1\text{ \ and \ }
m''_{j,1}=1\text{ \ for \ }j=0,2,\dots,p,\\
d_{\mathbf 1}(\mathbf m)&=m_{1,1}. \end{split} \end{equation}
Hence if $d_{\mathbf 1}(\mathbf m)<m_{0,1}$ and $d_{\mathbf 1}(\mathbf m)<m_{1,1}$, $\partial_{\mathbf 1}\mathbf m'$ and $\partial_{\mathbf 1}\mathbf m''$ are always well-defined and $\partial_{\mathbf 1}\mathbf m=\partial_{\mathbf 1}\mathbf m'\oplus \partial_{\mathbf 1}\mathbf m''$ and the direct decompositions \eqref{eq:DecC} of $\mathbf m$ correspond to those of $\partial_{\mathbf 1}\mathbf m$ and therefore Theorem~\ref{thm:GC} shows \eqref{eq:connection} by the induction because we may assume $d_{\mathbf 1}(\mathbf m)>0$. In fact, it follows from \eqref{eq:midinv} that the gamma factors in the denominator of the fraction in the right hand side of \eqref{eq:connection} don't change by the reduction and the change of the numerator just corresponds to the formula in Theorem~\ref{thm:GC}.
If $d_{\mathbf 1}(\mathbf m)=m_{0,1}$, there exists the direct decomposition \eqref{eq:DecC} with \eqref{eq:DecC0} which doesn't correspond to a direct decomposition of $\partial_{\mathbf 1}\mathbf m$ but corresponds to the term
$\Gamma(|\{\lambda_{\mathbf m'}\}|) =\Gamma(\lambda_{0,n_1}+\lambda_{1,1}+\cdots+\lambda_{p,1}) =\Gamma(\lambda'_{0,n_1}-\lambda'_{0,1}+1)$ in \eqref{eq:cid}. Similarly if $d_{\mathbf 1}(\mathbf m)=m_{1,1}$, there exists the direct decomposition \eqref{eq:DecC} with \eqref{eq:DecC1} and it corresponds to the term
$\Gamma(|\{\lambda_{\mathbf m'}\}|) = \Gamma(1-|\{\lambda_{\mathbf m''}\}|) =\Gamma(1-\lambda_{0,1}-\lambda_{1,n_1}-\lambda_{2,1}-\cdots-\lambda_{p,1}) =\Gamma(\lambda'_{1,1}-\lambda'_{1,n_1})$ (cf.~\eqref{eq:sum1}). Thus Theorem~\ref{thm:GC} assures \eqref{eq:connection} by the induction on $\ord\mathbf m$.
Note that the above proof with \eqref{eq:cdeqn} shows \eqref{eq:csum}. Hence \begin{align*}
\sum_{\substack{\mathbf m'\oplus\mathbf m''=\mathbf m\\
m'_{0,n_0}=m''_{1,n_1}=1}}|\{\lambda_{\mathbf m}\}|
&=\sum_{\nu=1}^{n_0-1}(\lambda_{0,n_0}-\lambda_{0,\nu}+1) +\sum_{\nu=1}^{n_1-1}(\lambda_{1,\nu}-\lambda_{1,n_1})\\[-8pt] & =(n_0-1)+(n_0-1)\lambda_{0,n_0}
-\sum_{\nu=1}^{n_0-1}
\lambda_{0,\nu}+\sum_{\nu=1}^{n_1-1}\lambda_{1,\nu}\\ &\quad +(n_1-1)\Bigl(\sum_{j=0}^p\sum_{\nu=1}^{n_j-\delta_{j,1}}
m_{j,\nu}\lambda_{j,\nu}-n+1\Bigr)\allowdisplaybreaks\\ &=
(n_0+n_1-2)\lambda_{0,n_0}
+
\sum_{\nu=1}^{n_0-1}\bigl((n_1-1)m_{0,\nu}-1\bigr)\lambda_{0,\nu}\\
&\quad+\sum_{\nu=1}^{n_1-1}\bigl((n_1-1)m_{1,\nu}+1\bigr)\lambda_{1,\nu}
+\sum_{j=2}^p\sum_{\nu=1}^{n_2}(n_1-1)m_{j,\nu}\lambda_{j,\nu}\\
&\quad+(n_0+n_1-2)-(n_1-1)\ord\mathbf m. \end{align*} The left hand side of the above first equation and the right hand side of the above last equation don't contain the term $\lambda_{1,n_1}$ and therefore the coefficients of $\lambda_{j,\nu}$ in the both sides are equal, which implies \eqref{eq:concob}. \end{proof} \begin{cor}\label{cor:C} Retain the notation in\/ {\rm Theorem~\ref{thm:c}.} We have \begin{align}
\#\{\mathbf m'\,;\,\mathbf m'\oplus\mathbf m''&=\mathbf m\text{ \ with \ } m'_{0,n_0}=m''_{1,n_1}=1\} = n_0+n_1-2,\label{eq:numdec} \allowdisplaybreaks\\
\sum_{\substack{\mathbf m'\oplus\mathbf m''=\mathbf m\\
m'_{0,n_0}=m''_{1,n_1}=1}}
\!\!\!\!\!\! \ord\mathbf m'&=(n_1-1)\ord\mathbf m,\label{eq:ordsum} \allowdisplaybreaks\\
\sum_{\substack{\mathbf m'\oplus\mathbf m''=\mathbf m\\
m'_{0,n_0}=m''_{1,n_1}=1}}
|\{\lambda_m'\}|&= \sum_{\nu=1}^{n_0-1}(\lambda_{0,n_0} - \lambda_{0,\nu} + 1) +\sum_{\nu=1}^{n_1-1}(\lambda_{1,\nu}-\lambda_{1,n_1}). \label{eq:csum} \end{align} Let $c(\lambda_{0,n_0}+t\!\rightsquigarrow\!\lambda_{1,n_1}-t)$ be the connection coefficient for the Riemann scheme $\bigl\{[\lambda_{j,\nu}+t(\delta_{j,0}\delta_{\nu,n_0}
-\delta_{j,1}\delta_{\nu,n_1})]_{(m_{j,\nu})}\bigr\}$. Then \begin{equation}\label{eq:clim}
\lim_{t\to+\infty}
c(0\!:\!\lambda_{0,n_0}+t\rightsquigarrow1\!:\!\lambda_{1,n_1}-t)=
\prod_{j=2}^{p-1}\bigl(1-c_j\bigr)^{\lambda(K)_{j,\ell(K)_j}}. \end{equation} Under the notation in\/ {\rm Theorem~\ref{thm:irrKac}} \begin{equation}\label{eq:dncKac} \begin{split}
&\{\mathbf m'\,;\,\mathbf m'\oplus\mathbf m''=\mathbf m\text{ \ with \ }
m'_{0,n_0}=m''_{1,n_1}=1\}\\
&=\{\mathbf m'\in\mathcal P\,;\,m'_{0,n_0}=1,\ m'_{1,n_1}=0,\
\alpha_{\mathbf m'}\text{ or }\alpha_{\mathbf m-\mathbf m'}
\in\Delta(\mathbf m)\}. \end{split} \end{equation} \end{cor} \begin{proof} We have \eqref{eq:csum} in the proof of Theorem~\ref{thm:GC} and then Stirling's formula and \eqref{eq:csum} prove \eqref{eq:clim}. Putting $(j,\nu)=(0,n_0)$ in \eqref{eq:concob} and considering the sum $\sum_\nu$ for \eqref{eq:concob} with $j=1$, we have \eqref{eq:numdec} and \eqref{eq:ordsum}, respectively.
Comparing the proof of Theorem~\ref{thm:c} with that of Theorem~\ref{thm:irrKac}, we have \eqref{eq:dncKac}. Proposition~\ref{prop:wm} also proves \eqref{eq:dncKac}. \end{proof}
\begin{rem}\label{rem:conn} {\rm i)\ } When we calculate a connection coefficient for a given rigid partition $\mathbf m$ by \eqref{eq:connection}, it is necessary to get all the direct decompositions $\mathbf m=\mathbf m'\oplus \mathbf m''$ satisfying $m'_{0,n_0}=m''_{1,n_1}=1$. In this case the equality \eqref{eq:numdec} is useful because we know that the number of such decompositions equals $n_0+n_1-2$, namely, the number of gamma functions appearing in the numerator equals that appearing in the denominator in \eqref{eq:connection}.
{\rm ii) } A direct decomposition $\mathbf m=\mathbf m'\oplus\mathbf m''$ for a rigid tuple $\mathbf m$ means that $\{\alpha_{\mathbf m'},\alpha_{\mathbf m''}\}$ is a fundamental system of a root system of type $A_2$ in $\mathbb R\alpha_{\mathbf m'} +\mathbb R\alpha_{\mathbf m''}$ such that $\alpha_{\mathbf m}= \alpha_{\mathbf m'}+\alpha_{\mathbf m''}$ and\\[2pt] \qquad$\begin{cases}
(\alpha_{\mathbf m'}|\alpha_{\mathbf m'})
=(\alpha_{\mathbf m''}|\alpha_{\mathbf m''})=2,\\
(\alpha_{\mathbf m'}|\alpha_{\mathbf m''})=-1.
\end{cases} $
\hspace{7cm} \begin{xy}{\ar (10,0)*+!L{\alpha_{\mathbf m'}}}, (.5,\halfrootthree): {\ar(0,0);(10,0)*+!D{\alpha_{\mathbf m}}}, (0,0),(.5,\halfrootthree): {\ar(0,0);(10,0)*+!D{\alpha_{\mathbf m''}}}, \end{xy}
{\rm iii)\ } In view of Definition~\ref{def:FRLM}, the condition $\mathbf m=\mathbf m'\oplus\mathbf m''$ in \eqref{eq:connection} means \begin{equation}\label{eq:sum1}
\bigl|\{\lambda_{\mathbf m'}\}\bigr|+
\bigl|\{\lambda_{\mathbf m''}\}\bigr|=1. \end{equation} Hence we have \begin{equation}\label{eq:cprod} \begin{split}
&c(\lambda_{0,n_0}\!\rightsquigarrow\!\lambda_{1,n_1})\cdot
c(\lambda_{1,n_1}\!\rightsquigarrow\!\lambda_{0,n_0})\\
&\qquad=\frac
{\displaystyle\prod_{\substack{\mathbf m'\oplus\mathbf m''=\mathbf m\\
m'_{0,n_0}=m''_{1,n_1}=1}}
\sin\bigl(|\{\lambda_{\mathbf m'}\}|\pi\bigr)
}
{\displaystyle\prod_{\nu=1}^{n_0-1}
\sin\bigl(\lambda_{0,\nu}-\lambda_{1,\nu}\bigr)\pi
\cdot\prod_{\nu=1}^{n_1-1}
\sin\bigl(\lambda_{1,\nu}-\lambda_{1,n_1}\bigr)\pi
}. \end{split} \end{equation}
{\rm iv)\ } By the aid of a computer, the author obtained the table of the concrete connection coefficients \eqref{eq:connection} for the rigid triplets $\mathbf m$ satisfying $\ord\mathbf m\le 40$ together with checking \eqref{eq:concob}, which contains 4,111,704 independent cases (cf.~\S\ref{sec:okubo}). \end{rem}
\subsection{An estimate for large exponents}\label{sec:estimate} The Gauss hypergeometric series \[
F(\alpha,\beta,\gamma;x) := \sum_{k=0}^\infty\frac{\alpha(\alpha+1)
\cdots(\alpha+k-1)\cdot
\beta(\beta+1)\cdots(\beta+k-1)}
{\gamma(\gamma+1)\cdots(\gamma+k-1)\cdot k!}x^k \] uniformly and absolutely converges for \begin{equation}
x\in \overline D:=\{x\in\mathbb C\,;\,|x|\le 1\} \end{equation} if $\RE\gamma>\RE(\alpha+\beta)$ and defines a continuous function on $\overline D$. The continuous function $F(\alpha,\beta,\gamma+n;x)$ on $\overline D$ uniformly converges to the constant function $1$ when $n\to+\infty$, which obviously implies \begin{equation}\label{eq:gammainf} \lim_{n\to\infty} F(\alpha,\beta,\gamma+n;1)=1 \end{equation} and proves Gauss's summation formula \eqref{eq:Gausssum} by using the recurrence relation \begin{equation}\label{eq:GCratio}
\frac{F(\alpha,\beta,\gamma;1)}{F(\alpha,\beta,\gamma+1;1)}
=\frac{(\gamma-\alpha)(\gamma-\beta)}{\gamma(\gamma-\alpha-\beta)}. \end{equation} We will generalize such convergence in a general system of ordinary differential equations of Schlesinger canonical form.
Under the condition \[
a>0,\ b>0\text{ and }c>a+b, \] the function $F(a,b,c;x)=\sum_{k=0}^\infty\frac{(a)_k(b)_k}{(c)_kk!}x^k$ is strictly increasing continuous function of $x\in [0,1]$ satisfying \[
1\le F(a,b,c;x)\le F(a,b,c;1)=
\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)} \] and it increases if $a$ or $b$ or $-c$ increases. In particular, if \[
0\le a\le N,\ 0\le b\le N \text{ and } c> 2N \] with a positive integer $N$, we have \begin{align*}
0&\le F(a,b,c;x) - 1\\
&\le \frac{\Gamma(c)\Gamma(c-2N)}{\Gamma(c-N)\Gamma(c-N)}-1
=\frac{(c-N)_N}{(c-2N)_N}-1
=\prod_{\nu=1}^N\frac{c-\nu}{c-N-\nu}-1\\
&\le\left(\frac{c-N}{c-2N}\right)^N-1
=\left(1+\frac{N}{c-2N}\right)^N-1\\
&\le N\left(1+\frac{N}{c-2N}\right)^{N-1}\frac{N}{c-2N}. \end{align*} Thus we have the following lemma.
\begin{lem}\label{lem:GHestim} For a positive integer $N$ we have \begin{equation}
|F(\alpha,\beta,\gamma;x) - 1|\le \left(1+\frac{N}{\RE\gamma-2N}\right)^N-1 \end{equation} if \begin{equation}
x\in\overline D,\ |\alpha|\le N,\ |\beta|\le N\text{\quad and\quad}\RE\gamma>2N. \end{equation} \end{lem}
\begin{proof} The lemma is clear because \begin{align*}
\Bigl|\sum_{k=1}^\infty\frac{(\alpha)_k(\beta)_k}{(\gamma)_kk!}x^k\Bigr|
&\le \sum_{k=1}^\infty\frac{(|\alpha|)_k(|\beta|)_k}{(\RE\gamma)_kk!}|x|^k
=F(|\alpha|,|\beta|,\RE\gamma-2N;|x|)-1 \end{align*}\\[-9mm] \end{proof}
For the Gauss hypergeometric equation \[x(1-x)u''+\bigl(\gamma-(\alpha+\beta+1)x\bigr)u'-\alpha\beta u=0\] we have \begin{align*}
(xu')'&=
u'+xu''=\frac{xu'}x
+\frac{((\alpha+\beta+1)x-\gamma)u'+\alpha\beta u}{1-x}\\
&=\frac{\alpha\beta}{1-x}u+\left(\frac1{x}-\frac{\gamma}{x(1-x)}
+\frac{\alpha+\beta+1}{1-x}\right)xu'\\
&=\frac{\alpha\beta}{1-x}u
+\left(\frac{1-\gamma}x+\frac{\alpha+\beta-\gamma+1}{1-x}\right)xu'. \end{align*} Putting \begin{equation}
\tilde u=\binom{u_0}{u_1}:=\binom u{\frac{xu'}\alpha} \end{equation} we have \begin{equation}
\begin{split}
\tilde u'&=
\frac{\begin{pmatrix}
0 & \alpha\\ 0 & 1-\gamma
\end{pmatrix}}x\tilde u
+\frac{\begin{pmatrix}
0 & 0\\ \beta& \alpha+\beta-\gamma+1
\end{pmatrix}}{1-x}\tilde u.
\end{split} \end{equation} In general, for \begin{align*}
v'&=\frac Axv+\frac B{1-x}v\allowdisplaybreaks \intertext{we have}\allowdisplaybreaks
xv'&=Av + \frac x{1-x}Bv\\
&= Av + x\bigl(xv'+(B-A)v\bigr). \end{align*} Thus \begin{equation}\label{eq:majgauss}
\begin{cases}
xu_0'=\alpha u_1,\\
xu_1'=(1-\gamma)u_1+x\bigl(xu_1'+\beta u_0+(\alpha+\beta)u_1\bigr)
\end{cases} \end{equation} and the functions \begin{equation}\label{eq:majGsol}
\begin{cases}
u_0 = F(\alpha,\beta,\gamma;x),\\
u_1 =\displaystyle\frac{\beta x }\gamma F(\alpha+1,\beta+1,\gamma+1;x)
\end{cases} \end{equation} satisfies \eqref{eq:majgauss}. \begin{thm}\label{thm:paralimits} Let $n$, $n_0$ and $n_1$ be positive integers satisfying $n=n_0+n_1$ and let $A=\begin{pmatrix}0 & A_0\\ 0 & A_1\end{pmatrix}$, $B= \begin{pmatrix}0 & 0 \\ B_0 & B_1\end{pmatrix}\in M(n,\mathbb C)$ such that $A_1$, $B_1\in M(n_1,\mathbb C)$, $A_0\in M(n_0,n_1,\mathbb C)$ and $B_0\in M(n_1,n_0,\mathbb C)$. Let $D({\mathbf 0},{\mathbf m})=D({\mathbf 0},m_1,\ldots,m_{n_1})$ be the diagonal matrix of size $n$ whose $k$-th diagonal element is\/ $m_{k-n_0}$ if\/ $k>n_0$ and\/ $0$ otherwise. Let\/ $u^{\mathbf m}$ be the local holomorphic solution of \begin{equation}
u =\frac {A-D({\mathbf 0},{\mathbf m})}xu
+ \frac{B-D({\mathbf 0},{\mathbf m})}{1-x}u \end{equation} at the origin. Then if\/ $\RE m_\nu$ are sufficiently large for $\nu=1,\dots,n_1$, the Taylor series of $u^{\mathbf m}$ at the origin
uniformly converge on $\overline D=\{x\in\mathbb C\,;\,|x|\le 1\}$ and for a positive number $C$, the function $u^{\mathbf m}$ and their derivatives uniformly converge to constants on $\overline D$ when\/
$\min\{\RE m_1,\ldots,\RE m_{n_1}\}\to+\infty$ with\/ $|A_{ij}|+|B_{ij}|\le C$. In particular, for $x\in\overline D$ and an integer $N$ satisfying \begin{equation}
\sum_{\nu=1}^{n_1}|(A_0)_{i\nu}|\le N,\ \sum_{\nu=1}^{n_1}|(A_1)_{i\nu}|\le N,\
\sum_{\nu=1}^{n_0}|(B_0)_{i\nu}|\le N,\ \sum_{\nu=1}^{n_1}|(B_1)_{i\nu}|\le N \end{equation} we have \begin{equation}
\max_{1\le\nu\le n}
\bigl|u_\nu^{\mathbf m}(x)-u_\nu^{\mathbf m}(0)\bigr|
\le
\max_{1\le\nu\le n_0}
|u^{\mathbf m}_\nu(0)|
\cdot\frac{2^N(N+1)^2}{\displaystyle\min_{1\le \nu\le n_1}\RE m_\nu-4N-1} \end{equation} if\/ $\RE m_\nu>5N+4$ for $\nu=1,\dots,n_1$. \end{thm} \begin{proof} Use the method of majorant series and compare to the case of Gauss hypergeometric series (cf.~\eqref{eq:majgauss} and \eqref{eq:majGsol}), namely, $\lim_{c\to+\infty}F(a,b,c;x)=1$ on $\overline D$ with a solution of the Fuchsian system \begin{align*}
u' &=\frac Axu + \frac B{1-x}u,\allowdisplaybreaks\\
A &=\begin{pmatrix}
0 & A_0\\ 0 & A_1
\end{pmatrix},\quad
B =\begin{pmatrix}
0 & 0\\ B_0 & B_1
\end{pmatrix},\quad
u =\binom{v_0}{v_1},\allowdisplaybreaks\\
xv_0' &= A_0v_1,\\
xv_1' &=x^2v_1'+(1-x)A_1v_1+xB_0v_0+xB_1v_1\\
&= A_1v_1+x\bigl(xv_1'+B_0v_0+(B_1-A_1)v_1\bigr) \end{align*} or the system obtained by the substitution $A_1\mapsto A_1-D(\mathbf m)$ and $B_1\mapsto B_1-D(\mathbf m)$. Fix positive real numbers $\alpha$, $\beta$ and $\gamma$ satisfying \begin{align*}
\alpha&\ge \sum_{\nu=1}^{n_1}|(A_0)_{i\nu}|\quad(1\le i\le n_0),
\quad
\beta\ge\sum_{\nu=1}^{n_0} |(B_0)_{i\nu}|\quad(1\le i\le n_1),\\
\alpha+\beta
&\ge \sum_{\nu=1}^{n_1}|(B_1-A_1)_{i\nu}|\quad(1\le i\le n_0),\\
\gamma&=
\min\{\RE m_1,\dots,\RE m_{n_1}\}
-2\max_{1\le i\le n_1}\sum_{\nu=1}^{n_1}|(A_1)_{i\nu}|-1
>\alpha+\beta. \end{align*} Then the method of majorant series with Lemma~\ref{lem:RSestim}, \eqref{eq:majgauss} and \eqref{eq:majGsol} imply \[
u^{\mathbf m}_i \ll
\begin{cases}
\max_{1\le \nu\le n_0}
|u^{\mathbf m}_\nu(0)|
\cdot F(\alpha,\beta,\gamma;x)&(1\le i\le n_0),\\
\frac\beta\gamma\cdot\max_{1\le\nu\le n_0}
|u^{\mathbf m}_\nu(0)|
\cdot F(\alpha+1,\beta+1,\gamma+1;x)&(n_0<i\le n),
\end{cases} \] which proves the theorem because of Lemma~\ref{lem:GHestim} with $\alpha=\beta=N$ as follows. Here $\sum_{\nu=0}^\infty a_\nu x^\nu \ll \sum_{\nu=0}^\infty b_\nu x^\nu$
for formal power series means $|a_\nu|\le b_\nu$ for $\nu\in\mathbb Z_{\ge 0}$.
Put $\bar m=\min\{\RE m_1,\dots,\RE m_{n_1}\}-2N-1$
and $L=\max_{1\le \nu\le n_0}|u^{\mathbf m}_\nu(0)|$. Then $\gamma\ge \bar m-2N-1$ and if $0\le i\le n_0$ and $x\le\overline D$, \begin{align*}
|u^{\mathbf m}_i(x) - u^{\mathbf m}_i(0)|
&\le L
\cdot\bigl(F(\alpha,\beta,\gamma;|x|)-1\bigr)\\
&\le L\biggl(\Bigl(1+\frac{N}{\bar m-4N-1}\Bigr)^N-1\biggr)\\
&\le L\Bigl(1 + \frac{N}{\bar m-4N-1}\Bigr)^{N-1}\frac{N^2}{\bar m-4N-1}
\le \frac{L2^{N-1}N^2}{\bar m - 4N-1}. \end{align*} If $n_0<i\le n$ and $x\in\overline D$, \begin{align*}
|u^{\mathbf m}_i(x)|
&\le \frac{\beta}{\gamma}\cdot LF(\alpha+1,\beta+1,\gamma+1;|x|)\\
&\le \frac{LN}{\bar m-2N-1}
\biggl(\Bigl(1+\frac{N+1}{\bar m-4N-3}\Bigr)^{N+1}+1\biggr)
\le \frac{LN(2^{N+1}+1)}{\bar m-2N-1}. \end{align*}\\[-.8cm] \end{proof}
\begin{lem}\label{lem:RSestim} Let $A\in M(n,\mathbb L)$ and put \begin{equation}
|A|:=\max_{1\le i\le n}
\sum_{\nu=1}^n|A_{i\nu}|. \end{equation} If positive real numbers $m_1,\ldots,m_n$ satisfy \begin{equation}
m_{\text{min}}:=\min\{m_1,\dots,m_n\} > 2|A|, \end{equation} we have \begin{equation}
|\bigl(kI_n+D(\mathbf m)-A\bigr)^{-1}|
\le (k+m_{\text{min}}-2|A|)^{-1}\qquad(\forall k\ge0). \end{equation} \end{lem} \begin{proof}Since \begin{align*}
\bigl|\bigl(D(\mathbf m)-A\bigr)^{-1}\bigr|
&= \bigl|D(\mathbf m)^{-1}(I_n-D(\mathbf m)^{-1}A)^{-1}\bigr|\\
&=\Bigl|D(\mathbf m)^{-1}\sum_{k=0}^\infty\bigl(D(\mathbf m)^{-1}A\bigr)^k\Bigr|\\
& \le m_{\text{min}}^{-1}\cdot\Bigl(1+\frac{2|A|}{m_{\text{min}}}\Bigr)
\le (m_{\min}-2|A|)^{-1}, \end{align*} we have the lemma by replacing $m_\nu$ by $m_\nu+k$ for $\nu=1,\dots,n$. \end{proof}
\subsection{Zeros and poles of connection coefficients}\label{sec:C2} In this subsection we examine the connection coefficients to calculate them in a different way from the one given in \S\ref{sec:C1}.
First review the connection coefficient $c(0\!:\!\lambda_{0,2}\!\rightsquigarrow\!1\!:\!\lambda_{1,2})$ for the solution of Fuchsian differential equation with the Riemann scheme $\begin{Bmatrix}
x=0 & 1 & \infty\\
\lambda_{0,1} & \lambda_{1,1} & \lambda_{2,1}\\
\lambda_{0,2} & \lambda_{1,2} & \lambda_{2,2} \end{Bmatrix}$. Denoting the connection coefficient $c(0\!:\!\lambda_{0,2}\!\rightsquigarrow\!1\!:\!\lambda_{1,2})$ by $
c(\left\{\begin{smallmatrix}
\lambda_{0,1}&&\lambda_{1,1}&\lambda_{2,1}\\
\lambda_{0,2}&\rightsquigarrow&\lambda_{1,2}&\lambda_{2,2}
\end{smallmatrix}\right\}), $ we have \begin{equation}
u_0^{\lambda_{0,2}}= c(\left\{\begin{smallmatrix}
\lambda_{0,1}&&\lambda_{1,1}&\lambda_{2,1}\\
\lambda_{0,2} &\rightsquigarrow& \lambda_{1,2}&\lambda_{2,2}
\end{smallmatrix}\right\})u_1^{\lambda_{1,2}}
+ c(\left\{\begin{smallmatrix}
\lambda_{0,1} && \lambda_{1,2}&\lambda_{2,1}\\
\lambda_{0,2} &\rightsquigarrow& \lambda_{1,1}&\lambda_{2,2}
\end{smallmatrix}\right\})u_1^{\lambda_{1,1}}. \end{equation} \begin{equation}
\begin{split}
&c(\left\{\begin{smallmatrix}
\lambda_{0,1} && \lambda_{1,1} &\lambda_{2,1}\\
\lambda_{0,2} &\rightsquigarrow& \lambda_{1,2} &\lambda_{2,2}
\end{smallmatrix}\right\})
=c(\left\{\begin{smallmatrix}
\lambda_{0,1}-\lambda_{0,2} &&
\lambda_{1,1}-\lambda_{1,2}&
\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,1}\\
0&\rightsquigarrow&
0&\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,2}
\end{smallmatrix}\right\})\\
&\quad= F(\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,1},
\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,2},
\lambda_{0,2}-\lambda_{0,1}+1;1)
\end{split}\label{eq:CGsum} \end{equation} under the notation in Definition~\ref{def:cc}. As was explained in the first part of \S\ref{sec:estimate}, the connection coefficient is calculated from \begin{gather}
\lim_{n\to\infty} c(\left\{\begin{smallmatrix}
\lambda_{0,1}-n \phantom{\rightsquigarrow} \lambda_{1,1}+n &\lambda_{2,1}\\
\lambda_{0,2}\ \ \rightsquigarrow \ \ \lambda_{1,2} & \lambda_{2,2}
\end{smallmatrix}\right\})=1\label{eq:GGlimit} \intertext{and}
\frac{c(\left\{\begin{smallmatrix}
\lambda_{0,1} \phantom{\rightsquigarrow} \lambda_{1,1} & \lambda_{2,1}\\
\lambda_{0,2} \rightsquigarrow \lambda_{1,2} & \lambda_{2,2}
\end{smallmatrix}\right\})}
{c(\left\{\begin{smallmatrix}
\lambda_{0,1}-1 \phantom{\rightsquigarrow} \lambda_{1,1}+1 & \lambda_{2,1}\\
\lambda_{0,2}\ \rightsquigarrow \ \lambda_{1,2} & \lambda_{2,2}
\end{smallmatrix}\right\})}
=\frac{(\lambda_{0,2}+\lambda_{1,1}+\lambda_{2,2})
(\lambda_{0,2}+\lambda_{1,1}+\lambda_{2,1})}
{(\lambda_{0,2}-\lambda_{0,1}+1)(\lambda_{1,1}-\lambda_{1,2})}.
\label{eq:GGratio} \end{gather} The relation \eqref{eq:GGlimit} is easily obtained from \eqref{eq:CGsum} and \eqref{eq:gammainf} or can be reduced to Theorem~\ref{thm:paralimits}.
We will examine \eqref{eq:GGratio}. For example, the relation \eqref{eq:GGratio} follows from the relation \eqref{eq:GCratio} which is obtained from \begin{multline*} \gamma\bigl(\gamma-1-(2\gamma-\alpha-\beta-1)x\bigr)
F(\alpha,\beta,\gamma;x)+(\gamma-\alpha)(\gamma-\beta) xF(\alpha,\beta,\gamma+1;x)\\ =\gamma(\gamma-1)(1-x)F(\alpha,\beta,\gamma-1;x) \end{multline*} by putting $x=1$ (cf.~\cite[\S14.1]{WW}). We may use a shift operator as follows. Since \begin{align*}
&\frac{d}{dx}F(\alpha,\beta,\gamma;x)=
\frac{\alpha\beta}{\gamma}F(\alpha+1,\beta+1,\gamma+1;x)\\\
&\quad=c(\left\{\begin{smallmatrix}
1-\gamma&& \gamma-\alpha-\beta&\alpha\\
0&\rightsquigarrow&0&\beta
\end{smallmatrix}\right\})\tfrac{d}{dx}u_1^0
+ c(\left\{\begin{smallmatrix}
1-\gamma&& 0&\alpha\\
0&\rightsquigarrow&\gamma-\alpha-\beta&\beta
\end{smallmatrix}\right\})\tfrac{d}{dx}u_1^{\gamma-\alpha-\beta} \end{align*} and \begin{equation*}
\tfrac{d}{dx}u_1^{\gamma-\alpha-\beta}\equiv
(\alpha+\beta-\gamma)(1-x)^{\gamma-\alpha-\beta-1}
\mod(1-x)^{\gamma-\alpha-\beta}\mathcal O_1, \end{equation*} we have \begin{equation*} \frac{\alpha\beta}{\gamma}c(\left\{\begin{smallmatrix}
-\gamma&& 0&\alpha+1\\
0&\rightsquigarrow&\gamma-\alpha-\beta-1&\beta+1
\end{smallmatrix}\right\}) =(\alpha+\beta-\gamma)
c(\left\{\begin{smallmatrix}
1-\gamma&& 0&\alpha\\
0&\rightsquigarrow&\gamma-\alpha-\beta&\beta
\end{smallmatrix}\right\}), \end{equation*} which also proves \eqref{eq:GGratio} because \begin{equation*}
\frac{c(\left\{\begin{smallmatrix}
\lambda_{0,1} && \lambda_{1,1} & \lambda_{2,1}\\
\lambda_{0,2} &\rightsquigarrow& \lambda_{1,2} &\lambda_{2,2}
\end{smallmatrix}\right\})}
{c(\left\{\begin{smallmatrix}
\lambda_{0,1}-1&& \lambda_{1,1}+1 & \lambda_{2,1}\\
\lambda_{0,2} & \rightsquigarrow & \lambda_{1,2} & \lambda_{2,2}
\end{smallmatrix}\right\})} =
\frac{c(\left\{\begin{smallmatrix}
\lambda_{0,1} -\lambda_{0,2}&&
0&\lambda_{0,2}+\lambda_{1,1}+\lambda_{2,1}\\
0&\rightsquigarrow&\lambda_{1,2}-\lambda_{1,1}
&\lambda_{0,2}+\lambda_{1,1}+\lambda_{2,2}
\end{smallmatrix}\right\})}
{c(\left\{\begin{smallmatrix}
\lambda_{0,1} -\lambda_{0,2}-1&&
0&\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,1}+1\\
0&\rightsquigarrow
&\lambda_{1,2}-\lambda_{1,1}-1& \lambda_{0,2}+\lambda_{1,2}+\lambda_{2,2}+1
\end{smallmatrix}\right\})}. \end{equation*} Furthermore each linear term appeared in the right hand side of \eqref{eq:GGratio} has own meaning, which is as follows.
Examine the zeros and poles of the connection coefficient $c(\left\{\begin{smallmatrix}
\lambda_{0,1}&&\lambda_{1,1} & \lambda_{2,1}\\
\lambda_{0,2}&\rightsquigarrow&\lambda_{1,2} & \lambda_{2,2}
\end{smallmatrix}\right\}) $. We may assume that the parameters $\lambda_{j,\nu}$ are generic in the zeros or the poles.
Consider the linear form $\lambda_{0,2}+\lambda_{1,1}+\lambda_{2,2}$. The local solution $u_0^{\lambda_{0,2}}$ corresponding to the characteristic exponent $\lambda_{0,2}$ at $0$ satisfies a Fuchsian differential equation of order 1 which has the characteristic exponents $\lambda_{2,2}$ and $\lambda_{1,1}$ at $\infty$ and $1$, respectively, if and only if the value of the linear form is $0$ or a negative integer. In this case $c(\left\{\begin{smallmatrix}
\lambda_{0,1}&&\lambda_{1,1}&\lambda_{2,1}\\
\lambda_{0,2}&\rightsquigarrow&\lambda_{1,2}&\lambda_{2,2}
\end{smallmatrix}\right\})$ vanishes. This explains the term $\lambda_{0,2}+\lambda_{1,1}+\lambda_{2,2}$ in the numerator of the right hand side of \eqref{eq:GGratio}. The term $\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,2}$ is similarly explained.
The normalized local solution $u_0^{\lambda_{0,2}}$ has poles where $\lambda_{0,1}-\lambda_{0,2}$ is a positive integer. The residue at the pole is a local solution corresponding to the exponent $\lambda_{0,2}$. This means that $c(\left\{\begin{smallmatrix}
\lambda_{0,1}&&\lambda_{1,1} & \lambda_{2,1}\\
\lambda_{0,2}&\rightsquigarrow&\lambda_{1,2} & \lambda_{2,2}
\end{smallmatrix}\right\})$ has poles where $\lambda_{0,1}-\lambda_{0,2}$ is a positive integer, which explains the term $\lambda_{0,2}-\lambda_{0,1}+1$ in the denominator of the right hand side of \eqref{eq:GGratio}.
There exists a local solution $a(\lambda)u_1^{\lambda_{1,1}}+b(\lambda)u_1^{\lambda_{1,2}}$ such that it is holomorphic for $\lambda_{j,\nu}$ and $b(\lambda)$ has a pole if the value of $\lambda_{1,1}-\lambda_{1,2}$ is a non-negative integer, which means $c(\left\{\begin{smallmatrix}
\lambda_{0,1}&&\lambda_{1,1}&\lambda_{2,1}\\
\lambda_{0,2}&\rightsquigarrow&\lambda_{1,2}&\lambda_{2,2}
\end{smallmatrix}\right\})$ has poles where $\lambda_{1,2}-\lambda_{1,1}$ is non-negative integer. This explains the term $\lambda_{1,1}-\lambda_{1,2}$ in the denominator of the right hand side of \eqref{eq:GGratio}. These arguments can be generalized, which will be explained in this subsection.
Fist we examine the possible poles of connection coefficients. \begin{prop}\label{prop:paradep} Let $Pu=0$ be a differential equation of order $n$ with a regular singularity at $x=0$ such that $P$ contains a holomorphic parameter $\lambda=(\lambda_1,\dots,\lambda_N)$ defined in a neighborhood of $\lambda^o=(\lambda_1^o,\dots,\lambda_N^o)$ in $\mathbb C^N$. Suppose that the set of characteristic exponents of $P$ at $x=0$ equals $\{[\lambda_1]_{(m_1)},\dots,[\lambda_N]_{(m_N)}\}$ with $n=m_1+\dots+m_N$ and \begin{equation}
\lambda_{2,1}^o:=\lambda_2^o-\lambda_1^o\in\mathbb Z_{\ge0}\text{ and\/ }
\lambda_i^o-\lambda_j^o\notin\mathbb Z\text{ if\/ }
1\le i< j\le N\text{ and\/ }j\ne 2. \end{equation} Let $u_{j,\nu}$ be local solutions of $Pu=0$ uniquely defined by \begin{equation}\label{eq:SolGen}
u_{j,\nu}\equiv x^{\lambda_j+\nu}\mod x^{\lambda_j+m_j}\mathcal O_0
\quad(j=1,\dots,m_j\text{ and }\nu=0,\dots,m_j-1). \end{equation} Note that $u_{j,\nu}=\sum_{k\ge0}a_{k,j,\nu}(\lambda)x^{\lambda_j+\nu+k}$ with meromorphic functions $a_{k,j,\nu}(\lambda)$ of $\lambda$ which are holomorphic in a neighborhood of $\lambda^o$ if $\lambda_2-\lambda_1\ne \lambda_{2,1}^o$. Then there exist solutions $v_{j,\nu}$ with holomorphic parameter $\lambda$ in a neighborhood of $\lambda^o$ which satisfy the following relations. Namely \begin{equation}
v_{j,\nu}=u_{j,\nu}\quad(3\le j\le N\text{ and }\nu=0,\dots,m_j-1) \end{equation} and when $\lambda_1^o+m_1\ge \lambda_2^o+m_2$, \begin{equation} \begin{split}
v_{1,\nu}&=u_{1,\nu}\qquad(0\le\nu < m_1),\\
v_{2,\nu}&=\frac{u_{2,\nu}-u_{1,\nu+\lambda_{2,1}^o}}
{\lambda_1-\lambda_2+\lambda_{2,1}^o}
- \sum_{m_2+\lambda_{2,1}^o\le i < m_1}
\frac{b_{\nu,i}u_{1,i}}
{\lambda_1-\lambda_2+\lambda_{2,1}^o}
\quad(0\le\nu < m_2) \end{split} \end{equation}
\noindent with the diagram \begin{xy}
\ar@{-} *++!D{\lambda_1^o} *{\circ} ;
(10,0) *++!D{\lambda_1^o+1} *{\circ}="B"
\ar@{-} "B";(18,0) \ar@{.} (18,0);(28,0)^*!U{\cdots}
\ar@{-} (28,0);(36,0) *++!D{\lambda_1^o+\lambda_{2,1}^o} *{\circ}="H"
\ar@{-} "H";(44,0) \ar@{.} (44,0);(54,0)
\ar@{-} (54,0);(62,0) *++!D{\lambda_1^o+\lambda_{2,1}^o+m_2-1} *{\circ}="D"
\ar@{} (36,-8) *++!D{\lambda_2^o} *{\circ}="C"
\ar@{-} "C";(44,-8) \ar@{.} (44,-8);(54,-8)^*!U{\cdots}
\ar@{-} (54,-8);(62,-8) *++!D{\lambda_2^o+m_2-1} *{\circ};
\ar@{-} "D";(70,0) \ar@{.} (70,0);(80,0)
\ar@{-} (80,0);(88,0) *++!D{\lambda_1^o+m_1-1} *{\circ} \end{xy}\\
\noindent which illustrates some exponents and when $\lambda_1^o+m_1< \lambda_2^o+m_2$, \begin{equation} \begin{split}
v_{2,\nu}&=u_{2,\nu}\qquad(0\le\nu<m_2),\\
v_{1,\nu}&=u_{1,\nu}-\!\!\!\!\!\!
\sum_{\max\{0,m_1-\lambda_{2,1}^o\}\le i< m_2}
\frac{b_{\nu,i} u_{2,i}}
{\lambda_1-\lambda_2+\lambda_{2,1}^o}
\qquad(0\le\nu<\min\{m_1,\lambda_{2,1}^o\}),\\
v_{1,\nu}&=\frac{u_{1,\nu}-u_{2,\nu-\lambda_{2,1}^o}}
{\lambda_1-\lambda_2+\lambda_{2,1}^o}-\!\!\!\!
\sum_{\max\{0,m_1-\lambda_{2,1}^o\}\le i< m_2}
\frac{b_{\nu,i} u_{2,i}}
{\lambda_1-\lambda_2+\lambda_{2,1}^o}
\quad (\lambda_{2,1}^o\le \nu< m_1) \end{split} \end{equation} with \quad\begin{xy}
\ar@{-} *++!D{\lambda_1^o} *{\circ} ;
(10,0) *++!D{\lambda_1^o+1} *{\circ}="B"
\ar@{-} "B";(18,0) \ar@{.} (18,0);(28,0)^*!U{\cdots}
\ar@{-} (28,0);(36,0) *++!D{\lambda_1^o+\lambda_{2,1}^o} *{\circ}="H"
\ar@{-} "H";(44,0) \ar@{.} (44,0);(54,0)^*!U{\cdots}
\ar@{-} (54,0);(62,0) *++!D{\lambda_1^o+m_1-1} *{\circ}
\ar@{} (36,-8) *++!D{\lambda_2^o} *{\circ}="C"
\ar@{-} "C";(44,-8) \ar@{.} (44,-8);(54,-8)
\ar@{-} (54,-8);(62,-8) *++!D{\lambda_2^o-\lambda_{2,1}^o+m_1-1} *{\circ}="D";
\ar@{-} "D";(70,-8) \ar@{.} (70,-8);(80,-8)
\ar@{-} (80,-8);(88,-8) *++!D{\lambda_2^o+m_2-1} *{\circ} \end{xy}\\[5pt]
\noindent and here\/ $b_{\nu,i}\in\mathbb C$. Note that\/ $v_{j,\nu}$ $(1\le j\le N,\ 0\le \nu< m_j)$ are linearly independent for any fixed\/ $\lambda$ in a neighborhood of\/ $\lambda^o$. \end{prop} \begin{proof} See \S\ref{sec:reg} and the proof of Lemma~\ref{lem:GRS} (and \cite[Theorem~6.5]{O1} in a more general setting) for the construction of local solutions of $Pu=0$.
Note that $u_{j,\nu}$ for $j\ge 3$ are holomorphic with respect to $\lambda$ in a neighborhood of $\lambda=\lambda^o$. Moreover note that the local monodromy generator $M_0$ of the solutions $Pu=0$ at $x=0$ satisfies $\prod_{j=1}^N(M_0-e^{2\pi\sqrt{-1}\lambda_j})=0$ and therefore the functions $(\lambda_1-\lambda_2-\lambda_{2,1}^o)u_{j,\nu}$ of $\lambda$ are holomorphically extended to the point $\lambda=\lambda^o$ for $j=1$ and $2$, and the values of the functions at $\lambda=\lambda^o$ are solutions of the equation $Pu=0$ with $\lambda=\lambda^o$.
Suppose $\lambda_1^o+m_1\ge \lambda_2^o+m_2$. Then $u_{j,\nu}$ $(j=1,2)$ are holomorphic with respect to $\lambda$ at $\lambda=\lambda^o$ and there exist $b_{j,\nu}\in\mathbb C$ such that \[
u_{2,\nu}|_{\lambda=\lambda^o}=u_{1,\nu+\lambda_{2,1}^o}|_{\lambda=\lambda^o}
+ \sum_{m_2+\lambda_{2,1}^o\le\nu < m_1}b_{\nu,i}
\bigl(u_{1,i}|_{\lambda=\lambda^o}\bigr) \] and we have the proposition. Here \[
u_{2,\nu}|_{\lambda=\lambda^o}\equiv x^{\lambda_2^o}+
\sum_{m_2+\lambda_{2,1}^o\le\nu < m_1}b_{\nu,i}x^{\lambda_1^o+\nu}
\mod x^{\lambda_1^o+m_1}\mathcal O_0. \]
Next suppose $\lambda_1^o+m_1< \lambda_2^o+m_2$. Then there exist $b_{j,\nu}\in\mathbb C$ such that \begin{align*}
\bigl((\lambda_1-\lambda_2+\lambda_{2,1}^o)u_{1,\nu}\bigr)|_{\lambda=\lambda^o}
&=\!\sum_{\max\{0,m_1-\lambda_{2,1}^o\}\le i< m_2} \!\!\!
{b_{\nu,i} \bigl(u_{2,i}|_{\lambda=\lambda^o}\bigr)}\\
&\qquad\qquad\qquad\qquad\qquad\qquad(0\le\nu<\min\{m_1,\lambda_{2,1}^o\}),\\
u_{1,\nu}|_{\lambda=\lambda^o}&=
\!\sum_{\max\{0,m_1-\lambda_{2,1}^o\}\le i< m_2}\!\!\!
{b_{\nu,i} \bigl(u_{2,i}|_{\lambda=\lambda^o}\bigr)}
\quad (\lambda_{2,1}^o\le \nu< m_1) \end{align*} and we have the proposition. \end{proof}
The proposition implies the following corollaries. \index{Wronskian} \begin{cor}\label{cor:zeropole} Retain the notation and the assumption in\/ {\rm Proposition~\ref{prop:paradep}.}
{\rm i)} \ Let\/ $W_j(\lambda,x)$ be the Wronskian of\/ $u_{j,1},\dots,u_{j,m_j}$ for $j=1,\dots,N$. Then\/ $(\lambda_1-\lambda_2+\lambda_{2,1}^o)^{\ell_1}W_1(\lambda)$ and\/ $W_j(\lambda)$ with $2\le j\le N$ are holomorphic with respect to $\lambda$ in a neighborhood of\/ $\lambda^o$ by putting \begin{align}
\ell_1&= \max\bigl\{0,\min\{m_1,m_2,\lambda_{2,1}^o,\lambda_{2,1}^o+m_2-m_1\}\bigr\}. \end{align}
{\rm ii)} \ Let \[
w_k=\sum_{j=1}^{N}\sum_{\nu=1}^{m_j}a_{j,\nu,k}(\lambda)u_{j,\nu,k} \] be a local solution defined in a neighborhood of\/ $0$ with a holomorphic $\lambda$ in a neighborhood of $\lambda^o$. Then \[
(\lambda_1-\lambda_2+\lambda_{2,1}^o)^{\ell_{2,j}}
\det\Bigl(a_{j,\nu,k}(\lambda)
\Bigr)_{\substack{1\le \nu\le m_j\\1\le k\le m_j}} \] with \[
\begin{cases}
\ell_{2,1}=\max\bigl\{0,\min\{m_1-\lambda_{2,1}^o,m_2\}\bigr\},\\
\ell_{2,2}=\min\{m_1,m_2\},\\
\ell_{2,j}=0\qquad(3\le j\le N)
\end{cases} \] are holomorphic with respect to $\lambda$ in a neighborhood of $\lambda^o$. \end{cor} \begin{proof} i) Proposition~\ref{prop:paradep} shows that $u_{j,\nu}$ ($2\le j \le N,\ 0\le \nu<m_j$) are holomorphic with respect to $\lambda$ at
$\lambda^o$. The functions $u_{1,\nu}$ for $\min\{m_1,\lambda_{2,1}^o\}\le \nu\le m_1$ are same. The functions $u_{1,\nu}$ for $0\le\nu<\min\{m_1,\lambda_{2,1}^o\}$ may have poles of order 1 along $\lambda_2-\lambda_1=\lambda_{2,1}^o$ and their residues are linear combinations of $u_{2,i}|_{\lambda_2=\lambda_1+\lambda_{2,1}^o}$ with $\max\{0,m_1-\lambda_{2,1}^o\}\le i< m_2$. Since \begin{align*} &\min\bigl
\{\#\{\nu\,;\,0\le\nu<\min\{m_1,\lambda_{2,1}^o\}\},\
\#\{i\,;\, \max\{0,m_1-\lambda_{2,1}^o\}\le i< m_2\}\bigr\}\\ &\quad=\max\bigl\{0,\min\{m_1,\lambda_{2,1}^o,m_2,m_2-m_1+\lambda_{2,1}^o
\}\bigr\}, \end{align*} we have the claim.
ii) A linear combination of $v_{j,\nu}$ ($1\le j\le N,\ 0\le\nu\le m_j$) may have a pole of order 1 along $\lambda_1-\lambda_2+\lambda_{2,1}^o$ and its residue is a linear combination of \begin{align*}
&\bigl(u_{1,\nu}+\sum_{m_2+\lambda_{2,1}^o\le i < m_1}
b_{\nu+\lambda_{2,1}^o,i}u_{1,i}\bigr)|_{\lambda_2=\lambda_1+\lambda_{2,1}^o}
\quad(\lambda_{2,1}^o\le \nu< \min\{m_1,m_2+\lambda_{2,1}^o\}),\\
&\bigl(u_{2,\nu}
+\sum_{\max\{0,m_1-\lambda_{2,1}^o\}\le i< m_2}b_{\nu+\lambda_{2,1}^o,i}
u_{2,i}\bigr)
|_{\lambda_2=\lambda_1+\lambda_{2,1}^o}
\quad(0\le\nu<m_1-\lambda_{2,1}^o),\\
&\sum_{\max\{0,m_1-\lambda_{2,1}^o\}\le i< m_2}b_{\nu,i}u_{2,i}|
_{\lambda_2=\lambda_1+\lambda_{2,1}^o}
\quad(0\le \nu<\min\{m_1,\lambda_{2,1}^o\}). \end{align*} Since \begin{align*}
&\#\bigl\{\nu\,;\,\lambda_{2,1}^o\le \nu< \min\{m_1,m_2+\lambda_{2,1}^o\}\bigr\}=
\max\bigl\{0,\min\{m_1-\lambda_{2,1}^o,m_2\}\bigr\},\\
&\#\{\nu\,;\,0\le\nu<m_1-\lambda_{2,1}^o\}\\
&\quad+\min\bigl\{
\#\{i\,;\,\max\{0,m_1-\lambda_{2,1}^o\}\le i<m_2\},
\#\{\nu\,;\,0\le\nu<\min\{m_1,\lambda_{2,1}^o\}\}
\bigr\}\\
&=\min\{m_1,m_2\}, \end{align*} we have the claim. \end{proof}
\begin{rem} If the local monodromy of the solutions of $Pu=0$ at $x=0$ is locally non-degenerate, the value of $(\lambda_1-\lambda_2+\lambda_{2,1}^o)^{\ell_1}W_1(\lambda)$ at $\lambda=\lambda^o$ does not vanish. \end{rem}
\index{Wronskian} \begin{cor}\label{cor:parapole} Let $Pu=0$ be a differential equation of order $n$ with a regular singularity at $x=0$ such that $P$ contains a holomorphic parameter $\lambda=(\lambda_1,\dots,\lambda_N)$ defined on $\mathbb C^N$. Suppose that the set of characteristic exponents of $P$ at $x=0$ equals $\bigl\{[\lambda_1]_{(m_1)},\dots,[\lambda_N]_{(m_N)}\bigr\}$ with $n=m_1+\dots+m_N$. Let $u_{j,\nu}$ be the solutions of $Pu=0$ defined by \eqref{eq:SolGen}.
{\rm i)} \ Let $W_1(x,\lambda)$ denote the Wronskian of $u_{1,1},\dots,u_{1,m_1}$. Then \begin{equation}\label{eq:W1hol}
\frac{W_1(x,\lambda)}
{\prod_{j=2}^N\prod_{0\le \nu< \min\{m_1,m_j\}}
\Gamma(\lambda_1-\lambda_j+m_1-\nu)} \end{equation} is holomorphic for $\lambda\in\mathbb C^N$.
{\rm ii)} \ Let \begin{equation}
v_k(\lambda)=\sum_{j=1}^N\sum_{\nu=1}^{m_j} a_{j,\nu,k}(\lambda)u_{j,\nu}
\quad(1\le k\le m_1) \end{equation} be local solutions of\/ $Pu=0$ defined in a neighborhood of\/ $0$ which have a holomorphic parameter $\lambda\in\mathbb C^N$. Then \begin{equation}\label{eq:solchol}
\frac{\det\Bigl(a_{1,\nu,k}(\lambda)
\Bigr)_{\substack{1\le \nu\le m_1\\1\le k\le m_1}}}
{\prod_{j=2}^N\prod_{1\le\nu\le\min\{m_1,m_j\}}
\Gamma(\lambda_j-\lambda_1-m_1+\nu)} \end{equation} is a holomorphic function of $\lambda\in\mathbb C^N$. \end{cor} \begin{proof} Let $\lambda_{j,1}^o\in\mathbb Z$. The order of poles of \eqref{eq:W1hol} and that of \eqref{eq:solchol} along $\lambda_j-\lambda_1=\lambda_{j,1}^o$ are \begin{align*} &\#\{\nu\,;\,0\le\nu<\min\{m_1,m_j\}\text{ \ and \ }m_1-\lambda_{j,1}^o-\nu\le 0\}
\\ &\quad=\#\{\nu\,;\,\max\{0,m_1-\lambda_{j,1}^o\}\le\nu<\min\{m_1,m_j\}\}\\ &\quad=\max\bigl\{0,\min\{m_1,m_j,\lambda_{j,1}^o,\lambda_{j,1}^o+m_j-m_1\}\bigr\} \intertext{and} &\#\{\nu\,;\, 1\le\nu\le\min\{m_1,m_j\}\text{ and }
\lambda_{j,1}^o-m_1+\nu\le 0\}\\ &\quad=\max\bigl\{0,\min\{m_1,m_j,m_1-\lambda_{j,1}^o\}\bigr\}, \end{align*} respectively. Hence Corollary~\ref{cor:zeropole} assures this corollary. \end{proof}
\begin{rem} The product of denominator of \eqref{eq:W1hol} and that of \eqref{eq:solchol} equals the periodic function \[
\prod_{j=2}^N(-1)^{[\frac{\min\{m_1,m_j\}}2]+1}
\Bigl(\frac\pi{\sin(\lambda_1-\lambda_j)\pi}\Bigr)
^{\min\{m_1,m_j\}}. \] \end{rem}
\index{connection coefficient!generalized} \begin{defn}[generalized connection coefficient]\label{def:GC} Let $P_{\mathbf m}u=0$ be the Fuchsian differential equation with the Riemann scheme \begin{equation}
\begin{Bmatrix}
x = c_0=0 & c_1=1 & c_2& \cdots & c_p=\infty\\
[\lambda_{0,1}]_{(m_{0,1})} & [\lambda_{1,1}]_{(m_{1,1})}
&[\lambda_{2,1}]_{(m_{2,1})}&\cdots
&[\lambda_{p,1}]_{(m_{p,1})}\\
\vdots & \vdots & \vdots &\vdots & \vdots\\
[\lambda_{0,n_0}]_{(m_{0,n_0})} & [\lambda_{1,n_1}]_{(m_{1,n_1})}&
[\lambda_{2,n_2}]_{(m_{2,n_2})}&\cdots
&[\lambda_{p,n_p}]_{(m_{p,n_p})}
\end{Bmatrix}. \end{equation} We assume $c_2,\dots,c_{p-1}\notin[0,1]$. Let $u_{0,\nu}^{\lambda_{0,\nu}+k}$ $(1\le\nu\le n_0,\ 0\le k<m_{0,\nu})$ and $u_{1,\nu}^{\lambda_{1,\nu}+k}$ $(1\le\nu\le n_1,\ 0\le k<m_{1,\nu})$ be local solutions of $P_{\mathbf m}u=0$ such that \begin{equation}
\begin{cases}
u_{0,\nu}^{\lambda_{0,\nu}+k}\equiv
x^{\lambda_{0,\nu}+k} &\mod x^{\lambda_{0,\nu}+m_{0,\nu}}\mathcal O_0,\\
u_{1,\nu}^{\lambda_{1,\nu}+k}\equiv
(1-x)^{\lambda_{1,\nu}+k} &\mod (1-x)^{\lambda_{1,\nu}+m_{1,\nu}}\mathcal O_1.
\end{cases} \end{equation} They are uniquely defined on $(0,1)\subset\mathbb R$ when $\lambda_{j,\nu}-\lambda_{j,\nu'}\notin\mathbb Z$ for $j=0,\,1$ and $1\le\nu<\nu'\le n_j$. Then the connection coefficients $c^{\nu',k'}_{\nu,k}(\lambda)$ are defined by \begin{equation}
u_{0,\nu}^{\lambda_{0,\nu}+k}
=\sum_{\nu',\,k'}c^{\nu',k'}_{\nu,k}(\lambda)
u_{1,\nu'}^{\lambda_{1,\nu'}+k'}. \end{equation} Note that $c^{\nu',k'}_{\nu,k}(\lambda)$ is a meromorphic function of $\lambda$ when $\mathbf m$ is rigid.
Fix a positive integer $n'$ and the integer sequences $1\le \nu^0_1<\nu^0_2<\dots<\nu^0_L\le n_0$ and $1\le \nu^1_1<\nu^1_2<\dots<\nu^1_{L'}\le n_1$ such that \begin{equation}
n'=m_{0,\nu^0_1}+\cdots+m_{0,\nu^0_L}=m_{1,\nu^1_1}+\cdots+m_{1,\nu^1_{L'}}. \end{equation} Then a \textsl{generalized connection coefficient} is defined by \begin{equation}\label{eq:GC} \begin{split}
&c\bigl(0:[\lambda_{0,\nu^0_1}]_{(m_{0,\nu^0_1})},\dots,
[\lambda_{0,\nu^0_L}]_{(m_{0,\nu^0_L})}\rightsquigarrow
1:[\lambda_{1,\nu^1_1}]_{(m_{1,\nu^1_1})},\dots,
[\lambda_{1,\nu^1_{L'}}]_{(m_{1,\nu^1_{L'}})}\bigr)\\
&\quad:= \det\Bigl(c^{\nu',k'}_{\nu,k}(\lambda)\Bigr)_{\substack{
\nu\in\{\nu^0_1,\dots,\nu^0_L\},\ 0\le k<m_{0,\nu}\\
\nu'\in\{\nu^1_1,\dots,\nu^1_{L'}\},\ 0\le k'<m_{1,\nu'}\\
}}. \end{split} \end{equation} The connection coefficient defined in \S\ref{sec:C1} corresponds to the case when $n'=1$. \end{defn}
\begin{rem} i) When $m_{0,1}=m_{1,1}$, Corollary~\ref{cor:parapole} assures that \[
\frac{c\bigl(0:[\lambda_{0,1}]_{(m_{0,1})}
\rightsquigarrow 1:[\lambda_{1,1}]_{(m_{1,1})}\bigr)}{
{\displaystyle
\prod_{\substack{2\le j\le n_0\\0\le k< \min\{m_{0,1},\,m_{0,j}\}}}
\!\!\!\!\!\!\!\!\!\!\!\!
\Gamma(\lambda_{0,1}-\lambda_{0,j}+m_{0,1}-k)}\cdot\!\!
\displaystyle
\prod_{\substack{2\le j\le n_1\\0< k\le \min\{m_{1,1},\,m_{1,j}\}}}
\!\!\!\!\!\!\!\!\!\!\!\!
\Gamma(\lambda_{1,j}-\lambda_{1,1}-m_{1,1}+k)} \] is holomorphic for $\lambda_{j,\nu}\in\mathbb C$.
ii) Let $v_1,\dots,v_{n'}$ be generic solutions of $P_{\mathbf m}u=0$.
Then the generalized connection coefficient in Definition~\ref{def:GC} corresponds to a usual connection coefficient of the Fuchsian differential equation satisfied by the Wronskian of the $n'$ functions $v_1,\dots,v_{n'}$. The differential equation is of order $\binom n{n'}$. In particular, when $n'=n-1$, the differential equation is isomorphic to the dual of the equation $P_{\mathbf m}=0$ (cf.~Theorem~\ref{thm:prod}) and therefore the result in \S\ref{sec:C1} can be applied to the connection coefficient. The precise result will be explained in another paper. \end{rem}
\begin{rem}\label{rem:Cproc} The following procedure has not been completed in general. But we give a procedure to calculate the generalized connection coefficient \eqref{eq:GC}, which we put $c(\lambda)$ here for simplicity when $\mathbf m$ is rigid. \begin{enumerate} \item Let $\bar\epsilon=\bigl(\bar\epsilon_{j,\nu}\bigr)$ be the shift of the Riemann scheme $\{\lambda_{\mathbf m}\}$ such that \begin{equation} \begin{cases}
\bar\epsilon_{0,\nu}
=-1&(\nu\in\{1,2,\dots,n_0\}\setminus\{\nu^0_1,\dots,\nu^0_L\}),\\
\bar\epsilon_{1,\nu}
=1&(\nu\in\{1,2,\dots,n_1\}\setminus\{\nu^1_1,\dots,\nu^1_{L'}\}),\\
\bar\epsilon_{j,\nu}
=0&(\text{otherwise}). \end{cases}\end{equation} Then for generic $\lambda$ we show that the connection coefficient \eqref{eq:GC} converges to a non-zero meromorphic function $\bar c(\lambda)$ of $\lambda$ by the shift $\{\lambda_{\mathbf m}\}\mapsto \{(\lambda+k\bar\epsilon)_{\mathbf m}\}$ when $\mathbb Z_{>0}\ni k\to \infty$.
\item Choose suitable linear functions $b_i(\lambda)$ of $\lambda$ by applying Proposition~\ref{prop:paradep} or Corollary~\ref{cor:parapole} to $c(\lambda)$ so that $e(\lambda):=\prod_{i=1}^N\Gamma\bigl(b_i(\lambda)\bigr)^{-1}\cdot c(\lambda) \bar c(\lambda)^{-1}$ is holomorphic for any $\lambda$.
In particular, when $L=L'=1$ and $\nu_1^0=\nu_1^1=1$, we may put \begin{equation*} \begin{split}
\{b_i\}&=\bigcup_{j=2}^{n_0}
\bigl\{\lambda_{0,1}-\lambda_{0,j}+m_{0,1}-\nu\,;\,
0\le\nu<\min\{m_{0,1},m_{0,j}\}\bigr\}\\
&\quad\cup\bigcup_{j=2}^{n_1}
\bigl\{\lambda_{1,j}-\lambda_{1,1}-m_{1,1}+\nu\,;\,
1\le\nu\le\min\{m_{1,1},m_{1,j}\}\bigr\}. \end{split} \end{equation*}
\item Find the zeros of $e(\lambda)$ some of which are explained by the reducibility or the shift operator of the equation $P_{\mathbf m}u=0$ and choose linear functions $c_i(\lambda)$ of $\lambda$ so that $f(\lambda):=\prod_{i=1}^{N'}\Gamma\bigl(c_i(\lambda)\bigr)\cdot e(\lambda)$ is still holomorphic for any $\lambda$.
\item If $N=N'$ and $\sum_i{d_i(\lambda)}=\sum_i{c_i(\lambda)}$, Lemma~\ref{lem:GammaRatio} assures $f(\lambda)=\bar c(\lambda)$ and \begin{equation}\label{eq:Ccj}
c(\lambda)=\frac{\prod_{i=1}^N\Gamma\bigl(b_i(\lambda)\bigr)}
{\prod_{i=1}^N\Gamma\bigl(c_i(\lambda)\bigr)}\cdot\bar c(\lambda) \end{equation} because $\frac{f(\lambda)}{f(\lambda+\epsilon)}$ is a rational function of $\lambda$, which follows from the existence of a shift operator assured by Theorem~\ref{thm:irredrigid}. \end{enumerate} \end{rem}
\begin{lem}\label{lem:GammaRatio} Let $f(t)$ be a meromorphic function of\/ $t\in\mathbb C$ such that $r(t)=\frac {f(t)}{f(t+1)}$ is a rational function and \begin{equation}
\lim_{\mathbb Z_{>0}\ni k\to\infty}f(t+k)=1. \end{equation} Then there exists $N\in\mathbb Z_{\ge 0}$ and $b_i$, $c_i\in\mathbb C$ for $i=1,\dots,n$ such that \begin{align}
b_1+\cdots+b_N&=c_1+\cdots+c_N,\label{eq:sumeq}\\
f(t)&=\frac{\prod_{i=1}^N\Gamma(t+b_i)}{\prod_{i=1}^N\Gamma(t+c_i)}.
\label{eq:Gammaq} \end{align} Moreover, if $f(t)$ is an entire function, then $f(t)$ is the constant function $1$. \end{lem} \begin{proof} Since $\lim_{k\to\infty}r(t+k)=1$, we may assume \[
r(t)=\frac{\prod_{i=1}^N(t+c_i)}{\prod_{i=1}^N(t+b_i)} \] and then \[
f(t)=\frac{\prod_{i=1}^N\prod_{\nu=0}^{n-1}(t+c_i+\nu)}{\prod_{i=1}^N
\prod_{\nu=0}^{n-1}(t+b_i+\nu)}f(t+n). \] Since \[
\lim_{n\to\infty}\frac{n!n^{x-1}}{\prod_{\nu=0}^{n-1}(x+\nu)}=\Gamma(x), \] the assumption implies \eqref{eq:sumeq} and \eqref{eq:Gammaq}.
We may assume $b_i\ne c_j$ for $1\le i\le N$ and $1\le j\le N$. Then the function \eqref{eq:Gammaq} with \eqref{eq:sumeq} has a pole if $N>0$. \end{proof}
We have the following proposition for zeros of $c(\lambda)$. \begin{prop}\label{prop:Czero} Retain the notation in Remark~\ref{rem:Cproc} and fix $\lambda$ so that \begin{equation}\label{eq:Cgeneric}
\lambda_{j,\nu}-\lambda_{j,\nu'}\notin\mathbb Z\quad
(j=0,\ 1\text{ \ and \ }0\le\nu<\nu'\le n_j). \end{equation}
{\rm i)} \ The relation $c(\lambda)=0$ is valid if and only if there exists a non-zero function \begin{equation*}
v = \sum_{\substack{\nu\in\{\nu^0_1,\dots,\nu^0_L\}\\ 0\le k<m_{0,\nu}}}
C_{\nu,k}u_0^{\lambda_{0,\nu}+k}
= \sum_{\substack{\nu\in\{1,\dots,n_1\}
\setminus\{\nu^1_1,\dots,\nu^1_{L'}\}\\0\le k<m_{1,\nu}}}
C'_{\nu,k}u_1^{\lambda_{1,\nu}+k} \end{equation*} on $(0,1)$ with $C_{\nu,k},\,C'_{\nu,k}\in\mathbb C$.
{\rm ii)} \ Fix a shift $\epsilon=(\epsilon_{j,\nu})$ compatible to $\mathbf m$ and let $R_{\mathbf m}(\epsilon,\lambda)$ be the shift operator in\/ {\rm Theorem~\ref{thm:irredrigid}.} Suppose $R_{\mathbf m}(\epsilon,\lambda)$ is bijective, namely, $c_{\mathbf m}(\epsilon;\lambda)\ne 0$ {\rm (cf.~Theorem~\ref{thm:shiftC}).} Then $c(\lambda+\epsilon)=0$ if and only if $c(\lambda)=0$ \end{prop} \begin{proof} Assumption \eqref{eq:Cgeneric} implies that $\{u_0^{\lambda_{0,\nu}+k}\}$ and $\{u_1^{\lambda_{1,\nu}+k}\}$ define sets of basis of local solutions of the equation $P_{\mathbf m}u=0$. Hence the claim i) is clear from the definition of $c(\lambda)$.
Suppose $c(\lambda)=0$ and $R_{\mathbf m}(\epsilon,\lambda)$ is bijective. Then applying the claim i) to $R_{\mathbf m}(\epsilon,\lambda)v$, we have $c(\lambda+\epsilon)=0$. If $R_{\mathbf m}(\epsilon,\lambda)$ is bijective, so is $R_{\mathbf m}(-\epsilon,\lambda+\epsilon)$ and $c(\lambda+\epsilon)=0$ implies $c(\lambda)=0$. \end{proof} \begin{cor}\label{cor:zeroC} Let\/ $\mathbf m=\mathbf m'\oplus\mathbf m''$ be a rigid decomposition of\/ $\mathbf m$ such that \begin{equation}
\sum_{\nu\in\{\nu^0_1,\dots,\nu^0_L\}}m'_{0,\nu}>
\sum_{\nu\in\{\nu^1_1,\dots,\nu^1_{L'}\}}m'_{1,\nu}. \end{equation} Then\/
$\Gamma(|\{\lambda_{\mathbf m'}\}|)\cdot c(\lambda)$ is holomorphic under the condition \eqref{eq:Cgeneric}. \end{cor} \begin{proof}
When $|\{\lambda_{\mathbf m'}\}|$=0, we have the decomposition $P_{\mathbf m}=P_{\mathbf m''}P_{\mathbf m'}$ and hence $c(\lambda)=0$. There exists a shift $\epsilon$ compatible to $\mathbf m$
such that $\sum_{j=0}^p\sum_{\nu=1}^{n_j}m'_{j,\nu}\epsilon_{j,\nu}=1$. Let $\lambda$ be generic under $|\{\lambda_{\mathbf m}\}|=0$ and
$|\{\lambda_{\mathbf m'}\}|\in\mathbb Z\setminus\{0\}$. Then Theorem~\ref{thm:isom}) ii) assures $c_{\mathbf m}(\epsilon;\lambda)\ne0$ and Proposition~\ref{prop:Czero} proves the corollary. \end{proof}
\begin{rem}\label{rem:Cgamma} Suppose that Remark~\ref{rem:Cproc} (1) is established. Then Proposition~\ref{prop:paradep} and Proposition~\ref{prop:Czero} with Theorem~\ref{thm:shiftC} assure that the denominator and the numerator of the rational function which equals $\frac{c(\lambda)}{c(\lambda+\bar\epsilon)}$ are products of certain linear functions of $\lambda$ and therefore \eqref{eq:Ccj} is valid with suitable linear functions $b_i(\lambda)$ and $c_i(\lambda)$ of $\lambda$ satisfying $\sum_{i=1}^Nb_i(\lambda)=\sum_{i=1}^Nc_i(\lambda)$. \end{rem}
\index{hypergeometric equation/function!generalized!connection coefficient} \begin{exmp}[generalized hypergeometric function] The generalized hypergeometric series \eqref{eq:IGHG} satisfies the equation $P_n(\alpha;\beta)u=0$ given by \eqref{eq:GHP} and \cite[\S4.1.2 Example~9]{Kh} shows that the equation is isomorphic to the Okubo system \begin{equation}
\begin{split}
&\Bigl(x-\left(\begin{smallmatrix}
1\\
& 0\ \\
& &\ \ddots\\
& & &\ \ \ddots\\
& & & & 0
\end{smallmatrix}\right)\Bigr)
\frac{d\tilde u}{dx}
= \left(\begin{smallmatrix}
-\beta_n & 1 \\
\alpha_{2,1} & 0 &1\\
\alpha_{3,1} & & 1 & 1\\[-3pt]
{\small \vdots} & & & \ {\small \ddots} &\ \
{\small\ddots}\!\!\!\!\!\!\!\\
\alpha_{n-1,1}& & & & n-3 & 1\\
\alpha_{n,1} & -c_{n-1} & -c_{n-2}&\cdots&-c_2 & -c_1+(n-2)
\end{smallmatrix}\right)\tilde u
\end{split} \end{equation} with \[
u=\begin{pmatrix}u_1\\ \vdots\\ u_n\end{pmatrix},\ u=u_1\text{ \ and \ }
\sum_{\nu=1}^n\alpha_\nu=\sum_{\nu=1}^n\beta_\nu. \] Let us calculate the connection coefficient \[
c(0\!:\!0\rightsquigarrow 1\!:\!-\beta_n)
=\lim_{x\to1-0}(1-x)^{\beta_n}
{}_nF_{n-1}(\alpha_1,\dots,\alpha_n;\beta_1,\dots,\beta_{n-1};x)
\quad(\RE\beta_n>0). \] Applying Theorem~\ref{thm:paralimits} to the system of Schlesinger canonical form satisfied by $\Ad\bigl((1-x)^{\beta_n}\bigr)$, the connection coefficient satisfies Remark~\ref{rem:Cproc} i) with $\bar c(\lambda)=1$, namely, \begin{equation}\label{eq:CGHGL}
\lim_{k\to+\infty}
c(0\!:\!0\rightsquigarrow 1\!:\!-\beta_n)|
_{\alpha_j\mapsto \alpha_j+k,\ \beta_j\mapsto \beta_j+k
\ \ (1\le j\le n)}=1. \end{equation} Then Remark~\ref{rem:Cproc} ii) shows that $\prod_{j=1}^n\Gamma(\beta_j)^{-1}\cdot c(0\!:\!0\rightsquigarrow 1\!:\!-\beta_n)$ is a holomorphic function of $(\alpha,\beta)\in\mathbb C^{n+(n-1)}$.
Corresponding to the Riemann scheme \eqref{eq:GRSGHG}, the existence of rigid decompositions \[
\overbrace{1\cdots1}^n;n-11;\overbrace{1\cdots1}^n=
\overbrace{0\cdots0}^{n-1}1;10;0\cdots\overset{\overset{i}\smallsmile}{1}
\cdots0
\oplus
\overbrace{1\cdots1}^{n-1}0;n-11;1\cdots\overset{\overset{i}\smallsmile}0\cdots1 \] for $i=1,\dots,n$ proves that $\prod_{i=1}^n\Gamma(\alpha_i)\cdot \prod_{j=1}^n\Gamma(\beta_j)^{-1}\cdot c(0\!:\!0\rightsquigarrow 1\!:\!-\beta_n)$ is also entire holomorphic. Then the procedure given in Remark~\ref{rem:Cproc} assures \begin{equation}\label{eq:CGHG}
c(0\!:\!0\rightsquigarrow 1\!:\!-\beta_n)
= \frac{\prod_{i=1}^n\Gamma(\beta_i)} {\prod_{i=1}^n\Gamma(\alpha_i)}. \end{equation}
We can also prove \eqref{eq:CGHG} as in the following way. Since \[\frac{d}{dx}F(\alpha;\beta;x)
=\frac{\alpha_1\cdots\alpha_n}{\beta_1\cdots\beta_{n-1}}
F(\alpha_1+1,\dots,\alpha_n+1;\beta_1+1,\dots,\beta_{n-1}+1;x) \] and \[\frac{d}{dx}\bigl(1-x\bigr)^{-\beta_n}
\bigl(1+(1-x)\mathcal O_1\bigr)=\beta_n
\bigl(1-x\bigr)^{-\beta_n-1}\bigl(1+(1-x)\mathcal O_1\bigr), \] we have \[ \frac{c(0\!:\!0\rightsquigarrow 1\!:\!-\beta_n)}
{c(0\!:\!0\rightsquigarrow 1\!:\!-\beta_n)|
_{\alpha_j\mapsto \alpha_j+1,\ \beta_j\mapsto \beta_j+1}}
=\frac{\alpha_1\dots\alpha_n}{\beta_1\dots\beta_n}, \] which proves \eqref{eq:CGHG} because of \eqref{eq:CGHGL}. \end{exmp}
A further study of generalized connection coefficients will be developed in another paper. In this paper we will only give some examples in \S\ref{sec:EOEx} and \S\ref{sec:eq211}.
\section{Examples}\label{sec:ex} When we classify tuples of partitions in this section, we identify the tuples which are isomorphic to each other. For example, $21,111,111$ is isomorphic to any one of $12,111,111$ and $111,21,111$ and $21,3,111,111$.
Most of our results in this note are constructible and can be implemented in computer programs. Several reductions and constructions and decompositions of tuples of partitions and connections coefficients associated with Riemann schemes etc.\ can be computed by a program \texttt{okubo} written by the author (cf.~\S\ref{sec:okubo}).
In \S\ref{sec:basicEx} and \S\ref{sec:rigidEx} we list fundamental and rigid tuples respectively, most of which are obtained by the program \texttt{okubo}.
In \S\ref{sec:PoEx} and \S\ref{sec:GHG} we apply our fractional calculus to Jordan-Pochhammer equations and a hypergeometric family (generalized hypergeometric equations), respectively. Most of the results in these sections are known but it will be useful to understand our unifying interpretation and apply it to general Fuchsian equations.
In \S\ref{sec:EOEx} we study an even family and an odd family corresponding to Simpson's list \cite{Si}. The differential equations of an even family appear in suitable restrictions of Heckman-Opdam hypergeometric systems and in particular the explicit calculation of a connection coefficient for an even family was the original motivation for the study of Fuchsian differential equations developed in this note (cf.~\cite{OS}). We also calculate a generalized connection coefficient for an even family of order 4.
In \S\ref{sec:4Ex}, \S\ref{sec:ord6Ex} and \S\ref{sec:Rob} we study the rigid Fuchsian differential equations of order not larger than 4 and those of order 5 or 6 and the equations belonging to 12 maximal series and some minimal series classified by \cite{Ro} which include the equations in Yokoyama's list \cite{Yo}. We list sufficient data from which we get some connection coefficients and the necessary and sufficient conditions for the irreducibility of the equations as is explained in \S\ref{sec:RobEx}.
In \S\ref{sec:TriEx} we give some interesting identities of trigonometric functions as a consequence of the explicit value of connection coefficients.
We examine Appell hypergeometric equations in \S\ref{sec:ApEx}, which will be further discussed in another paper.
In \S\ref{sec:okubo} we explain computer programs \texttt{okubo} and a library of \texttt{risa/asir} which calculate the results described in this paper.
\subsection{Basic tuples}\label{sec:basicEx}\label{sec:Exbasic} \index{tuple of partitions!basic} The number of basic tuples and fundamental tuples (cf.~Definition~\ref{def:fund}) with a given $\Pidx$ are as follows.
\noindent {
\begin{tabular}{|c|r|r|r|r|r|r|r|r|r|r|r|r|}\hline $\Pidx$ &\ \,0\ & \ \,1\ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ \\ \hline\hline $\#$ fund.\ tuples & \ \,1\ & \ \,4\ & 13 & 36 & 67 & 103 & 162 & 243 & 305 & 456 & 578 & 720 \\ \hline $\#$ basic tuples & \ \,0\ & \ \,4\ & 13 & 36 & 67 & 90 & 162 & 243 & 305 & 420 & 565 & 720 \\ \hline $\#$ basic triplets & \ \,0\ & \ \,3\ & \ \,9&24& 44 & 56 & \ \,97& 144 & 163 & 223 & 291 & 342 \\ \hline $\#$ basic 4-tuples& \ \,0\ & \ \,1\ & \ \,3& \ \,9& 17 & \,24 & \ \,45& \ \,68 & \ \,95 & 128 & 169 & 239 \\ \hline maximal order & \ \,6\ & 12 & 18 & 24 & 30 & 36 & 42 & 48 & 54 & 60 & 66 & 72 \\ \hline \end{tabular}}
Note that if $\mathbf m$ is a basic tuple with $\idx\mathbf m<0$, then \begin{equation}
\Pidx k\mathbf m= 1+k^2(\Pidx\mathbf m -1)\qquad(k=1,2,\ldots). \end{equation} Hence the non-trivial fundamental tuple $\mathbf m$ with $\Pidx\mathbf m\le 4$ or equivalently $\idx\mathbf m\ge-6$ is always basic.
The tuple $2\mathbf m$ with a basic tuple $\mathbf m$ satisfying $\Pidx\mathbf m=2$ is a fundamental tuple and $\Pidx2\mathbf m=5$. The tuple $422,44,44,44$ is this example.
\subsubsection{$\Pidx\mathbf m=1$} \index{tuple of partitions!index of rigidity!$=0$} There exist 4 basic tuples: (cf.~\cite{Ko3}, Corollary~\ref{cor:idx0}) \index{00D4@$\tilde D_4,\ \tilde E_6,\ \tilde E_7,\ \tilde E_8$}
$\tilde D_4$:\ 11,11,11,11\ \ \ \ $\tilde E_6$:\ 111,111,111\ \ \ \ $\tilde E_7$:\ 22,1111,1111\ \ \ \ $\tilde E_8$:\ 33,222,111111
They are not of Okubo type. The tuples of partitions of Okubo type with minimal order which are reduced to the above basic tuples are as follows.
$\tilde D_4$:\ 21,21,21,111\ \ \ \ $\tilde E_6$:\ 211,211,1111\ \ \ \ $\tilde E_7$:\ 32,2111,11111\ \ \ \ $\tilde E_8$:\ 43,322,1111111
The list of simply reducible tuples of partitions whose indices of rigidity equal $0$ is given in Example~\ref{ex:SR0}.
We list the number of realizable tuples of partitions whose indices of rigidity equal 0 according to their orders and the corresponding fundamental tuple.
\begin{tabular}{|r|r|r|r|r|r|}\hline ord & {\small 11,11,11,11} & {\small 111,111,111} & {\small 22,1111,1111} & {\small 33,222,111111} & total\\ \hline\hline 2 & 1 & & & & 1 \\ \hline 3 & 1 & 1 & & & 2 \\ \hline 4 & 4 & 1 & 1 & & 6 \\ \hline 5 & 6 & 3 & 1 & & 10 \\ \hline 6 & 21 & 8 & 5 & 1 & 35 \\ \hline 7 & 28 & 15 & 6 & 1 & 50 \\ \hline 8 & 74 & 31 & 21 & 4 & 130 \\ \hline 9 & 107 & 65 & 26 & 5 & 203 \\ \hline 10& 223 & 113 & 69 & 12 & 417 \\ \hline 11& 315 & 204 & 90 & 14 & 623 \\ \hline 12& 616 & 361 & 205 & 37 & 1219 \\ \hline 13& 808 & 588 & 256 & 36 & 1688 \\ \hline 14& 1432 & 948 & 517 & 80 & 2977 \\ \hline 15& 1951 & 1508 & 659 & 100 & 4218 \\ \hline 16& 3148 & 2324 & 1214 & 179 & 6865 \\ \hline 17& 4064 & 3482 & 1531 & 194 & 9271 \\ \hline 18& 6425 & 5205 & 2641 & 389 &14660 \\ \hline 19& 8067 & 7503 & 3246 & 395 &19211 \\ \hline 20& 12233 & 10794 & 5400 & 715 &29142 \\ \hline \end{tabular}{}
\subsubsection{$\Pidx\mathbf m=2$}\label{sec:idx-2} \index{tuple of partitions!basic!index of rigidity$\ \ge-6$} There are 13 basic tuples (cf.~Proposition~\ref{prop:bas2}, \cite[Proposition~8.4]{O3}):\\[-10pt] \begin{verbatim}
+2:11,11,11,11,11 3:111,111,21,21 *4:211,22,22,22
4:1111,22,22,31 4:1111,1111,211 5:11111,11111,32
5:11111,221,221 6:111111,2211,33 *6:2211,222,222
*8:22211,2222,44 8:11111111,332,44 10:22222,3331,55 *12:2222211,444,66 \end{verbatim} Here the number preceding to a tuple is the order of the tuple and the sign \texttt{*} means that the tuple is the one given in Example~\ref{eq:Qsp} ($D_4^{(m)}$, $E_6^{(m)}$, $E_7^{(m)}$ and $E_8^{(m)}$) and the sign $+$ means $d(\mathbf m)<0$.
\subsubsection{$\Pidx\mathbf m=3$} There are 36 basic tuples\\[-10pt] { \begin{verbatim}
+2:11,11,11,11,11,11 3:111,21,21,21,21 4:22,22,22,31,31
+3:111,111,111,21 +4:1111,22,22,22 4:1111,1111,31,31
4:211,211,22,22 4:1111,211,22,31 *6:321,33,33,33
6:222,222,33,51 +4:1111,1111,1111 5:11111,11111,311
5:11111,2111,221 6:111111,222,321 6:111111,21111,33
6:21111,222,222 6:111111,111111,42 6:222,33,33,42
6:111111,33,33,51 6:2211,2211,222 7:1111111,2221,43
7:1111111,331,331 7:2221,2221,331 8:11111111,3311,44
8:221111,2222,44 8:22211,22211,44 *9:3321,333,333
9:111111111,333,54 9:22221,333,441 10:1111111111,442,55
10:22222,3322,55 10:222211,3331,55 12:22221111,444,66 *12:33321,3333,66 14:2222222,554,77 *18:3333321,666,99 \end{verbatim}}
\subsubsection{$\Pidx\mathbf m=4$} There are 67 basic tuples\\[-10pt] {\small \begin{verbatim}
+2:11,11,11,11,11,11,11 3:21,21,21,21,21,21 +3:111,111,21,21,21
+4:22,22,22,22,31 4:211,22,22,31,31 4:1111,22,31,31,31
+3:111,111,111,111 +4:1111,1111,22,31 4:1111,211,22,22
4:211,211,211,22 4:1111,211,211,31 5:11111,11111,41,41
5:11111,221,32,41 5:221,221,221,41 5:11111,32,32,32
5:221,221,32,32 6:3111,33,33,33 6:2211,2211,2211
+6:222,33,33,33 6:222,33,33,411 6:2211,222,33,51
*8:431,44,44,44 8:11111111,44,44,71 5:11111,11111,221
5:11111,2111,2111 +6:111111,111111,33 +6:111111,222,222
6:111111,111111,411 6:111111,222,3111 6:21111,2211,222
6:111111,2211,321 6:2211,33,33,42 7:1111111,1111111,52
7:1111111,322,331 7:2221,2221,322 7:1111111,22111,43
7:22111,2221,331 8:11111111,3221,44 8:11111111,2222,53
8:2222,2222,431 8:2111111,2222,44 8:221111,22211,44
9:33111,333,333 9:3222,333,333 9:22221,22221,54
9:222111,333,441 9:111111111,441,441 10:22222,33211,55
10:1111111111,433,55 10:1111111111,4411,55 10:2221111,3331,55
10:222211,3322,55 12:222111111,444,66 12:333111,3333,66
12:33222,3333,66 12:222222,4431,66 *12:4431,444,444
12:111111111111,552,66 12:3333,444,552 14:33332,4442,77
14:22222211,554,77 15:33333,555,771 *16:44431,4444,88
16:333331,5551,88 18:33333111,666,99 18:3333222,666,99 *24:4444431,888,cc \end{verbatim}}
Here $\verb|a|,\verb|b|,\verb|c|,\ldots$ represent 10,11,12,\ldots, respectively.
\subsubsection{Dynkin diagrams of basic tuples whose indices of rigidity equals $-2$} We express the basic root $\alpha_{\mathbf m}$ for $\Pidx\mathbf m=2$ using the Dynkin diagram (See \eqref{eq:Dynkinidx0} for $\Pidx\mathbf m=1$). The circles in the diagram represent the simple roots in $\supp\alpha_{\mathbf m}$ and two circles are connected by a line if the inner product of the corresponding simple roots is not zero. The number attached to a circle is the corresponding coefficient $n$ or $n_{j,\nu}$ in the expression \eqref{eq:Kazpart}.
For example, if $\mathbf m=22,22,22,211$, then $\alpha_{\mathbf m}=4\alpha_0+2\alpha_{0,1}+2\alpha_{1,1} +2\alpha_{2,1}+2\alpha_{3,1}+\alpha_{3,2}$, which corresponds to the second diagram in the following.
The circle with a dot at the center means a simple root whose inner product with $\alpha_{\mathbf m}$ does not vanish. Moreover the type of the root system $\Pi(\mathbf m)$ (cf.~\eqref{eq:Pim}) corresponding to the simple roots without a dot is given.
\index{tuple of partitions!basic!Dynkin diagram}\index{Dynkin diagram} {\small\begin{gather*} \begin{xy}
\ar@{-} (10,0) *++!D{1} *\cir<4pt>{};
(0,0) *+!D+!L{2}*{\cdot}*\cir<4pt>{}="A", \ar@{-} "A"; (3.09,9.51) *++!L{1} *\cir<4pt>{}; \ar@{-} "A"; (-8.09,5.88) *++!D{1} *\cir<4pt>{}; \ar@{-} "A"; (3.09,-9.51) *++!L{1} *\cir<4pt>{}; \ar@{-} "A"; (-8.09,-5.88)*++!D{1} *\cir<4pt>{}; \ar@{} (0,-14) *{11,11,11,11,11\quad 5A_1} \end{xy}\qquad \begin{xy}
\ar@{-} *++!D{2} *\cir<4pt>{};
(10,0) *+!L+!D{4}*\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *++!D{2} *{\cdot} *\cir<4pt>{}="B", \ar@{-} "B"; (30,0) *++!D{1} *\cir<4pt>{}, \ar@{-} "A"; (10,-10) *++!L{2} *\cir<4pt>{}, \ar@{-} "A"; (10,10) *++!L{2} *\cir<4pt>{}; \ar@{} (15,-14) *{22,22,22,211\quad D_4+A_1} \end{xy} \\[.1cm] \begin{xy}
\ar@{-} *++!D{1} *\cir<4pt>{};
(10,0) *++!D{2} *\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *+!L+!D{3}*\cir<4pt>{}="B", \ar@{-} "B"; (30,0) *++!D{2} *\cir<4pt>{}="C", \ar@{-} "C"; (40,0) *++!D{1} *\cir<4pt>{}, \ar@{-} "B"; (20,-10) *++!L{1} *{\cdot}*\cir<4pt>{}="C", \ar@{-} "B"; (20,10) *++!L{1} *{\cdot}*\cir<4pt>{}; \ar@{} (22,-14) *{21,21,111,111\quad A_5} \end{xy} \qquad \begin{xy}
\ar@{-} *++!D{1} *{\cdot}*\cir<4pt>{};
(10,0) *+!L+!D{4}*\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *++!D{3} *\cir<4pt>{}="B", \ar@{-} "B"; (30,0) *++!D{2} *\cir<4pt>{}="C", \ar@{-} "C"; (40,0) *++!D{1} *\cir<4pt>{}, \ar@{-} "A"; (10,-10) *++!L{2} *\cir<4pt>{}, \ar@{-} "A"; (10,10) *++!L{2} *\cir<4pt>{}, \ar@{} (15,-14) *{31,22,22,1111\quad D_6} \end{xy} \\%[.1cm] \begin{xy}
\ar@{-} *++!D{2} *\cir<4pt>{};
(10,0) *++!D{4} *\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *+!L+!D{6}*\cir<4pt>{}="B", \ar@{-} "B"; (30,0) *++!D{4} *\cir<4pt>{}="C", \ar@{-} "C"; (40,0) *++!D{2} *{\cdot}*\cir<4pt>{}="D", \ar@{-} "D"; (50,0) *++!D{1} *\cir<4pt>{}, \ar@{-} "B"; (20,10) *++!L{4} *\cir<4pt>{}="F", \ar@{-} "F"; (20,20) *++!L{2} *\cir<4pt>{}, \ar@{} (25,-5) *{222,222,2211\quad E_6+A_1} \end{xy}\quad \begin{xy}
\ar@{-} *++!D{1} *\cir<4pt>{};
(10,0) *++!D{2} *\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *++!D{3} *\cir<4pt>{}="B", \ar@{-} "B"; (30,0) *+!L+!D{4}*\cir<4pt>{}="C", \ar@{-} "C"; (40,0) *++!D{3} *\cir<4pt>{}="D", \ar@{-} "D"; (50,0) *++!D{2} *\cir<4pt>{}="E", \ar@{-} "E"; (60,0) *++!D{1} *\cir<4pt>{}, \ar@{-} "C"; (30,10) *++!L{2} *{\cdot}*\cir<4pt>{}="F", \ar@{-} "F"; (30,20) *++!L{1} *\cir<4pt>{}, \ar@{} (35,-4) *{211,1111,1111\quad A_7+A_1} \end{xy} \allowdisplaybreaks\\[-.1cm] \begin{xy}
\ar@{-} *++!D{1} *{\cdot}*\cir<4pt>{};
(10,0) *++!D{3} *\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *+!L+!D{5}*\cir<4pt>{}="B", \ar@{-} "B"; (30,0) *++!D{4} *\cir<4pt>{}="C", \ar@{-} "C"; (40,0) *++!D{3} *\cir<4pt>{}="D", \ar@{-} "D"; (50,0) *++!D{2} *\cir<4pt>{}="E", \ar@{-} "E"; (60,0) *++!D{1} *\cir<4pt>{}, \ar@{-} "B"; (20,10) *++!L{3} *\cir<4pt>{}="F", \ar@{-} "F"; (20,20) *++!L{1} *{\cdot}*\cir<4pt>{}, \ar@{} (25,-5) *{221,221,11111\quad D_7} \end{xy}\allowdisplaybreaks\\ \begin{xy}
\ar@{-} *++!D{2} *\cir<4pt>{};
(10,0) *++!D{4} *\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *++!D{6} *\cir<4pt>{}="B", \ar@{-} "B"; (30,0) *+!L+!D{8}*\cir<4pt>{}="C", \ar@{-} "C"; (40,0) *++!D{6} *\cir<4pt>{}="D", \ar@{-} "D"; (50,0) *++!D{4} *\cir<4pt>{}="E", \ar@{-} "E"; (60,0) *++!D{2} *{\cdot}*\cir<4pt>{}="F", \ar@{-} "F"; (70,0) *++!D{1} *\cir<4pt>{}. \ar@{-} "C"; (30,10) *++!L{4} *\cir<4pt>{}, \ar@{} (35,-4) *{44,2222,22211\quad E_7+A_1} \end{xy}\allowdisplaybreaks\\ \begin{xy}
\ar@{-} *++!D{1} *{\cdot}*\cir<4pt>{};
(10,0) *++!D{4} *\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *++!D{7} *\cir<4pt>{}="B", \ar@{-} "B"; (30,0) *+!L+!D{10}*\cir<4pt>{}="C", \ar@{-} "C"; (40,0) *++!D{8} *\cir<4pt>{}="D", \ar@{-} "D"; (50,0) *++!D{6} *\cir<4pt>{}="E", \ar@{-} "E"; (60,0) *++!D{4} *\cir<4pt>{}="F", \ar@{-} "F"; (70,0) *++!D{2} *\cir<4pt>{}. \ar@{-} "C"; (30,10) *++!L{5} *\cir<4pt>{}, \ar@{} (35,-4) *{55,3331,22222\quad E_8} \end{xy}\allowdisplaybreaks\\ \begin{xy}
\ar@{-} *++!D{1} *\cir<4pt>{};
(10,0) *++!D{2} *\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *++!D{3} *\cir<4pt>{}="B", \ar@{-} "B"; (30,0) *++!D{4} *\cir<4pt>{}="C", \ar@{-} "C"; (40,0) *+!L+!D{5}*\cir<4pt>{}="D", \ar@{-} "D"; (50,0) *++!D{4} *\cir<4pt>{}="E", \ar@{-} "E"; (60,0) *++!D{3} *\cir<4pt>{}="F", \ar@{-} "F"; (70,0) *++!D{2} *\cir<4pt>{}="G". \ar@{-} "G"; (80,0) *++!D{1} *\cir<4pt>{}. \ar@{-} "D"; (40,10) *++!L{2} *{\cdot}*\cir<4pt>{}, \ar@{} (45,-4) *{32,11111,111111\quad A_9} \end{xy}\allowdisplaybreaks\\ \begin{xy}
\ar@{-} *++!D{1} *\cir<4pt>{};
(10,0) *++!D{2} *{\cdot}*\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *++!D{4} *\cir<4pt>{}="B", \ar@{-} "B"; (30,0) *+!L+!D{6}*\cir<4pt>{}="C", \ar@{-} "C"; (40,0) *++!D{5} *\cir<4pt>{}="D", \ar@{-} "D"; (50,0) *++!D{4} *\cir<4pt>{}="E", \ar@{-} "E"; (60,0) *++!D{3} *\cir<4pt>{}="F", \ar@{-} "F"; (70,0) *++!D{2} *\cir<4pt>{}="G". \ar@{-} "G"; (80,0) *++!D{1} *\cir<4pt>{}. \ar@{-} "C"; (30,10) *++!L{3} *\cir<4pt>{}, \ar@{} (35,-4) *{33,2211,111111\quad D_8+A_1} \end{xy}\allowdisplaybreaks\\ \begin{xy}
\ar@{-} *++!D{4} *\cir<4pt>{};
(10,0) *++!D{8} *\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *+!L+!D{12}*\cir<4pt>{}="B", \ar@{-} "B"; (30,0) *++!D{10} *\cir<4pt>{}="C", \ar@{-} "C"; (40,0) *++!D{8} *\cir<4pt>{}="D", \ar@{-} "D"; (50,0) *++!D{6} *\cir<4pt>{}="E", \ar@{-} "E"; (60,0) *++!D{4} *\cir<4pt>{}="F", \ar@{-} "F"; (70,0) *++!D{2} *{\cdot}*\cir<4pt>{}="G". \ar@{-} "G"; (80,0) *++!D{1} *\cir<4pt>{}. \ar@{-} "B"; (20,10) *++!L{6} *\cir<4pt>{}, \ar@{} (25,-4) *{66,444,2222211\quad E_8+A_1} \end{xy}\allowdisplaybreaks\\ \begin{xy}
\ar@{-} *++!D{2} *{\cdot}*\cir<4pt>{};
(10,0) *++!D{5} *\cir<4pt>{}="A", \ar@{-} "A"; (20,0) *+!L+!D{8}*\cir<4pt>{}="B", \ar@{-} "B"; (30,0) *++!D{7} *\cir<4pt>{}="C", \ar@{-} "C"; (40,0) *++!D{6} *\cir<4pt>{}="D", \ar@{-} "D"; (50,0) *++!D{5} *\cir<4pt>{}="E", \ar@{-} "E"; (60,0) *++!D{4} *\cir<4pt>{}="F", \ar@{-} "F"; (70,0) *++!D{3} *\cir<4pt>{}="G". \ar@{-} "G"; (80,0) *++!D{2} *\cir<4pt>{}="H". \ar@{-} "H"; (90,0) *++!D{1} *\cir<4pt>{}. \ar@{-} "B"; (20,10) *++!L{4} *\cir<4pt>{}, \ar@{} (25,-4) *{44,332,11111111\quad D_{10}} \end{xy} \end{gather*}}
\subsection{Rigid tuples}\label{sec:rigidEx} \index{tuple of partitions!rigid} \subsubsection{Simpson's list} Simpson \cite{Si} classified the rigid tuples containing the partition $11\cdots1$ into 4 types (Simpson's list), which follows from Proposition~\ref{prop:sred}. They are $H_n$, $EO_{2m}$, $EO_{2m+1}$ and $X_6$ in the following table. \index{Simpson's list} \index{tuple of partitions!rigid!Simpson's list}
See Remark~\ref{rem:length} ii) for $[\Delta(\mathbf m)]$ with these rigid tuples $\mathbf m$.
\index{tuple of partitions!rigid!21111,222,33} The simply reducible rigid tuple (cf.~\S\ref{sec:simpred}) which is not in Simpson's list is isomorphic to $21111,222,33$.
\begin{tabular}{|c|c|c|c|}\hline order&type &name &partitions\\ \hline\hline $n$&$H_n$&hypergeometric family&$1^n,1^n,n-11$\\ \hline $2m$&$EO_{2m}$&even family& $1^{2m},mm-11,mm$\\ \hline $2m+1$&$EO_{2m+1}$&odd family& $1^{2m+1},mm1,m+1m$\\ \hline $6$&$X_6=\gamma_{6,2}$&extra case&$111111,222,42$\\ \hline $6$&$\gamma_{6,6}$&&$21111,222,33$\\ \hline $n$&$P_n$&Jordan Pochhammer&$n-11,n-11,\ldots\in\mathcal P_{n+1}^{(n)}$\\ \hline \end{tabular}
$H_1=EO_1$, $H_2=EO_2=P_2$, $H_3=EO_3$.
\subsubsection{Isomorphic classes of rigid tuples} Let $\mathcal R_{p+1}^{(n)}$ be the set of rigid tuples in $\mathcal P^{(n)}_{p+1}$. Put $\mathcal R_{p+1}=\bigcup_{n=1}^\infty \mathcal R_{p+1}^{(n)}$, $\mathcal R^{(n)}=\bigcup_{p=2}^\infty \mathcal R_{p+1}^{(n)}$ and $\mathcal R=\bigcup_{n=1}^\infty\mathcal R^{(n)}$. The sets of isomorphic classes of the elements of $\mathcal R_{p+1}^{(n)}$ (resp.~$\mathcal R_{p+1}$, $\mathcal R^{(n)}$ and $\mathcal R$) are denoted $\bar{\mathcal R}_{p+1}^{(n)}$ (resp.~$\bar{\mathcal R}_{p+1}$, $\bar{\mathcal R}^{(n)}$ and $\bar{\mathcal R}$). Then the number of the elements of $\bar{\mathcal R}^{(n)}$ are as follows. \nopagebreak
{
\begin{tabular}{|r|r|r||r|r|r||r|r|r|}\hline {$n$} & ${\#\bar{\mathcal R}}_3^{(n)}$
& ${\#\bar{\mathcal R}^{(n)}}$ & {$n$} & ${\#\bar{\mathcal R}^{(n)}_3}$ & $\#\bar{\mathcal R}^{(n)}$ & {$n$} & ${\#\bar{\mathcal R}^{(n)}_3}$ & $\#\bar{\mathcal R}^{(n)}$
\\ \hline 2 & 1 & 1 & 15&1481&2841 &28&114600&190465\\ \hline 3 & 1 & 2 & 16&2388&4644 &29&143075&230110\\ \hline 4 & 3 & 6 & 17&3276&6128 &30&190766&310804\\ \hline 5 & 5 & 11& 18&5186&9790 &31&235543&371773\\ \hline 6 & 13& 28& 19&6954&12595&32&309156&493620\\ \hline 7 & 20& 44& 20&10517&19269&33&378063&588359\\ \hline 8 & 45& 96& 21&14040&24748&34&487081&763126\\ \hline 9 & 74&157& 22&20210&36078&35&591733&903597\\ \hline 10&142&306& 23&26432&45391&36&756752&1170966\\ \hline 11&212&441& 24&37815&65814&37&907150&1365027\\ \hline 12&421&857& 25&48103&80690&38&1143180&1734857\\ \hline 13&588&1177& 26&66409&112636&39&1365511&2031018\\ \hline 14&1004&2032& 27&84644&139350&40&1704287&2554015\\ \hline \end{tabular}}
\subsubsection{Rigid tuples of order at most 8} \index{tuple of partitions!rigid!order$\ \le8$} We show all the rigid tuples whose orders are not larger than 8. \noindent{\begin{longtable}{ll} \texttt{2:11,11,11} \ ($H_2$: Gauss) \\ \\ \texttt{3:111,111,21} \ ($H_3: {}_3F_2$) & \texttt{3:21,21,21,21} \ ($P_3$) \\ \\ \texttt{4:1111,1111,31} \ ($H_4: {}_4F_3)$ & \texttt{4:1111,211,22} \ ($EO_4$: even)\\ \texttt{4:211,211,211} \ ($B_4$, $\text{II}_2$, $\alpha_4$)& \texttt{4:211,22,31,31} \ ($I_4,\, \text{II}^*_2$)\\ \texttt{4:\underline{22,22,22,31}} \ ($P_{4,4}$) & \texttt{4:31,31,31,31,31} \ ($P_4$) \\ \\ \texttt{5:11111,11111,41} \ ($H_5: {}_5F_4$) & \texttt{5:11111,221,32} \ ($EO_5$: odd) \\ \texttt{5:2111,2111,32} \ ($C_5$) & \texttt{5:2111,221,311} \ ($B_5$, $\text{III}_2$) \\ \texttt{5:\underline{221,221,221}} \ ($\alpha_5$) & \texttt{5:221,221,41,41} \ ($J_5$) \\ \texttt{5:221,32,32,41} & \texttt{5:311,311,32,41} \ ($I_5,\, \text{III}^*_2$) \\ \texttt{5:\underline{32,32,32,32}} \ ($P_{4,5}$) & \texttt{5:\underline{32,32,41,41,41}} \ ($M_5$) \\ \texttt{5:41,41,41,41,41,41} \ ($P_5$)\\ \\ \texttt{6:111111,111111,51} \ ($H_6: {}_6F_5$)\phantom{A} &\texttt{6:111111,222,42} ($D_6=X_6$: extra)\\ \texttt{6:111111,321,33} \ ($EO_6$: even)&\texttt{6:21111,2211,42} \ ($E_6$)\\ \texttt{6:\underline{21111,222,33}} \ ($\gamma_{6,6}$)&\texttt{6:21111,222,411} \ ($F_6$, $\text{IV}$)\\ \texttt{6:21111,3111,33} \ ($C_6$) &\texttt{6:\underline{2211,2211,33}} \ ($\beta_6$)\\ \texttt{6:2211,2211,411} \ ($G_6$) &\texttt{6:2211,321,321}\\ \texttt{6:\underline{222,222,321}} \ ($\alpha_6$)&\texttt{6:222,3111,321}\\ \texttt{6:3111,3111,321} \ ($B_6$, $\text{II}_3$) &\texttt{6:2211,222,51,51} \ ($J_6$)\\ \texttt{6:2211,33,42,51} &\texttt{6:\underline{222,33,33,51}}\\ \texttt{6:222,33,411,51} &\texttt{6:3111,33,411,51}\ ($I_6,\,\text{II}^*_3$)\\ \texttt{6:321,321,42,51} &\texttt{6:321,42,42,42}\\ \texttt{6:\underline{33,33,33,42}} \ ($P_{4,6}$)
&\texttt{6:\underline{33,33,411,42}}\\ \texttt{6:33,411,411,42} &\texttt{6:411,411,411,42} \ ($N_6,\,\text{IV}^*$)\\ \texttt{6:33,42,42,51,51} \ ($M_6$)&\texttt{6:321,33,51,51,51} \ ($K_6$)\\ \texttt{6:411,42,42,51,51} &\texttt{6:51,51,51,51,51,51,51} \ ($P_6$)\\ \\
\texttt{7:1111111,1111111,61} \ ($H_7$)& \texttt{7:1111111,331,43} \ ($EO_7$)\\ \texttt{7:211111,2221,52} \ ($D_7$) & \texttt{7:211111,322,43} \ ($\gamma_7$)\\ \texttt{7:22111,22111,52} \ ($E_7$) & \texttt{7:22111,2221,511} \ ($F_7$)\\ \texttt{7:22111,3211,43} & \texttt{7:22111,331,421}\\ \texttt{7:\underline{2221,2221,43}} \ ($\beta_7$) & \texttt{7:2221,31111,43} \\ \texttt{7:2221,322,421} & \texttt{7:\underline{2221,331,331}}\\ \texttt{7:2221,331,4111} & \texttt{7:31111,31111,43} \ ($C_7$)\\ \texttt{7:31111,322,421} & \texttt{7:31111,331,4111} \ ($B_7,\ \text{III}_3$)\\ \texttt{7:3211,3211,421} & \texttt{7:\underline{3211,322,331}}\\ \texttt{7:3211,322,4111} & \texttt{7:\underline{322,322,322}} \ ($\alpha_7$)\\ \texttt{7:2221,2221,61,61} \ ($J_7$) & \texttt{7:2221,43,43,61} \\ \texttt{7:3211,331,52,61} & \texttt{7:322,322,52,61}\\ \texttt{7:322,331,511,61} & \texttt{7:322,421,43,61}\\ \texttt{7:322,43,52,52} & \texttt{7:\underline{331,331,43,61}}\\ \texttt{7:331,43,511,52} & \texttt{7:4111,4111,43,61} \ ($I_7,\,\text{III}_3^*$) \\ \texttt{7:4111,43,511,52} & \texttt{7:421,421,421,61} \\ \texttt{7:421,421,52,52} & \texttt{7:\underline{421,43,43,52}} \\ \texttt{7:\underline{43,43,43,43}} \ ($P_{4,7}$) & \texttt{7:421,43,511,511} \\ \texttt{7:331,331,61,61,61} \ ($L_7$) & \texttt{7:421,43,52,61,61} \\ \texttt{7:\underline{43,43,43,61,61}} & \texttt{7:43,52,52,52,61} \\
\texttt{7:511,511,52,52,61} \ ($N_7$) & \texttt{7:43,43,61,61,61,61} \ ($K_7$)\\ \texttt{7:52,52,52,61,61,61} \ ($M_7$) & \texttt{7:61,61,61,61,61,61,61,61} \ ($P_7$)\\ \\
\texttt{8:11111111,11111111,71} \ ($H_8$) & \texttt{8:11111111,431,44} \ ($EO_8$)\\ \texttt{8:2111111,2222,62} \ ($D_8$) & \texttt{8:2111111,332,53} \\ \texttt{8:2111111,422,44}& \texttt{8:221111,22211,62} \ ($E_8$) \\ \texttt{8:221111,2222,611} \ ($F_8$)& \texttt{8:221111,3311,53} \\ \texttt{8:\underline{221111,332,44}} \ ($\gamma_8$)& \texttt{8:221111,4211,44} \\ \texttt{8:22211,22211,611} \ ($G_8$)& \texttt{8:22211,3221,53}\\ \texttt{8:\underline{22211,3311,44}} & \texttt{8:22211,332,521}\\ \texttt{8:22211,41111,44}& \texttt{8:22211,431,431} \\ \texttt{8:22211,44,53,71}& \texttt{8:\underline{2222,2222,53}} \ ($\beta_{8,2}$) \\ \texttt{8:2222,32111,53}& \texttt{8:\underline{2222,3221,44}} \ ($\beta_{8,4}$) \\ \texttt{8:2222,3311,521}& \texttt{8:2222,332,5111}\\ \texttt{8:2222,422,431}& \texttt{8:311111,3221,53} \\ \texttt{8:311111,332,521} & \texttt{8:311111,41111,44} \ ($C_8$)\\ \texttt{8:32111,32111,53} & \texttt{8:\underline{32111,3221,44}}\\ \texttt{8:32111,3311,521} & \texttt{8:32111,332,5111}\\ \texttt{8:32111,422,431} & \texttt{8:3221,3221,521}\\ \texttt{8:3221,3311,5111} & \texttt{8:\underline{3221,332,431}}\\ \texttt{8:\underline{332,332,332}} \ ($\alpha_8$)& \texttt{8:\underline{332,332,4211}}\\ \texttt{8:332,41111,422} & \texttt{8:332,4211,4211} \\ \texttt{8:3221,4211,431} & \texttt{8:\underline{3311,3311,431}} \\ \texttt{8:\underline{3311,332,422}} & \texttt{8:3221,422,422} \\ \texttt{8:3311,4211,422} & \texttt{8:41111,41111,431} \ ($B_8,\,II_4$) \\ \texttt{8:41111,4211,422} & \texttt{8:4211,4211,4211} \\ \texttt{8:22211,2222,71,71} \ ($J_8$) & \texttt{8:\underline{2222,44,44,71}} \\ \texttt{8:3221,332,62,71} & \texttt{8:3221,44,521,71} \\ \texttt{8:3221,44,62,62} & \texttt{8:3311,3311,62,71} \\ \texttt{8:3311,332,611,71} & \texttt{8:3311,431,53,71} \\ \texttt{8:3311,44,611,62} & \texttt{8:332,422,53,71} \\ \texttt{8:\underline{332,431,44,71}} & \texttt{8:332,44,611,611}\\ \texttt{8:332,53,53,62} & \texttt{8:41111,44,5111,71} \ ($I_8,\,II_4^*$) \\ \texttt{8:41111,44,611,62} & \texttt{8:4211,422,53,71} \\ \texttt{8:4211,44,611,611} & \texttt{8:4211,53,53,62} \\ \texttt{8:\underline{422,422,44,71}} & \texttt{8:422,431,521,71} \\ \texttt{8:422,431,62,62} & \texttt{8:\underline{422,44,53,62}} \\ \texttt{8:\underline{431,44,44,62}} & \texttt{8:\underline{431,44,53,611}}\\ \texttt{8:422,53,53,611} & \texttt{8:431,431,611,62}\\ \texttt{8:431,521,53,62} & \texttt{8:\underline{44,44,44,53}} \ ($P_{4,8}$) \\ \texttt{8:44,5111,521,62} & \texttt{8:44,521,521,611} \\ \texttt{8:\underline{44,521,53,53}} & \texttt{8:5111,5111,53,62} \\ \texttt{8:5111,521,53,611} & \texttt{8:521,521,521,62} \\ \texttt{8:332,332,71,71,71} & \texttt{8:332,44,62,71,71} \\ \texttt{8:4211,44,62,71,71} & \texttt{8:422,44,611,71,71} \\ \texttt{8:431,53,53,71,71} & \texttt{8:\underline{44,44,62,62,71}} \\ \texttt{8:44,53,611,62,71} & \texttt{8:521,521,53,71,71} \\ \texttt{8:521,53,62,62,71} & \texttt{8:53,53,611,611,71} \\ \texttt{8:53,62,62,62,62} & \texttt{8:611,611,611,62,62} \ ($N_8$) \\ \texttt{8:53,53,62,71,71,71} & \texttt{8:431,44,71,71,71,71} \ ($K_8$) \\ \texttt{8:611,62,62,62,71,71} \ ($M_8$)& \texttt{8:71,71,71,71,71,71,71,71,71} \ ($P_8$) \end{longtable}}
\vspace*{-10pt} Here the underlined tuples are not of Okubo type (cf.~\eqref{eq:OkuboT}).
The tuples $H_n$, $EO_n$ and $X_6$ are tuples in Simpson's list. The series $A_n=EO_n$, $B_n$, $C_n$, $D_n$, $E_n$, $F_n$, $G_{2m}$, $I_n$, $J_n$, $K_n$, $L_{2m+1}$, $M_n$ and $N_n$ are given in \cite{Ro} and called submaximal series. The Jordan-Pochhammer tuples are denoted by $P_n$ and the series $H_n$ and $P_n$ are called maximal series by \cite{Ro}. The series $\alpha_n,\beta_n,\gamma_n$ and $\delta_n$ are given in \cite{Ro} and called minimal series. See \S\ref{sec:Rob} for these series introduced by \cite{Ro}. Then $\delta_n=P_{4,n}$ and they are generalized Jordan-Pochhammer tuples (cf.~Example~\ref{ex:JPH} and \S\ref{sec:minseries}). Moreover $\text{II}_n$, $\text{II}^*_n$, $\text{III}_n$, $\text{III}^*_n$, IV and $\text{IV}^*$ are in Yokoyama's list in \cite{Yo} (cf.~\S\ref{sec:Bseries}).
\centerline{\bf Hierarchy of rigid triplets} \nopagebreak \noindent {\small\xymatrix{ {1^2,1^2,1^2}\ar[r]\ar[ddrr]&{21,1^3,1^3}\ar[r]\ar[dr]\ar[ddr]
&{31,1^4,1^4}\ar[r]\ar[ddr]&{41,1^5,1^5}\ar[r]
&{51,1^6,1^6}\\ {1,1,1}\ar[u]&&{2^2,21^2,1^4}\ar[r]\ar[dr]\ar[ddr]
&{32,2^21,1^5}\ar[r]\ar[dr]\ar[ddr]&{3^2,321,1^6}\\ &&21^2,21^2,21^2\ar[dr]\ar[ddr]\ar[drr]&32,21^3,21^3\ar[dr]
&{42,2^3,1^6}\\ && &31^2,2^21,21^3\ar[dr]&321,31^3,2^3\\ && &\underline{2^21,2^21,2^21}\ar[r]&321,321,2^21^2 } \\[-.2cm] \rightline{$\vdots$\phantom{ABCDEFGH}} }
Here the arrows represent certain operations $\partial_\ell$ of tuples given by Definition~\ref{def:pell}.
\subsection{Jordan-Pochhammer family} \label{sec:PoEx} $P_n$\index{Jordan-Pochhammer} \index{000Delta1@$[\Delta(\mathbf m)]$} \index{00P@$P_{p+1,n},\ P_n$}
We have studied the the Riemann scheme of this family in Example~\ref{ex:midconv} iii).
$\mathbf m=(p-11,p-11,\dots,p-11)\in\mathcal P_{p+1}^{(p)}$ \begin{align*} &\begin{Bmatrix}
x=0 &1=\frac1{c_1} & \cdots & \frac1{c_{p-1}}&\infty\\
[0]_{(p-1)} & [0]_{(p-1)} & \cdots & [0]_{(p-1)}&[1-\mu]_{(p-1)}\\
\lambda_0+\mu& \lambda_1+\mu & \cdots & \lambda_{p-1}+\mu&-\lambda_0-\dots-\lambda_{p-1}-\mu
\end{Bmatrix}\\
\Delta(\mathbf m)&=\{\alpha_0,\,\alpha_0+\alpha_{j,1}\,;\,j=0,\dots,p\}\\ [\Delta(\mathbf m)]&=1^{p+1}\cdot(p-1)\\ P_p&=H_1\oplus P_{p-1}:p+1=(p-1)H_1\oplus H_1:1 \end{align*} Here the number of the decompositions of a given type is shown after the decompositions. For example, $P_p=H_1\oplus P_{p-1}:p+1=(p-1)H_1\oplus H_1:1$ represents the decompositions \[
\begin{split}
\mathbf m&=10,\dots,\overset{\underset{\smallsmile}\nu}{01},\dots,10
\oplus p-21,\dots,\overset{\underset{\smallsmile}\nu}{p-10},\ldots, p-21
\qquad(\nu=0,\dots,p)\\
&=(p-1)(10,\dots,10)\oplus 01,\dots,01.
\end{split} \]
The differential equation $P_{P_p}(\lambda,\mu)u=0$ with this Riemann scheme is given by \[ P_{P_{p}}(\lambda,\mu):=\RAd\bigl(\partial^{-\mu}\bigr)\circ
\RAd\Bigl(x^{\lambda_0}\prod_{j=1}^{p-1}(1-c_jx)^{\lambda_j}\Bigr)\partial \] and then \begin{equation}\label{eq:Poch}
\begin{split}
P_{P_{p}}(\lambda,\mu)&=\sum_{k=0}^p p_k(x)\partial^{p-k},\\
p_k(x):\!&=\binom{-\mu+p-1}{k}
p_0^{(k)}(x) +
\binom{-\mu+p-1}{k-1}
q^{(k-1)}(x) \end{split}\end{equation} with \begin{equation} p_0(x)=x\prod_{j=1}^{p-1}(1-c_jx),\quad q(x)=p_0(x)\Bigl(-\frac{\lambda_0}{x}+\sum_{j=1}^{p-1}
\frac{c_j\lambda_j}{1-c_jx}\Bigr). \end{equation} It follows from Theorem~\ref{thm:irred} that the equation is irreducible if and only if \begin{equation}
\lambda_j\notin\mathbb Z\ \ (j=0,\dots,p-1),\
\mu\notin\mathbb Z\text{ \ and \ } \lambda_0+\cdots+\lambda_{p-1}
+\mu\notin\mathbb Z. \end{equation} It follows from Proposition~\ref{prop:shift} that the shift operator defined by the map $u\mapsto \partial u$ is bijective if and only if \begin{equation}
\mu\notin\{1,2,\dots,p-1\}\text{ \ and \ }
\lambda_0+\cdots+\lambda_{p-1}+\mu\ne0. \end{equation} The normalized solution at $0$ corresponding to the exponent $\lambda_0+\mu$ is \begin{align*} u_0^{\lambda_0+\mu}(x)&=\frac{\Gamma(\lambda_0+\mu+1)}
{\Gamma(\lambda_0+1)\Gamma(\mu)}\int_0^x\Bigl(t^{\lambda_0}
\prod_{j=1}^{p-1}(1-c_jt)^{\lambda_j}\Bigr)(x-t)^{\mu-1}dt \allowdisplaybreaks\\ &=\frac{\Gamma(\lambda_0+\mu+1)}
{\Gamma(\lambda_0+1)\Gamma(\mu)}\int_0^x\sum_{m_1=0}^\infty
\cdots\sum_{m_{p-1}=0}^\infty
\frac{(-\lambda_1)_{m_1}\cdots(-\lambda_{p-1})_{m_{p-1}}}
{m_1!\cdots m_{p-1}!}\\
&\qquad{}c_2^{m_2}\cdots c_{p-1}^{m_{p-1}}
t^{\lambda_0+m_1+\cdots+m_{p-1}}(x-t)^{\mu-1}dt \allowdisplaybreaks\\
&=\sum_{m_1=0}^\infty
\cdots\sum_{m_{p-1}=0}^\infty
\frac{(\lambda_0+1)_{m_1+\cdots+m_{p-1}}
(-\lambda_1)_{m_1}\cdots(-\lambda_{p-1})_{m_{p-1}}}
{(\lambda_0+\mu+1)_{m_1+\cdots+m_{p-1}}m_1!\cdots m_{p-1}!}\\
&\qquad
c_2^{m_2}\cdots c_{p-1}^{m_{p-1}}
x^{\lambda_0+\mu+m_1+\cdots+m_{p-1}}\\
&=x^{\lambda_0+\mu}\Bigl
(1-\frac{(\lambda_0+1)(\lambda_1c_1+\cdots+\lambda_{p-1}c_{p-1})}
{\lambda_0+\mu+1}x+\cdots\Bigr). \end{align*} This series expansion of the solution is easily obtained from the formula in \S\ref{sec:series} (cf.~Theorem~\ref{thm:expsol}) and Theorem~\ref{thm:shifm1} gives the recurrence relation \begin{equation}
u_0^{\lambda_0+\mu}(x)= u_0^{\lambda_0+\mu}(x)
\!\bigm|_{\lambda_1\mapsto\lambda_1-1}
- \Bigl(\frac{\lambda_0}{\lambda_0+\mu}u_0^{\lambda_0+\mu}(x)\Bigr)
\!\Bigm|_{\substack{\lambda_0\mapsto\lambda_0+1\\\lambda_1\mapsto\lambda_1-1}}. \end{equation} Lemma~\ref{lem:conn} with $a=\lambda_0$, $b=\lambda_1$ and $u(x)=\prod_{j=2}^{p-1}(1-c_jx)^{\lambda_j}$ gives the following connection coefficients \begin{align*} c(0:\lambda_0+\mu\!\rightsquigarrow\!1:\lambda_1+\mu)&=
\frac{\Gamma(\lambda_0+\mu+1)\Gamma(-\lambda_1-\mu)}
{\Gamma(\lambda_0+1)\Gamma(-\lambda_1)}\prod_{j=2}^{p-1}(1-c_j)^{\lambda_j},
\allowdisplaybreaks\\ c(0:\lambda_0+\mu\!\rightsquigarrow\!1:0)&=
\frac{\Gamma(\lambda_0+\mu+1)}{\Gamma(\mu)\Gamma(\lambda_0+1)}\int_0^1
t^{\lambda_0}(1-t)^{\lambda_1+\mu-1}\prod_{j=2}^{p-1}(1-c_jt)^{\lambda_j}dt\\
&\hspace{-80pt}=\frac{\Gamma(\lambda_0+\mu+1)\Gamma(\lambda_1+\mu)}
{\Gamma(\mu)\Gamma(\lambda_0+\lambda_1+\mu+1)}
F(\lambda_0+1,-\lambda_2,\lambda_0+\lambda_1+\mu+1;c_2)\qquad(p=3). \end{align*} Here we have \begin{equation}
u_0^{\lambda_0+\mu}(x)=\sum_{k=0}^\infty
C_k(x-1)^k+ \sum_{k=0}^\infty C'_k(x-1)^{\lambda_1+\mu+k} \end{equation} for $0< x< 1$ with $C_0=c(0:\lambda_0+\mu\!\rightsquigarrow\!1:0)$ and $C'_0=c(0:\lambda_0+\mu\!\rightsquigarrow\!1:\lambda_1+\mu)$. Since $\frac {d^ku_0^{\lambda_0+\mu}}{dx^k}$ is a solution of the equation $P_{P_p}(\lambda,\mu-k)u=0$, we have \begin{equation}
C_k = \frac{\Gamma(\lambda_0+\mu+1)}{\Gamma(\mu-k)\Gamma(\lambda_0+1)k!}
\int_0^1
t^{\lambda_0}(1-t)^{\lambda_1+\mu-k-1}\prod_{j=2}^{p-1}(1-c_jt)^{\lambda_j}dt. \end{equation} When $p=3$, \[C_k=\frac{\Gamma(\lambda_0+\mu+1)\Gamma(\lambda_1+\mu-k)}
{\Gamma(\mu-k)\Gamma(\lambda_0+\lambda_1+\mu+1-k)k!}
F(\lambda_0+1,-\lambda_2,\lambda_0+\lambda_1+\mu+1-k;c_2). \]
Put \[ \begin{split}
u_{\lambda,\mu}(x)&=\frac1{\Gamma(\mu)}
\int_0^x\Bigl(t^{\lambda_0}
\prod_{j=1}^{p-1}(1-c_jt)^{\lambda_j}\Bigr)(x-t)^{\mu-1}dt
=\partial^{-\mu}v_{\lambda},\\
v_{\lambda}(x):\!&= x^{\lambda_0}\prod_{j=1}^{p-1}(1-c_jx)^{\lambda_j}. \end{split} \] We have \begin{equation}\label{eq:PoCon0} \begin{split}
u_{\lambda,\mu+1}&=\partial^{-\mu-1}v_{\lambda}=\partial^{-1}\partial^{-\mu}v_{\lambda}
=\partial^{-1}u_{\lambda,\mu},\\
u_{\lambda_0+1,\lambda_1,\ldots,\mu}&
=\partial^{-\mu}v_{\lambda_0+1,\lambda_1,\ldots}=\partial^{-\mu}xv_\lambda
=-\mu\partial^{-\mu-1}v_{\lambda}+x\partial^{-\mu}v_\lambda\\
&=-\mu\partial^{-1}u_{\lambda,\mu}+xu_{\lambda,\mu},\\
u_{\ldots,\lambda_j+1,\ldots}&=\partial^{-\mu}(1-c_jx)v_{\lambda}=
\partial^{-\mu}v_\lambda +c_j\mu\partial^{-\mu-1}v_\lambda - c_jx\partial^{-\mu}v_\lambda\\
&=(1-c_jx)u_{\lambda,\mu}+c_j\mu\partial^{-1}u_{\lambda,\mu}. \end{split}\end{equation} From these relations with $P_{P_p}u_{\lambda,\mu}=0$ we have all the contiguity relations. For example \begin{align}
\partial u_{\lambda_0,\dots,\lambda_{p-1},\mu+1}&= u_{\lambda,\mu},\\
\partial u_{\lambda_0+1,\ldots,\lambda_{p-1},\mu}&=
(x\partial +1-\mu) u_{\lambda,\mu},\notag\\
\partial u_{\ldots,\lambda_j+1,\ldots,\mu}&=
\bigl((1-c_jx)\partial - c_j(1-\mu)\bigr)u_{\lambda,\mu}\notag \end{align} and \begin{align*}
P_{P_p}(\lambda,\mu+1)&=\sum_{j=0}^{p-1}p_j(x)\partial^{p-j} + p_n\\[-5pt]
p_n &= (-1)^{p-1}c_1\dots c_{p-1}\Bigl((-\mu-1)_p+
(-\mu)_{p-1} \sum_{j=0}^{p-1}\lambda_j\Bigr)\\
&=c_1\cdots c_{p-1}(\mu+2-p)_{p-1}
(\lambda_0+\cdots+\lambda_{p-1}-\mu-1) \end{align*} and hence \begin{align*}
\Bigl(\sum_{j=0}^{p-1}p_j(x)\partial^{p-j-1}\Bigr)u_{\lambda,\mu}&=
-p_n u_{\lambda,\mu+1} = -p_n \partial^{-1}u_{\lambda,\mu}. \end{align*} Substituting this equation to \eqref{eq:PoCon0}, we have $Q_j\in W(x;\lambda,\mu)$ such that $Q_ju_{\lambda,\mu}$ equals $u_{(\lambda_\nu+\delta_{\nu,j})_{\nu=0,\dots,p-1},\mu}$ for $j=0,\dots,p-1$, respectively. The operators $R_j\in W(x;\lambda,\mu)$ satisfying $R_jQ_ju_{\lambda,\mu}=u_{\lambda,\mu}$ are calculated by the Euclid algorithm, namely, we find $S_j\in W(x;\lambda,\mu)$ so that $R_jQ_j+S_jP_{P_p}=1$. Thus we also have $T_j\in W(x;\lambda,\mu)$ such that $T_ju_{\lambda,\mu}$ equals $u_{(\lambda_\nu-\delta_{\nu,j})_{\nu=0,\dots,p-1},\mu}$ for $j=0,\dots,p-1$, respectively.
As is shown in \S\ref{sec:Versal} the \textsl{Versal Jordan-Pochhammer operator} $\tilde P_{P_p}$ is given by \eqref{eq:Poch} with \index{Jordan-Pochhammer!versal} \begin{equation}
p_0(x)=\prod_{j=1}^{p}(1-c_jx),\quad
q(x)=\sum_{k=1}^p\lambda_kx^{k-1}\prod_{j=k+1}^p(1-c_jx). \end{equation} If $c_1,\dots,c_p$ are different to each other, the Riemann scheme of $\tilde P_{P_p}$ is \[
\begin{Bmatrix}
x=\frac1{c_j}\ (j=1,\dots,p) & \infty\\
[0]_{(p-1)} & [1-\mu]_{(p-1)}\\
\displaystyle\sum_{k=j}^p
\frac{\lambda_k}{c_j\prod_{\substack{1\le\nu\le k\\\nu\ne j}}
(c_j-c_\nu)}+\mu&
\displaystyle\sum_{k=1}^p\frac{(-1)^k\lambda_k}{c_1\dots c_k}-\mu
\end{Bmatrix}. \] The solution of $\tilde P_{P_p}u=0$ is given by \[
u_C(x) = \int_C\Bigl(\exp
\int_0^t\sum_{j=1}^p\frac{-\lambda_j s^{j-1}}{\prod_{1\le\nu\le j}
(1-c_\nu s)}ds\Bigr)(x-t)^{\mu-1}dt. \] Here the path $C$ starting from a singular point and ending at a singular point is chosen so that the integration has a meaning. In particular when $c_1=\cdots=c_p=0$, we have \[
u_C(x)=\int_C\exp\Bigl(-\sum_{j=1}^p\frac{\lambda_jt^j}{j!}\Bigr)
(x-t)^{\mu-1}dt \] and if $\lambda_p\ne0$, the path $C$ starts from $\infty$ to one of the $p$ independent directions $\lambda_p^{-1}e^{\frac{2\pi\nu\sqrt{-1}}{p}+t}$ $(t\gg1,\ \nu=0,1,\dots,p-1)$ and ends at $x$.
Suppose $n=2$. The corresponding Riemann scheme for the generic characteristic exponents and its construction from the Riemann scheme of the trivial equation $u'=0$ is as follows: \begin{align*}\begin{split}
&\begin{Bmatrix} x=0 & 1& \infty\\
b_0 & c_0 & a_0\\
b_1 & c_1 & a_1\\ \end{Bmatrix}\qquad(\text{Fuchs relation: } a_0+a_1+b_0+b_1+c_0+c_1=1)\\ &\quad\xleftarrow{x^{b_0}(1-x)^{c_0}\partial^{-a_1-b_1-c_1}}
\begin{Bmatrix} x=0 & 1& \infty\\
-a_1-b_0-c_1 & -a_1-b_1-c_0 & -a_0+a_1+1\\ \end{Bmatrix}\\ &\quad\xleftarrow{x^{-a_1-b_0-c_1}(1-x)^{-a_1-b_1-c_0}}
\begin{Bmatrix} x=0 & 1& \infty\\
0 & 0 & 0\\ \end{Bmatrix}. \end{split}\end{align*} Then our fractional calculus gives the corresponding equation
\begin{align}\begin{split}
&x^2(1-x)^2u''-x(1-x)\bigl((a_0+a_1+1)x+b_0+b_1-1\bigr)u'\\
&\quad{}+\bigl(a_0a_1x^2-(a_0a_1+b_0b_1-c_0c_1)x+b_0b_1\bigr)u=0, \end{split}\end{align}
the connection formula \begin{align}\begin{split}
&c(0\!:\!b_1 \rightsquigarrow 1\!:\!c_1)=\frac{
\Gamma(c_0-c_1)
\Gamma(b_1-b_0+1)
}{
\Gamma(a_0+b_1+c_0)
\Gamma(a_1+b_1+c_0)
} \end{split}\end{align} and expressions of its solution by the integral representation \begin{align}\label{eq:GGI}\begin{split}
&\int_0^{x}x^{b_0}(1-x)^{c_0}(x-s)^{a_1+b_1+c_1-1}
s^{-a_1-c_1-b_0}(1-s)^{-a_1-b_1-c_0-}ds\\
&\quad=\frac{
\Gamma(a_0+b_1+c_0)
\Gamma(a_1+b_1+c_1)
}{
\Gamma(b_1-b_0+1)
}x^{b_1}\phi_{b_1}(x) \end{split}\end{align} and the series expansion \begin{align}\label{eq:GGS}\begin{split}
&\sum_{n\ge0}\frac{(a_0+b_1+c_0)_n(a_1+b_1+c_0)_n}{(b_1-b_0+1)_nn!}(1-x)^{c_0}x^{b_1+n}\\
&\quad=(1-x)^{c_0}x^{b_1}F(a_0+b_1+c_0,a_1+b_1+c_0,b_1-b_0-1;x). \end{split}\end{align} Here $\phi_{b_1}(x)$ is a holomorphic function in a neighborhood of $0$ satisfying $\phi_{b_1}(0)=1$ for generic spectral parameters. We note that the transposition of $c_0$ and $c_1$ in \eqref{eq:GGS} gives a nontrivial equality, which corresponds to Kummer's relation of Gauss hypergeometric function and the similar statement is true for \eqref{eq:GGI}. In general, different procedures of reduction of a equation give different expressions of its solution.
\subsection{Hypergeometric family} $H_n$\label{sec:GHG} \index{hypergeometric equation/function!generalized} \index{00Hn@$H_n$}
We examine the hypergeometric family which corresponds to the equations satisfied by the generalized hypergeometric series \eqref{eq:IGHG}. Its spectral type is in the Simpson's list (cf.~\S\ref{sec:rigidEx}).
$\mathbf m=(1^n,n-11,1^n)$ : ${}_nF_{n-1}(\alpha,\beta;z)$ \begin{align*}
1^n,n-11,1^n&=1,10,1\oplus 1^{n-1},n-21,1^{n-1}\\
\Delta(\mathbf m)&=\{\alpha_0+\alpha_{0,1}
+\cdots+\alpha_{0,\nu}+\alpha_{2,1}+\cdots+\alpha_{2,\nu'}\,;\,\\
&\qquad
0\le\nu<n,\ 0\le\nu'< n\}\\
[\Delta(\mathbf m)]&=1^{n^2}\\
H_n &= H_1 \oplus H_{n-1}:n^2\\
H_n &\overset{1}{\underset{R2E0}\longrightarrow} H_{n-1} \end{align*}
Since $\mathbf m$ is of Okubo type, we have a system of Okubo normal form with the spectral type $\mathbf m$. Then the above $R2E0$ represents the reduction of systems of equations of Okubo normal form due to Yokoyama \cite{Yo2}. The number $1$ on the arrow represents a reduction by a middle convolution and the number shows the difference of the orders. \begin{gather}\label{eq:HnR} \begin{Bmatrix}
x=0 & 1 & \infty\\
\lambda_{0,1} & [\lambda_{1,1}]_{(n-1)} & \lambda_{2,1}\\
\vdots & & \vdots \\
\lambda_{0,n-1}& &\lambda_{2,n-1}\\
\lambda_{0,n} & \lambda_{1,2} &\lambda_{2,n}
\end{Bmatrix} ,\quad \begin{Bmatrix}
x = 0 & 1 & \infty\\
1-\beta_1 & [0]_{(n-1)} & \alpha_1\\
\vdots & & \vdots \\
1-\beta_{n-1}& & \alpha_{n-1}\\
0 & -\beta_n & \alpha_n
\end{Bmatrix}\\
\sum_{\nu=1}^n(\lambda_{0,\nu}+\lambda_{2,\nu})
+(n-1)\lambda_{1,1}+\lambda_{1,2}=n-1,\notag\\
\alpha_1+\cdots+\alpha_n=\beta_1+\cdots+\beta_n.\notag \end{gather}
It follows from Theorem~\ref{thm:sftUniv} that the universal operators \[
P_{H_1}^0(\lambda),\ P_{H_1}^{2}(\lambda),\ P_{H_{n-1}}^0(\lambda),\
P_{H_{n-1}}^1(\lambda),\ P_{H_{n-1}}^2(\lambda). \] are shift operators for the universal model $P_{H_n}(\lambda)u=0$.
The Riemann scheme of the operator \[ P=\RAd(\partial^{-\mu_{n-1}})\circ\RAd(x^{\gamma_{n-1}})\circ\cdots
\circ\RAd(\partial^{-\mu_1})\circ\RAd(x^{\gamma_1}(1-x)^{\gamma'})\partial \] equals \begin{equation}\label{eq:HGRS0} {
\begin{Bmatrix}
x=0 & 1 & \infty\\
0 & [0]_{(n-1)} & 1-\mu_{n-1}\\
(\gamma_{n-1}+\mu_{n-1}) & & 1-(\gamma_{n-1} + \mu_{n-1}) - \mu_{n-2}\\
\displaystyle\sum_{j={n-2}}^{n-1}(\gamma_j+\mu_j)&&
1 - \displaystyle\sum_{j={n-2}}^{n-1}(\gamma_j+\mu_j) - \mu_{n-3}\\
\vdots && \vdots\\
\displaystyle\sum_{j=2}^{n-1}(\gamma_j+\mu_j)&&
1 - \displaystyle\sum_{j=2}^{n-1}(\gamma_j+\mu_j) -\mu_1\\
\displaystyle\sum_{j=1}^{n-1}(\gamma_j+\mu_j)&
\gamma'+\displaystyle\sum_{j=1}^{n-1}\mu_j&
-\gamma'-\displaystyle\sum_{j=1}^{n-1}(\gamma_j+\mu_j)
\end{Bmatrix},} \end{equation} which is obtained by the induction on $n$ with Theorem~\ref{thm:GRSmid} and corresponds to the second Riemann scheme in \eqref{eq:HnR} by putting \begin{equation}\begin{aligned}
\gamma_j&=\alpha_{j+1}-\beta_j&&(j=1,\dots,n-2),&
\gamma'&=-\alpha_1+\beta_1-1,\\
\mu_j &= -\alpha_{j+1}+\beta_{j+1}&&(j=1,\dots,n-1),&
\mu_{n-1} &= 1-\alpha_n. \end{aligned}\end{equation} The integral representation of the local solutions at $x=0$ (resp.\ 1 and $\infty$) corresponding to the exponents $\sum_{j=1}^{n-1}(\gamma_j+\mu_j)$ (resp.\ $\gamma'+\sum_{j=1}^{n-1}\mu_j$ and $-\gamma'-\sum_{j=1}^{n-1}(\gamma_j+\mu_j)$ are given by \begin{equation}
I_c^{\mu_{n-1}}x^{\gamma_{n-1}}I_c^{\mu_{n-2}}\cdots
I_c^{\mu_1}x^{\gamma_1}(1-x)^{\gamma'} \end{equation} by putting $c=0$ (resp.\ 1 and $\infty$).
For simplicity we express this construction using additions and middle convolutions by \begin{equation}
u=\partial^{-\mu_{n-1}}x^{\gamma_{n-1}}\cdots\partial^{-\mu_2}x^{\gamma_2}
\partial^{-\mu_2}x^{\gamma_1}(1-x)^{\gamma'}. \end{equation}
For example, when $n=3$, we have the solution \[
\int_c^x t^{\alpha_3-\beta_2}(x-t)^{1-\alpha_3}dt
\int_c^t s^{\alpha_2-\beta_1}(1-s)^{-\alpha_1+\beta_1-1} (t-s)^{-\alpha_2-\beta_2}ds. \]
The operator corresponding to the second Riemann scheme is \begin{equation}\label{eq:GHP}
P_n(\alpha;\beta):=\prod_{j=1}^{n-1}(\vartheta-\beta_j)\cdot \partial - \prod_{j=1}^{n}(\vartheta-\alpha_j). \end{equation} This is clear when $n=1$. In general, we have \begin{align*}
&\RAd(\partial^{-\mu})\circ \RAd(x^\gamma)P_n(\alpha,\beta)\\
&=
\RAd(\partial^{-\mu})\circ \Ad(x^\gamma)\Bigl(\prod_{j=1}^{n-1}x(\vartheta+\beta_j)\cdot \partial - \prod_{j=1}^{n}x(\vartheta+\alpha_j)\Bigr)\\
&= \RAd(\partial^{-\mu})\Bigl(\prod_{j=1}^{n-1}(\vartheta+\beta_j-1-\gamma)
(\vartheta-\gamma) - \prod_{j=1}^{n}x(\vartheta+\alpha_j-\gamma)\Bigr)\\
&=\Ad(\partial^{-\mu})\Bigl(\prod_{j=1}^{n-1}(\vartheta+\beta_j-\gamma)\cdot
(\vartheta-\gamma+1)\partial - \prod_{j=1}^{n}(\vartheta+1)(\vartheta+\alpha_j-\gamma)\Bigr)\\
&=\prod_{j=1}^{n-1}(\vartheta+\beta_j-\gamma-\mu)\cdot
(\vartheta-\gamma-\mu+1)\partial
-\prod_{j=1}^{n}(\vartheta+1-\mu)\cdot(\vartheta+\alpha_j-\gamma-\mu) \end{align*} and therefore we have \eqref{eq:GHP} by the correspondence of the Riemann schemes with $\gamma=\gamma_n$ and $\mu=\mu_n$.
Suppose $\lambda_{1,1}=0$. We will show that \index{hypergeometric equation/function!generalized!local solution} \begin{equation}\label{eq:HGS} \begin{split}
&\sum_{k=0}^\infty
\frac{\prod_{j=1}^n(\lambda_{2,j}-\lambda_{0,n})_k}
{\prod_{j=1}^{n-1}(\lambda_{0,n}-\lambda_{0,j}+1)_k k!}
x^{\lambda_{0,n}+k}\\
&\quad= x^{\lambda_{0,n}}{}_{n}F_{n-1}\bigl(
(\lambda_{2,j}-\lambda_{0,n})_{j=1,\dots,n},
(\lambda_{0,n}-\lambda_{0,j}+1)_{j=1,\dots,n-1};x\bigr) \end{split} \end{equation} is the local solution at the origin corresponding to the exponent $\lambda_{0,n}$. Here \begin{equation}
{}_nF_{n-1}(\alpha_1,\dots,\alpha_n,\beta_1,\dots,\beta_{n-1};x)
=\sum_{k=0}^\infty
\frac{(\alpha_1)_k\cdots(\alpha_{n-1})_k(\alpha_n)_k}{
(\beta_1)_k\cdots(\beta_{n-1})_kk!}x^k. \end{equation}
We may assume $\lambda_{0,1}=0$ for the proof of \eqref{eq:HGS}. When $n=1$, the corresponding solution equals $(1-x)^{-\lambda_{2,1}}$ and we have \eqref{eq:HGS}. Note that \begin{align*}
&I_0^\mu x^\gamma \sum_{k=0}^\infty
\frac{\prod_{j=1}^n(\lambda_{2,j}-\lambda_{0,n})_k}
{\prod_{j=1}^{n-1}(\lambda_{0,n}-\lambda_{0,j}+1)_k k!}
x^{\lambda_{0,n}+k}\\
&=\sum_{k=0}^\infty
\frac{\prod_{j=1}^n(\lambda_{2,j}-\lambda_{0,n})_k}
{\prod_{j=1}^{n-1}(\lambda_{0,n}-\lambda_{0,j}+1)_k k!}
\frac{\Gamma(\lambda_{0,n}+\gamma+k+1)}
{\Gamma(\lambda_{0,n}+\gamma+\mu+k+1)}x^{\lambda_{0,n}+\gamma+\mu+k}\\
&=\frac{\Gamma(\lambda_{0,n}+\gamma+1)}
{\Gamma(\lambda_{0,n}+\gamma+\mu+1)}\sum_{k=0}^\infty
\frac{\prod_{j=1}^n(\lambda_{2,j}-\lambda_{0,n})_k
\cdot(\lambda_{0,n}+\gamma+1)_k
\cdot x^{\lambda_{0,n}+\gamma+\mu+k}}
{\prod_{j=1}^{n-1}(\lambda_{0,n}-\lambda_{0,j}+1)_k
\cdot(\lambda_{0,n}+\gamma+\mu+1)_k k!}. \end{align*} Comparing \eqref{eq:HGRS0} with the first Riemann scheme under $\lambda_{0,1}=\lambda_{1,1}=0$ and $\gamma=\gamma_n$ and $\mu=\mu_n$, we have the solution \eqref{eq:HGS} by the induction on $n$. The recurrence relation in Theorem~\ref{thm:shifm1} corresponds to the identity \index{hypergeometric equation/function!generalized!recurrence relation} \begin{equation} \begin{split} &{}_nF_{n-1}(\alpha_1,\dots,\alpha_{n-1},\alpha_n+1;\beta_1,\dots,\beta_{n-1};x)\\ &\quad={}_nF_{n-1}(\alpha_1,\dots,\alpha_n;\beta_1,\dots,\beta_{n-1};x)\\ &\quad\quad{}+\frac{\alpha_1\cdots\alpha_{n-1}}{\beta_1\cdots\beta_{n-1}} x\cdot{}_nF_{n-1}(\alpha_1+1,\dots,\alpha_n+1;\beta_1+1,\dots,\beta_{n-1}+1;x). \end{split} \end{equation}
The series expansion of the local solution at $x=1$ corresponding to the exponent $\gamma'+\mu_1+\cdots+\mu_{n-1}$ is a little more complicated.
For the Riemann scheme \begin{align*} &\qquad \begin{Bmatrix}
x=\infty&0&1\\
-\mu_2+1 & [0]_{(2)} & 0\\
1-\gamma_2-\mu_1-\mu_2 & &\gamma_2+\mu_2\\
-\gamma'-\gamma_1-\gamma_2-\mu_1-\mu_2
&\underline{\gamma'+\mu_1+\mu_2}&\gamma_1+\gamma_2+\mu_1+\mu_2
\end{Bmatrix}\allowdisplaybreaks, \intertext{we have the local solution at $x=0$}
&I_0^{\mu_2}(1-x)^{\gamma_2}I_0^{\mu_1}x^{\gamma'}(1-x)^{\gamma_1}
=I_0^{\mu_2}(1-x)^{\gamma_2}\sum_{n=0}^\infty
\frac{(-\gamma_1)_n}{n!}x^n
\\&=
I_0^{\mu_2}\sum_{n=0}^\infty \frac{\Gamma(\gamma'+1+n)
(-\gamma_1)_n}{\Gamma(\gamma'+\mu_1+1+n)n!}
x^{\gamma'+\mu_1+n}(1-x)^{\gamma_2} \allowdisplaybreaks\\
&=I_0^{\mu_2}\sum_{m,n=0}^\infty
\frac{\Gamma(\gamma'+1+n)(-\gamma_1)_n(-\gamma_2)_m}
{\Gamma(\gamma'+\mu_1+1+n)m!n!}
x^{\gamma'+\mu_1+m+n} \allowdisplaybreaks\\
&=\sum_{m,n=0}^\infty
\frac{\Gamma(\gamma'+\mu_1+1+m+n)\Gamma(\gamma'+1+n)
(-\gamma_1)_n(-\gamma_2)_m x^{\gamma'+\mu_1+\mu_2+m+n}}
{\Gamma(\gamma'+\mu_1+\mu_2+1+m+n)\Gamma(\gamma'+\mu_1+1+n)
m!n!} \allowdisplaybreaks\\
&=\frac{\Gamma(\gamma'+1)x^{\gamma'+\mu_1+\mu_2}}
{\Gamma(\gamma'+\mu_1+\mu_2+1)}
\sum_{m,n=0}^\infty
\frac{(\gamma'+\mu_1+1)_{m+n}(\gamma'+1)_n(-\gamma_1)_n
(-\gamma_2)_m x^{m+n}}
{(\gamma'+\mu_1+\mu_2+1)_{m+n}(\gamma'+\mu_1+1)_nm!n!}.
\intertext{Applying the last equality in \eqref{eq:I0H} to the above second equality, we have}
&I_0^{\mu_2}(1-x)^{\gamma_2}I_0^{\mu_1}x^{\gamma'}(1-x)^{\gamma_1}
\\
&=\sum_{n=0}^\infty \frac{\Gamma(\gamma'+1+n)(-\gamma_1)_n}
{\Gamma(\gamma'+\mu_1+1+n)n!}
x^{\gamma'+\mu_1+\mu_2+n}(1-x)^{-\gamma_2} \allowdisplaybreaks\\
&\quad\cdot\sum_{m=0}^\infty
\frac{\Gamma(\gamma'+\mu_1+1+n)}
{\Gamma(\gamma'+\mu_1+\mu_2+1+n)}
\frac{(\mu_2)_m(-\gamma_2)_m}
{(\gamma'+\mu_1+n+\mu_2+1)_mm!}
\Bigl(\frac{x}{x-1}\Bigr)^m \allowdisplaybreaks\\
&=\frac{\Gamma(\gamma'+1)
x^{\gamma'+\mu_1+\mu_2}(1-x)^{-\gamma_2}}
{\Gamma(\gamma'+\mu_1+\mu_2+1)}\!
\sum_{m,n=0}^\infty\!\!
\frac{(\gamma'+1)_n(-\gamma_1)_n(-\gamma_2)_m(\mu_2)_m}
{(\gamma'+\mu_1+\mu_2+1)_{m+n}m!n!}x^n
\Bigl(\frac x{x-1}\Bigr)^m \allowdisplaybreaks\\
&=\frac{\Gamma(\gamma'+1)}
{\Gamma(\gamma'+\mu_1+\mu_2+1)}\\
&\quad\cdot
x^{\gamma'+\mu_1+\mu_2}(1-x)^{-\gamma_2}
F_3\bigl(-\gamma_2,-\gamma_1,\mu_2,\gamma'+1;\gamma'+\mu_1+\mu_2+1;
x,\frac{x}{x-1}\bigr), \end{align*} where $F_3$ is Appell's hypergeometric function \eqref{eq:F3}.
Let $u_1^{-\beta_n}(\alpha_1,\dots,\alpha_n;\beta_1,\dots,\beta_{n-1};x)$ be the local solution of $P_n(\alpha,\beta)u=0$ at $x=1$ such that $u_1^{-\beta_n} (\alpha;\beta;x)\equiv (x-1)^{-\beta_n}\mod (x-1)^{1-\beta_n}\mathcal O_1$ for generic $\alpha$ and $\beta$. Since the reduction \[
\begin{Bmatrix}
\lambda_{0,1} & [0]_{(n-1)} & \lambda_{2,1}\\
\vdots & & \vdots\\
\lambda_{0,n} & \lambda_{1,2} & \lambda_{2,n}
\end{Bmatrix}
\xrightarrow{\partial_{max}}
\begin{Bmatrix}
\lambda_{0,1}' & [0]_{(n-2)} & \lambda_{2,1}'\\
\vdots & & \vdots\\
\lambda_{0,n-1}' & \lambda_{1,2}' & \lambda_{2,n-1}'
\end{Bmatrix} \] satisfies $\lambda_{1,2}'=\lambda_{1,2}+\lambda_{0,1}+\lambda_{0,2}-1$ and $\lambda_{0,j}'+\lambda_{2,j}'=\lambda_{0,j+1}+\lambda_{2,j+1}$ for $j=1,\dots,n-1$, Theorem~\ref{thm:shifm1} proves \index{hypergeometric equation/function!generalized!recurrence relation} \begin{equation} \begin{split}
u_1^{-\beta_n}(\alpha;\beta;x)&=
u_1^{-\beta_n}(\alpha_1,\dots,\alpha_n+1;\beta_1,\dots,\beta_{n-1}+1;x)\\
&\quad{}+ \frac{\beta_{n-1}-\alpha_n}{1-\beta_n}
u_1^{1-\beta_n}(\alpha;\beta_1,\dots,\beta_{n-1}+1;x). \end{split} \end{equation}
The condition for the irreducibility of the equation equals \begin{equation}
\lambda_{0,\nu}+\lambda_{1,1}+\lambda_{2,\nu'}\notin\mathbb Z
\qquad(1\le\nu\le n,\ 1\le \nu'\le n), \end{equation} which is easily proved by the induction on $n$ (cf.~Example~\ref{exmp:irred} ii)). The shift operator under a compatible shift $(\epsilon_{j,\nu})$ is bijective if and only if \begin{equation}
\lambda_{0,\nu}+\lambda_{1,1}+\lambda_{2,\nu'} \text{ \ and \ } \lambda_{0,\nu}+\epsilon_{0,\nu}+\lambda_{1,1}+\epsilon_{1,1}+\lambda_{2,\nu'} +\epsilon_{2,\nu'} \end{equation} are simultaneously not integers or positive integers or non-positive integers for each $\nu\in\{1,\dots,n\}$ and $\nu'\in\{1,\dots,n\}$.
Connection coefficients in this example are calculated by \cite{Le} and \cite{OTY} etc. In this paper we get them by Theorem~\ref{thm:c}.
There are the following direct decompositions $(\nu=1,\dots,n)$.
\begin{gather*} \begin{split} 1\dots1\overline{1};n-1\underline{1};1\dots1 &= 0\dots0\overline{1};\ \ 1\ \ \ \,\underline{0};0\dots0\overset{\underset{\smallsmile}\nu}10\dots0\\[-5pt]
&\,\oplus 1\dots1\overline{0};n-2\underline{1};1\dots101\dots1. \end{split} \end{gather*} These $n$ decompositions $\mathbf m=\mathbf m'\oplus\mathbf m''$ satisfy the condition $m'_{0,n_0}=m''_{1,n_1}=1$ in \eqref{eq:connection}, where $n_0=n$ and $n_1=2$. Since $n_0+n_1-2=n$, Remark~\ref{rem:conn} i) shows that these decompositions give all the decompositions appearing in \eqref{eq:connection}. Thus we have \index{hypergeometric equation/function!generalized!connection coefficient} \begin{gather*} c(\lambda_{0,n}\rightsquigarrow\lambda_{1,2}) =\frac{\displaystyle\prod_{\nu=1}^{n-1}
\Gamma({\lambda_{0,n}}-\lambda_{0,\nu}+1)
\cdot\Gamma(\lambda_{1,1}-{\lambda_{1,2}})}
{\displaystyle\prod_{\nu=1}^n\Gamma({\lambda_{0,n}}+\lambda_{1,1}+\lambda_{2,\nu})} =\displaystyle\prod_{\nu=1}^n\frac{\Gamma(\beta_\nu)}
{\Gamma(\alpha_\nu)}\\ \qquad\qquad\ \ =\lim_{x\to 1-0}(1-x)^{\beta_n}
{}_nF_{n-1}(\alpha,\beta;x)
\qquad(\RE \beta_n > 0). \end{gather*}
Other connection coefficients are obtained by the similar way. \begin{align*} c(\lambda_{0,n}\rightsquigarrow\lambda_{2,n}&):\quad \text{When }n=3, \text{ we have} \\
11\overline{1},21,11\underline{1}
&\!\!=\!00{1},10,10{0}\!
\quad\,00{1},10,01{0}\!
\quad\,10{1},11,11{0}\!
\quad\,01{1},11,11{0}\\[-5pt]
&\!\oplus\!11{0},11,01{1}\!
=\!11{0},11,10{1}\!
=\!01{0},10,00{1}\!
=\!10{0},10,00{1} \end{align*} In general, by the rigid decompositions \begin{align*}
1\cdots1\overline{1}\,,\,n-11\,,\,1\cdots1\underline{1}
&\!=0\cdots0\overline{1}\,,\,\ \ \ 1\,\ \ 0\,,\,0\ldots0\overset{\underset{\smallsmile}i}10\cdots0\underline{0} \\[-8pt]
&\oplus1\cdots1\overline{0}\,,\,n-21\,,\,1\cdots101\cdots1\underline{1}\\
&\!=1\cdots1\overset{\underset\smallsmile i}01\cdots1\overline{1}\,,\,n-21\,,\,1\cdots1\underline{0} \\[-8pt]
&\oplus0\ldots010\cdots0\overline{0}\,,\,\ \ \ 1\ \ \,0\,,\,0\cdots0\underline{1} \end{align*} for $i=1,\dots,n-1$ we have \begin{align*} c(\lambda_{0,n}\rightsquigarrow\lambda_{2,n}) &=\prod_{k=1}^{n-1}\frac{
\Gamma(\lambda_{2,k}-\lambda_{2,n})}
{\Gamma\bigl(
\left|\begin{Bmatrix}
\lambda_{0,n}
& \lambda_{1,1}
& \lambda_{2,k}
\end{Bmatrix}\right|
\bigr)}\\
&\quad\cdot\prod_{k=1}^{n-1}\frac{
\Gamma(\lambda_{0,n}-\lambda_{0,k}+1)}
{\Gamma\bigl(
\left|\begin{Bmatrix}
(\lambda_{0,\nu})_{\substack{1\le\nu\le n\\ \nu\ne k}}
& [\lambda_{1,1}]_{(n-2)} &
& \!\!\!\!(\lambda_{2,\nu})_{1\le\nu\le n-1}\\
& \lambda_{1,2}
\end{Bmatrix}\right|
\bigr)}\\
&=\prod_{k=1}^{n-1}\frac{\Gamma(\beta_k)\Gamma(\alpha_k-\alpha_n)}
{\Gamma(\alpha_k)\Gamma(\beta_k-\alpha_n)}. \intertext{Moreover we have}
c(\lambda_{1,2}\!\rightsquigarrow\!\lambda_{0,n})
&=\frac{\Gamma\bigl(\lambda_{1,2}-\lambda_{1,1}+1\bigr)\cdot
\prod_{\nu=1}^{n-1}\Gamma\bigl(\lambda_{0,\nu}-\lambda_{0,n}
\bigr)}
{\displaystyle\prod_{j=1}^n
\Gamma\bigl(
\left|\begin{Bmatrix}
(\lambda_{0,\nu})_{1\le\nu\le n-1}
& [\lambda_{1,1}]_{(n-2)} &
& \!\!\!\!(\lambda_{2,\nu})_{1\le\nu\le n,\ \nu\ne j}\\
& \lambda_{1,2}
\end{Bmatrix}\right|
\bigr)}\\
&=\prod_{\nu=1}^n\frac{\Gamma(1-\beta_\nu)}{\Gamma(1-\alpha_\nu)}. \end{align*} Here we denote \[
(\mu_\nu)_{1\le\nu\le n}
=\left(\begin{smallmatrix}\mu_1\\ \mu_2\\ \vdots\\ \mu_n\end{smallmatrix}\right)\in\mathbb C^n
\text{\ \ and \ \ }
(\mu_\nu)_{\substack{1\le\nu\le n\\ \nu\ne i}}
=\left(\begin{smallmatrix}\mu_1\\ \vdots\\ \mu_{i-1}\\ \mu_{i+1}\\ \vdots\\ \mu_n\
\end{smallmatrix}\right)\in\mathbb C^{n-1} \] for complex numbers $\mu_1,\dots,\mu_n$.
These connection coefficients were obtained by \cite{Le} and \cite{Yos} etc.
We have \begin{equation}
\begin{split}
{}_nF_{n-1}(\alpha,\beta;x)&=\sum_{k=0}^\infty C_k(1-x)^k +
\sum_{k=0}^\infty C'_k(1-x)^{k-\beta_n},\\
C_0 &= {}_nF_{n-1}(\alpha,\beta;1)\quad(\RE\beta_n < 0),\\
C'_0 &= \prod_{\nu=1}^n\frac{\Gamma(\beta_\nu)}{\Gamma(\alpha_\nu)}
\end{split} \end{equation} for $0<x<1$ if $\alpha$ and $\beta$ are generic. Since \begin{multline*} \frac{d^k}{dx^k}{}_nF_{n-1}(\alpha,\beta;x)\\=
\frac{(\alpha_1)_k\cdots(\alpha_n)_k} {(\beta_1)_k\cdots(\beta_{n-1})_k} {}_nF_{n-1}(\alpha_1+k,\dots,\alpha_n+k,\beta_1+k,\dots,\beta_{n-1}+k;x), \end{multline*} we have \begin{equation}
C_k =
\frac{(\alpha_1)_k\cdots(\alpha_n)_k} {(\beta_1)_k\cdots(\beta_{n-1})_kk!}
{}_nF_{n-1}(\alpha_1+k,\dots,\alpha_n+k,\beta_1+k,\dots,\beta_{n-1}+k;1). \end{equation}
We examine the monodromy generators for the solutions of the generalized hypergeometric equation. For simplicity we assume $\beta_i\notin\mathbb Z$ and $\beta_i-\beta_j\notin\mathbb Z$ for $i\ne j$. Then $u=(u_0^{\lambda_{0,1}},\ldots,u_0^{\lambda_{0,n}})$ is a base of local solution at $0$ and the corresponding monodromy generator around $0$ with respect to this base equals \[
M_0=\begin{pmatrix}
e^{2\pi\sqrt{-1}\lambda_{0,1}}\\
&\ddots\\
&&e^{2\pi\sqrt{-1}\lambda_{0,n}}\\
\end{pmatrix} \] and that around $\infty$ equals \begin{align*}
M_\infty &=\biggl(\sum_{k=1}^n
e^{2\pi\sqrt{-1}\lambda_{2,\nu}}
c(\lambda_{0,i}\rightsquigarrow\lambda_{2,k})
c(\lambda_{2,k}\rightsquigarrow\lambda_{k,j})
\biggr)_{\substack{1\le i\le n\\1\le j\le n}}\\
&=\biggl(\sum_{k=1}^n
e^{2\pi{\sqrt{-1}\lambda_{2,\nu}}}
\prod_{\nu\in\{1,\dots,n\}\setminus\{k\}}
\frac{\sin2\pi
(\lambda_{0,i}+\lambda_{1,1}+\lambda_{2,\nu})}
{\sin2\pi(\lambda_{0,k}-\lambda_{0,\nu})}\\
&\quad\cdot
\prod_{\nu\in\{1,\dots,n\}\setminus\{j\}}\!\!
\frac{\sin2\pi(\lambda_{0,i}+\lambda_{1,1}+\lambda_{2,\nu})}
{\sin2\pi(\lambda_{2,j}-\lambda_{2,\nu})}
\biggr)_{\substack{1\le i\le n\\1\le j\le n}}. \end{align*}
Lastly we remark that the versal generalized hypergeometric operator is \index{hypergeometric equation/function!generalized!versal} \[ \begin{split} \tilde P&=\RAd(\partial^{-\mu_{n-1}})\circ
\RAd\bigl((1-c_1x)^{\frac{\gamma_{n-1}}{c_1}}\bigr)
\circ\cdots
\circ\RAd(\partial^{-\mu_1})\\
&\quad\circ
\RAd\left((1-c_1x)^{\frac{\gamma_1}{c_1}+\frac{\gamma'}{c_1(c_1-c_2)}}
(1-c_2x)^{\frac{\gamma'}{c_2(c_2-c_1)}}\right)\partial\\
&=\RAd(\partial^{-\mu_{n-1}})\circ
\RAdei\bigl(\frac{\gamma_{n-1}}{1-c_1x}\bigr)
\circ\cdots
\circ\RAd(\partial^{-\mu_1})\\
&\quad\circ
\RAdei\left(\frac{\gamma_1}{1-c_1x}
+ \frac{\gamma'x}{(1-c_1x)(1-c_2x)}\right)\partial \end{split} \] and when $n=3$, we have the integral representation of the solutions \[ \begin{split}
\int_c^x\int_c^t\exp\Bigl(
-\int_c^s\frac{\gamma_1(1-c_2u)+\gamma'u}{(1-c_1u)(1-c_2u)}du\Bigr)
(t-s)^{\mu_1-1}\bigl(1-c_1t\bigr)^{\frac{\gamma_2}{c_1}}
(x-t)^{\mu_2-1} ds\,dt. \end{split} \] Here $c$ equals $\frac1{c_1}$ or $\frac1{c_2}$ or $\infty$. \subsection{Even/Odd family} $EO_n$ \label{sec:EOEx} \index{00EOn@$EO_n$}
The system of differential equations of Schlesinger canonical form belonging to an even or odd family is concretely given by \cite{Gl}. We will examine concrete connection coefficients of solutions of the single differential equation belonging to an even or odd family. The corresponding tuples of partitions and their reductions and decompositions are as follows. \begin{align*}
m+1m,m^21,1^{2m+1}&=10,10,1\oplus m^2,mm-11,1^{2m}\\
&=1^2,1^20,1^2\oplus mm-1,(m-2)^21,1^{2m-1}\\
m^2,mm-11,1^{2m}&=1,100,1\oplus mm-1,(m-1)^21,1^{2m-1}\\
&=1^2,110,1^2\oplus (m-1)^2,m-1m-21,1^{2m-2}\\
EO_n &= H_1\oplus EO_{n-1}:2n = H_2\oplus EO_{n-2}:\binom n2\\ [\Delta(\mathbf m)]&=1^{\binom n2+2n}\\
EO_n &\overset{1}{\underset{R1E0R0E0}\longrightarrow}EO_{n-1}\\
EO_2&=H_2,\quad EO_3=H_3 \end{align*} The following operators are shift operators of the universal model $P_{EO_n}(\lambda)u=0$: \[
P_{H_1}^2(\lambda),\ P_{EO_{n-1}}^1(\lambda),\ P_{EO_{n-1}}^2(\lambda),\
P_{H_2}^2(\lambda),\ P_{EO_{n-2}}^1(\lambda),\ P_{EO_{n-2}}^2(\lambda). \] \noindent $EO_{2m}$ ($\mathbf m=(1^{2m},mm-11,mm)$ :\ even family) \index{even family} \begin{gather*}
\begin{Bmatrix}
x=\infty & 0 & 1\\
\lambda_{0,1} & [\lambda_{1,1}]_{(m)} & [\lambda_{2,1}]_{(m)}\\
\vdots & [\lambda_{1,2}]_{(m-1)} & [\lambda_{2,2}]_{(m)}\\
\lambda_{0,2m} & \lambda_{1,3}
\end{Bmatrix},\\ \sum_{\nu=1}^{2m}\lambda_{0,\nu} +m(\lambda_{1,1}+\lambda_{2,1}+\lambda_{2,2}) +(m-1)\lambda_{1,2}+\lambda_{1,3}=2m-1. \end{gather*} The rigid decompositions \begin{align*}
&1\cdots1\overline1\,,\,mm-1\underline1\,,\,mm\\
&=0\cdots0\overline1\,,\,10\underline0\,,\,
\overset{\underset{\smallsmile} i}10\oplus
1\cdots1\overline0\,,\,m-1m-1\underline1\,,
\,\overset{\underset{\smallsmile} i}01\\[-3pt]
&=0\cdots\overset{\underset{\smallsmile} j}1
\overline1\,,\,11\underline0\,,\,11\oplus
1\cdots\overset{\underset{\smallsmile} j}0
\overline0\,,\,m-1m-2\underline1\,,\,m-1m-1, \end{align*} which are expressed by $EO_{2m}=H_1\oplus EO_{2m-1}=H_2\oplus EO_{2m-2}$, give \index{even family!connection coefficient} \begin{align*}
c(\lambda_{0,2m}\rightsquigarrow\lambda_{1,3})
&=\prod_{i=1}^2\frac
{\Gamma\bigl(\lambda_{1,i}-\lambda_{1,3}\bigr)
}{\Gamma\bigl(
\left|\begin{Bmatrix}
\lambda_{0,2m} &\ \lambda_{1,1}\ &\lambda_{2,i}
\end{Bmatrix}\right|
\bigr)}
\cdot
\prod_{j=1}^{2m-1}\frac
{\Gamma\bigl(\lambda_{0,2m}-\lambda_{0,j}+1)}
{\Gamma\bigl(
\left|\begin{Bmatrix}
\lambda_{0,j} & \lambda_{1,1} & \lambda_{2,1}\\
\lambda_{0,2m} &\ \lambda_{1,2}\ & \lambda_{2,2}
\end{Bmatrix}\right|
\bigr)},\allowdisplaybreaks\\
c(\lambda_{1,3}\rightsquigarrow\lambda_{0,2m})
&=\displaystyle\prod_{i=1}^2\frac
{\Gamma\bigl(\lambda_{1,3}-\lambda_{1,i}+1\bigr)
}{\Gamma\bigl(
\left|\begin{Bmatrix}
&[\lambda_{1,1}]_{(m-1)}&[\lambda_{2,\nu}]_{(m)}\\
(\lambda_{0,\nu})_{1\le\nu\le 2m-1}
&\ [\lambda_{1,2}]_{(m-1)}\ &[\lambda_{2,3-i}]_{(m-1)}
\\
&\lambda_{1,3}
\end{Bmatrix}\right|
\bigr)}
\\
&\quad
\cdot
\prod_{j=1}^{2m-1}\frac
{\Gamma\bigl(\lambda_{0,j}-\lambda_{0,2m})}
{\Gamma\bigl(
\left|\begin{Bmatrix}
& [\lambda_{1,1}]_{(m-1)}&[\lambda_{2,1}]_{(m-1)}\\
(\lambda_{0,\nu})_{\substack{1\le\nu\le 2m-1\\ \nu\ne j}}
&\ [\lambda_{1,2}]_{(m-2)}\ &[\lambda_{2,2}]_{(m-1)}\\[-4pt]
&\lambda_{1,3}&
\end{Bmatrix}\right|
\bigr)}. \end{align*} These formulas were obtained by the author in 2007 (cf.~\cite{O3}), which is a main motivation for the study in this paper. The condition for the irreducibility is \index{even family!reducibility} \begin{equation*}
\begin{cases}
\lambda_{0,\nu}+\lambda_{1,1}+\lambda_{2,k}\notin\mathbb Z
&(1\le\nu\le 2m,\ k=1,2),\\
\lambda_{0,\nu}+\lambda_{0,\nu'}+\lambda_{1,1}+\lambda_{1,2}+\lambda_{2,1}
+\lambda_{2,2}-1\notin\mathbb Z
&(1\le\nu<\nu'\le 2m,\ k=1,2).
\end{cases} \end{equation*} The shift operator for a compatible shift $(\epsilon_{j,\mu})$ is bijective if and only if the values of each linear function in the above
satisfy \eqref{eq:non-pos}.
For the Fuchsian equation $\tilde Pu=0$ of type $EO_4$ with the Riemann scheme\index{even family!$EO_4$} \begin{align}\label{eq:EO4GRS} \begin{split}
&\begin{Bmatrix} x=\infty & 0 & 1\\ [a_1]_{(2)} & b_1 & [0]_{(2)}&\!\!;x\\ [a_2]_{(2)} & b_2 & c_1\\
& b_3 & c_2\\
& 0 & \\ \end{Bmatrix} \end{split}\end{align} and the Fuchs relation \begin{align}\label{eq:e4Fuchs}\begin{split}
&2a_1+2a_2+b_1+b_2+b_3+c_1+c_2=3 \end{split}\end{align} we have the connection formula \begin{align}\begin{split}
&c(0\!:\!0 \rightsquigarrow 1\!:\!c_2)
=\frac{
\Gamma(c_1-c_2)
\Gamma(-c_2)
\prod_{\nu=1}^3\Gamma(1-b_\nu)
}{
\Gamma(a_1)
\Gamma(a_2)
\prod_{\nu=1}^3\Gamma(a_1+a_2+b_\nu+c_1-1)
}. \end{split}\label{eq:Ceo4}\end{align}
Let $\tilde Q$ be the Gauss hypergeometric operator with the Riemann scheme \[
\begin{Bmatrix}
x=\infty & 0 & 1\\
a_1 & 1-a_1-a_2-c_1 & 0\\
a_2 & 0 & c_1
\end{Bmatrix}. \] We may normalize the operators by \[
\tilde P=x^3(1-x)\partial^4+\cdots\text{ \ and \ }
\tilde Q=x(1-x)\partial^2+\cdots. \] Then \[
\begin{split}
\tilde P&=\tilde S\tilde Q -\prod_{\nu=1}^3(a_1+a_2+b_\nu+c_1-1)\cdot\partial\\
\tilde Q&=\bigl(x(1-x)\partial+(a_1+a_2+c_1-(a_1+a_2+1)x)\bigr)\partial-a_1a_2
\end{split} \] with a suitable $\tilde S,\ \tilde T\in W[x]$ and $e\in\mathbb C$ and as is mentioned in Theorem~\ref{thm:sftUniv}, $\tilde Q$ is a shift operator satisfying\index{even family!$EO_4$!shift operator} \index{shift operator} \begin{equation}\label{eq:EO4sht} \begin{Bmatrix} x=\infty & 0 & 1\\ [a_1]_{(2)} & b_1 & [0]_{(2)}&\!\!;x\\ [a_2]_{(2)} & b_2 & c_1\\
& b_3 & c_2\\
& 0 & \\ \end{Bmatrix} \ \xrightarrow{\tilde Q} \ \begin{Bmatrix} x=\infty & 0 & 1\\ [a_1+1]_{(2)} & b_1-1 & [0]_{(2)}&\!\!;x\\ [a_2+1]_{(2)} & b_2-1 & c_1\\
& b_3-1 & c_2-1\\
& 0 & \\ \end{Bmatrix}. \end{equation} Let $u_0^0=1+\cdots$ and $u_1^{c_2}=(1-x)^{c_2}+\cdots$ be the normalized local solutions of $\tilde Pu=0$ corresponding to the characteristic exponents $0$ at $0$ and $c_2$ at $1$, respectively. Then the direct calculation shows \[
\begin{split}
\tilde Q u_0^0&=\frac{a_1a_2\prod_{\nu=1}^3(a_1+a_2+b_\nu+c_1-1)}
{\prod_{\nu=1}^3(1-b_\nu)}+\cdots,\\
\tilde Qu_1^{c_2}&=c_2(c_2-c_1)(1-x)^{c_2-1}+\cdots.
\end{split} \] Denoting by $c(a_1,a_2,b_1,b_2,b_3,c_1,c_2)$ the connection coefficient $c(0\!:\!0 \rightsquigarrow 1\!:\!c_2)$ for the equation with the Riemann scheme \eqref{eq:EO4GRS}, we have \[
\frac{c(a_1,a_2,b_1,b_2,b_3,c_1,c_2)}
{c(a_1+1,a_2+1,b_1-1,b_2-1,b_3-1,c_1,c_2-1)}
=\frac{a_1a_2\displaystyle\prod_{\nu=1}^3(a_1+a_2+b_\nu+c_1-1)}
{(c_1-c_2)(-c_2)\displaystyle\prod_{\nu=1}^3(1-b_\nu)}, \] which proves \eqref{eq:Ceo4} since $\lim_{k\to\infty}c(a_1+k,a_2+k,b_1-k,b_2-k,b_3-k,c_1,c_2-k)=1$. Note that the shift operator \eqref{eq:EO4sht} is not bijective if and only if \[
\tilde Qu=\prod_{\nu=1}^3(a_1+a_2+b_\nu+c_1-1)\cdot\partial u=0 \] has a non-zero solution, which is equivalent to \[
a_1a_2\prod_{\nu=1}^3(a_1+a_2+b_\nu+c_1-1)=0. \]
By the transformation $x\mapsto\frac{x}{x-1}$ we have \begin{align*} &\begin{Bmatrix}
x = \infty & 0 & 1\\
[0]_{(2)} & 0 & [a_1]_{(2)}\\
c_1 & b_1 & [a_2]_{(2)}\\
c_2 & b_2\\
& b_3
\end{Bmatrix}\\[-.3cm] &\qquad \xrightarrow{(1-x)^{a_1}\partial^{1-a_1}(1-x)^{-a_1}}
\begin{Bmatrix}
x = \infty & 0 & 1\\
2-2a_1 & & a_1\\
1+c_1-a_1 & a_1+b_1-1 & [a_1+a_2-1]_{(2)}\\
1+c_2-a_1 & a_1+b_2-1 \\
& a_1+b_3-1
\end{Bmatrix}\\ &\qquad \xrightarrow{x^{1-a_1-b_1}(1-x)^{1-a_1-a_2}} \begin{Bmatrix}
x = \infty & 0 & 1\\
a_2+b_1 & & 1-a_2\\
a_1+a_2+b_1+c_1-1 & 0 & [0]_{(2)}\\
a_1+a_2+b_1+c_2-1 & b_2-b_1 \\
& b_3-b_1
\end{Bmatrix} \end{align*} and therefore Theorem~\ref{thm:GC} gives the following connection formula for \eqref{eq:EO4GRS}: \begin{align*}
c(0\!:\!b_1\rightsquigarrow \infty\!:\!a_2)&=
\frac{\Gamma(b_1+1)\Gamma(a_1-a_2)}{\Gamma(a_1+b_1)\Gamma(1-a_2)}\cdot
{}_3F_2(a_2+b_1, a_1+a_2+b_1+c_1-1,
\\&\qquad\qquad a_1+a_2+b_1+c_2-1; b_1-b_2-1, b_1-b_3-1;1). \intertext{In the same way, we have}
c(1\!:\!c_1\rightsquigarrow \infty\!:\!a_2)&=
\frac{\Gamma(c_1+1)\Gamma(a_1-a_2)}{\Gamma(a_1+c_1)\Gamma(1-a_2)}\cdot
{}_3F_2(b_1-c_1,b_2-c_1,b_3-c_1;
\\&\qquad\qquad a_1+c_1, c_1-c_2+1;1). \end{align*}
\index{Wronskian} \index{connection coefficient!generalized} \index{even family!connection coefficient!generalized} We will calculate generalized connection coefficients defined in Definition~\ref{def:GC}. In fact, we get \begin{align}
c(1\!:\![0]_{(2)}\rightsquigarrow \infty\!:\![a_2]_{(2)})&=
\frac{\prod_{\nu=1}^2\Gamma(2-c_\nu)\cdot\prod_{i=1}^2\Gamma(a_1-a_2+i)}
{\Gamma(a_1)\prod_{\nu=1}^3\Gamma(a_1+b_\nu)},\label{eq:e4dc1}\\
c(\infty\!:\![a_2]_{(2)}\rightsquigarrow1\!:\![0]_{(2)})&=
\frac{\prod_{\nu=1}^2\Gamma(c_\nu-1)\cdot\prod_{i=0}^1\Gamma(a_2-a_1-i)}
{\Gamma(1-a_1)\prod_{\nu=1}^3\Gamma(1-a_1-b_\nu)}\label{eq:e4dc2} \end{align} according to the procedure given in Remark~\ref{rem:Cproc}, which we will explain.
The differential equation with the Riemann scheme $\begin{Bmatrix}
x=\infty & 0 & 1\\
\alpha_1 &[0]_{(2)}&[0]_{(2)}\\
\alpha_2 &[\beta]_{(2)}&\gamma_1\\
\alpha_3 &&\gamma_2\\
\alpha_4 \\ \end{Bmatrix}$ is $Pu=0$ with \index{tuple of partitions!rigid!1111,211,22} \begin{equation}\begin{split}
P=&\prod_{j=1}^4\bigl(\vartheta+\alpha_j\bigr)
+\partial\bigl(\vartheta-\beta\bigr)\bigl((\partial-2\vartheta+\gamma_1+\gamma_2-1)(\vartheta-\beta)\\
&+\sum_{1\le i<j\le 3}\!\!\alpha_i\alpha_j
-(\beta-2\gamma_1-2\gamma_2-4)(\beta-1)
-\gamma_1\gamma_2+1\bigr). \end{split}\end{equation} The equation $Pu=0$ is isomorphic to the system \begin{equation} \begin{gathered}
\frac{d\tilde u}{dx}=\frac{A}x\tilde u+\frac{B}{x-1}\tilde u,\\
A=\begin{pmatrix}
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & c & 0\\
0 & 0 & 0 & c
\end{pmatrix},\
B=\begin{pmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
s & 1 & a & 0\\
r & t & 0 & b
\end{pmatrix},\
\tilde u=\begin{pmatrix}
u_1\\u_2\\u_3\\u_4
\end{pmatrix} \end{gathered} \end{equation} by the correspondence \begin{align*}
&\begin{cases}u_1=u,\\
u_2=(x-1)xu''+\bigl((1-a-c)x+a-1\bigr)u'-su,\\
u_3=xu',\\
u_4=x^2(x-1)u'''+\bigl((3-a-c)x^2+(a-2)x\bigr)u''+(1-a-c-s)xu',
\end{cases} \end{align*} where we may assume $\RE\gamma_1\ge\RE\gamma_2$ and \[
\begin{split}
\beta&=c,\
\gamma_1=a+1,\
\gamma_2=b+2,\\
\prod_{\nu=1}^4(\xi-\alpha_\nu)&=\xi^4+(a+b+2c)\xi^3
+\bigl((a+c)(b+c)-s-t\bigr)\xi^2\\
&\quad{}-\bigl((b+c)s+(a+c)t\bigr)\xi+st-r.
\end{split} \] Here $s$, $t$ and $r$ are uniquely determined from $\alpha_1,\alpha_2,\alpha_3,\alpha_4,\beta,\gamma_1,\gamma_2$ because $b+c\ne a+c$. We remark that $\Ad(x^{-c})\tilde u$ satisfies a system of Okubo normal form.
Note that the shift of parameters $(\alpha_1,\dots,\alpha_4,\beta,\gamma_1,\gamma_2) \mapsto(\alpha_1,\dots,\alpha_4,\beta-1,\gamma_1+1,\gamma_2+1)$ corresponds to the shift $(a,b,c,s,t,r)\mapsto(a+1,b+1,c-1,s,t,r)$.
Let $u^j_{\alpha_1,\dots,\alpha_4,\beta,\gamma_1,\gamma_2}(x)$ be local holomorphic solutions of $Pu=0$ in a neighborhood of $x=0$ determined by \begin{align*}
u^j_{\alpha_1,\dots,\alpha_4,\beta,\gamma_1,\gamma_2}(0)&=\delta_{j,0},\\
\bigl(\tfrac{d}{dx}
u^j_{\alpha_1,\dots,\alpha_4,\beta,\gamma_1,\gamma_2}\bigr)
(0)&=\delta_{j,1} \end{align*} for $j=0$ and $1$. Then Theorem~\ref{thm:paralimits} proves \begin{align*}
\lim_{k\to\infty}\tfrac{d^\nu}{dx^\nu}
u^0_{\alpha,\beta-k,\gamma_1+k,\gamma_1+k}(x)&=\delta_{0,\nu}
\quad(\nu=0,1,2,\ldots) \end{align*}
uniformly on $\overline D=\{x\in\mathbb C\,;\,|x|\le 1\}$.
Put $u=v_{\alpha,\beta,\gamma_1,\gamma_2}= (\gamma_1-2)^{-1}u^1_{\alpha,\beta,\gamma}$. Then Theorem~\ref{thm:paralimits} proves \begin{align*}
&\lim_{k\to\infty}\tfrac{d^\nu}{dx^\nu}
v_{\alpha,\beta-k,\gamma_1+k,\gamma_2+k}(x)=0\quad(\nu=0,1,2,\ldots),\\
&\lim_{k\to\infty}\Bigl( (x-1)x\tfrac{d^2}{dx^2}+\bigl((2-\beta-\gamma_1)x+\gamma_1+k-2\bigr)\tfrac{d}{dx}
-s\Bigr)
v_{\alpha,\beta-k,\gamma_1+k,\gamma_2+k}(x)=1 \end{align*} uniformly on $\overline D$. Hence \begin{align*}
\lim_{k\to\infty}\bigl(\tfrac{d}{dx}u^1_{\alpha,\beta-k,\gamma+1,\gamma_1+k}\bigr)
(x)=1 \end{align*} uniformly on $\overline D$. Thus we obtain \[
\lim_{k\to\infty}
c(\infty\!:\![a_2]_{(2)}\rightsquigarrow1\!:\![0]_{(2)})
|_{a_1\mapsto a_1-k,\ c_1\mapsto c_1+k,\ c_2\mapsto c_2+k}=1 \] for the connection coefficient in \eqref{eq:e4dc2}. Then the procedure given in Remark~\ref{rem:Cproc} and Corollary~\ref{cor:zeroC} with the rigid decompositions \begin{align*} 2\underline{2},1111,\overline{2}11
&=1\underline{2},0111,\overline{1}11
\oplus 1\underline{0},1000,100
=1\underline{2},1011,\overline{1}11
\oplus 1\underline{0},0100,\overline{1}00\\
&=1\underline{2},1101,\overline{1}11
\oplus 1\underline{0},0010,\overline{1}00
=1\underline{2},1101,\overline{1}11
\oplus 1\underline{0},0010,\overline{1}00 \end{align*} prove \eqref{eq:e4dc2}. Corresponding to Remark~\ref{rem:Cproc} (4), we note \[ \sum_{\nu=1}^2(c_\nu-1)+\sum_{i=0}^1(a_2-a_1-i)=
(1-a_1)+\sum_{\nu=1}^3(1-a_1-b_\nu) \] because of the Fuchs relation \eqref{eq:e4Fuchs}. We can similarly obtain \eqref{eq:e4dc1}.
The holomorphic solution of $\tilde Pu=0$ at the origin is given by \begin{align*}\begin{split}
&u_0(x)=\sum_{m\ge0,\ n\ge0}\\
&\frac{(a_1+a_2+b_3+c_2-1)_{n}
\prod_{\nu=1}^2\bigl((a_\nu)_{m+n}(a_1+a_2+b_\nu+c_1-1)_{m}\bigr) }
{(1-b_1)_{m+n}(1-b_2)_{m+n}(1-b_3)_{m}m!n!}x^{m+n} \end{split}\end{align*} and it has the integral representation \begin{align*}\begin{split}
u_0(x)&=\frac{
\prod_{\nu=1}^3\Gamma(1-b_\nu)
}{
\prod_{\nu=1}^2\bigl(\Gamma(a_\nu)\Gamma(1-a_\nu-b_\nu)
\Gamma(b_\nu+c_\nu+a_1+a_2-1)\bigr)}\\
&\quad\int_0^{x}\int_0^{s_0}\int_0^{s_1}x^{b_1}(x-s_0)^{-b_1-a_1}
s_0^{b_2+a_1-1}(s_0-s_1)^{-b_2-a_2}\\
&\quad\cdot s_1^{b_3+a_2-1}(1-s_1)^{-b_3-c_1-a_2-a_1+1}
(s_1-s_2)^{c_1+b_1+a_2+a_1-2}\\
&\quad\cdot s_2^{b_2+c_2+a_2+a_1-2}(1-s_2)^{-c_2-b_1-a_2-a_1+1}ds_2ds_1ds_0. \end{split}\end{align*} The equation is irreducible if and only if any value of the following linear functions is not an integer. \index{even family!reducibility} \begin{align*}\begin{split}
&a_1\quad a_2\\%
&a_1+b_1\quad a_1+b_2\quad a_1+b_3\quad a_2+b_1\quad a_2+b_2\quad a_2+b_3\\%
&a_1+a_2+b_1+c_1-1\quad a_1+a_2+b_1+c_2-1\quad a_1+a_2+b_2+c_1-1\\
&a_1+a_2+b_2+c_2-1\quad a_1+a_2+b_3+c_1-1\quad a_1+a_2+b_2+c_2-1. \end{split}\end{align*}
In the same way we have the connection coefficients for odd family.
\noindent \index{odd family!connection coefficient} $EO_{2m+1}$ ($\mathbf m=(1^{2m+1},mm1,m+1m)$ :\ odd family) \index{odd family} \[
\begin{Bmatrix}
x=\infty & 0 & 1\\
\lambda_{0,1} & [\lambda_{1,1}]_{(m)} & [\lambda_{2,1}]_{(m+1)}\\
\vdots & [\lambda_{1,2}]_{(m)} & [\lambda_{2,2}]_{(m)}\\
\lambda_{0,2m+1} & \lambda_{1,3}
\end{Bmatrix} \]
$\sum_{\nu=1}^{2m+1}\lambda_{0,\nu} +m(\lambda_{1,1}+\lambda_{1,2}+\lambda_{2,2}) +(m+1)\lambda_{2,1}+\lambda_{1,3}=2m.$
\phantom{.} \begin{align*}
c(\lambda_{0,2m+1}\rightsquigarrow\lambda_{1,3})
&=\prod_{k=1}^2
\frac{\Gamma\bigl(\lambda_{1,k}-\lambda_{1,3}\bigr)
}{\Gamma\bigl(
\left|\begin{Bmatrix}
\lambda_{0,2m+1} & \lambda_{1,k} &\lambda_{2,1}
\end{Bmatrix}\right|
\bigr)}
\\
&\quad
\cdot
\prod_{k=1}^{2m}\frac
{\Gamma\bigl(\lambda_{0,2m+1}-\lambda_{0,k}+1)}
{\Gamma\bigl(
\left|\begin{Bmatrix}
\lambda_{0,k} & \lambda_{1,1} & \lambda_{2,1}\\
\lambda_{0,2m+1} & \lambda_{1,2} & \lambda_{2,2}
\end{Bmatrix}\right|
\bigr)},\allowdisplaybreaks\\
c(\lambda_{1,3}\rightsquigarrow\lambda_{0,2m+1})
&=\displaystyle\prod_{k=1}^2\frac
{\Gamma\bigl(\lambda_{1,3}-\lambda_{1,k}+1\bigr)
}{\Gamma\bigl(
\left|\begin{Bmatrix}
&[\lambda_{1,k}]_{(m)}&[\lambda_{2,1}]_{(m)}\\
(\lambda_{0,\nu})_{1\le \nu\le 2m}
&[\lambda_{1,3-k}]_{(m-1)}&[\lambda_{2,2}]_{(m)}
\\
&\lambda_{1,3}
\end{Bmatrix}\right|
\bigr)}
\\
&\quad
\cdot
\prod_{k=1}^{2m}\frac
{\Gamma\bigl(\lambda_{0,k}-\lambda_{0,2m+1})}
{\Gamma\bigl(
\left|\begin{Bmatrix}
& [\lambda_{1,1}]_{(m-1)}&[\lambda_{2,1}]_{(m)}\\
(\lambda_{0,\nu})_{\substack{1\le \nu\le 2m\\ \nu\ne k}}
&[\lambda_{1,2}]_{(m-1)}&[\lambda_{2,2}]_{(m-1)}\\
&\lambda_{1,3}&
\end{Bmatrix}\right|
\bigr)}. \end{align*}
The condition for the irreducibility is \index{odd family!reducibility} \begin{equation*}
\begin{cases}
\lambda_{0,\nu}+\lambda_{1,k}+\lambda_{2,1}\notin\mathbb Z
&(1\le\nu\le 2m+1,\ k=1,2),\\
\lambda_{0,\nu}+\lambda_{0,\nu'}+\lambda_{1,1}+\lambda_{1,2}+\lambda_{2,1}
+\lambda_{2,2}-1\notin\mathbb Z
&(1\le\nu<\nu'\le 2m+1,\ k=1,2).
\end{cases} \end{equation*} The same statement using the above linear functions as in the case of even family is valid for the bijectivity of the shift operator with respect to compatible shift $(\epsilon_{j,\nu})$.
We note that the operation $\RAd(\partial^{-\mu})\circ \RAd\bigl(x^{-\lambda_{1,2}}(1-x)^{-\lambda_{2,2}}\bigr)$ transforms the operator and solutions with the above Riemann scheme of type $EO_n$ into those of type $EO_{n+1}$: \begin{align*}
&\begin{Bmatrix}
\lambda_{0,1} & [\lambda_{1,1}]_{([\frac n2])}
&[\lambda_{2,1}]_{([\frac{n+1}2])}\\
\vdots & [\lambda_{1,2}]_{([\frac {n-1}2])}
&[\lambda_{2,2}]_{([\frac{n}2])}\\
\lambda_{0,n}&\lambda_{1,3}\\
\end{Bmatrix} \allowdisplaybreaks\\ &\xrightarrow{x^{-\lambda_{1,2}}(1-x)^{-\lambda_{2,2}}} \begin{Bmatrix}
\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,2} & [\lambda_{1,1}-\lambda_{1,2}]_{([\frac n2])}
&[\lambda_{2,1}-\lambda_{2,2}]_{([\frac{n+1}2])}\\
\vdots & [0]_{([\frac {n-1}2])}
&[0]_{([\frac{n}2])}\\
\lambda_{0,n}+\lambda_{1,2}+\lambda_{2,2}&\lambda_{1,3}-\lambda_{1,2}\\
\end{Bmatrix} \allowdisplaybreaks\\ &\xrightarrow{\partial^{-\mu}} \begin{Bmatrix}
\lambda_{0,1}+\lambda_{1,2}+\lambda_{2,2}-\mu
& [\lambda_{1,1}-\lambda_{1,2}+\mu]_{([\frac n2])}
&[\lambda_{2,1}-\lambda_{2,2}+\mu]_{([\frac{n+1}2])}\\
\vdots & [\mu]_{([\frac {n+1}2])}
&[\mu]_{([\frac{n+2}2])}\\
\lambda_{0,n}+\lambda_{1,2}+\lambda_{2,2}-\mu
&\lambda_{1,3}-\lambda_{1,2}+\mu\\
1-\mu
\end{Bmatrix}. \end{align*} \subsection{Trigonometric identities}\label{sec:TriEx} The connection coefficients corresponding to the Riemann scheme of the hypergeometric family in \S\ref{sec:GHG} satisfy \[ \begin{split} \sum_{\nu=1}^nc(1:\lambda_{1,2}\!\rightsquigarrow\!0:\lambda_{0,\nu}) \cdot c(0:\lambda_{1,\nu}\!\rightsquigarrow\!1:\lambda_{1,2})=1,\\ \sum_{\nu=1}^nc(\infty:\lambda_{2,i}\!\rightsquigarrow\!0:\lambda_{0,\nu}) \cdot c(0:\lambda_{0,\nu}\!\rightsquigarrow\!\infty:\lambda_{2,j})=\delta_{ij}. \end{split} \] These equations with Remark~\ref{rem:conn} iii) give the identities \begin{gather*} \sum_{k=1}^n\frac{\prod_{\nu\in\{1,\dots,n\}}\sin(x_k-y_\nu)}
{\prod_{\nu\in\{1,\ldots,n\}\setminus\{k\}}\sin(x_k-x_\nu)}
= \sin\Bigl(\sum_{\nu=1}^n x_\nu-\sum_{\nu=1}^n y_\nu\Bigr),\\ \sum_{k=1}^n\prod_{\nu\in\{1,\dots,n\}\setminus\{k\}}
\frac{\sin(y_i-x_\nu)}{\sin(x_k-x_\nu)}
\prod_{\nu\in\{1,\dots,n\}\setminus\{j\}}
\frac{\sin(x_k-y_\nu)}{\sin(y_j-y_\nu)}
= \delta_{ij}\quad(1\le i,\,j\le n). \end{gather*} We have the following identity from the connection coefficients of even/odd families. \begin{align*}
&\sum_{k=1}^n\sin(x_k+s)\cdot\sin(x_k+t)\cdot
\prod_{\nu\in\{1,\ldots,n\}\setminus\{k\}}
\frac{\sin(x_k+x_\nu+2u)}{\sin(x_k-x_\nu)}\\
&=\begin{cases}
\sin\Bigl(nu+\displaystyle\sum_{\nu=1}^nx_\nu\Bigr)\cdot
\sin\Bigl(s+t+(n-2)u+\sum_{\nu=1}^nx_\nu\Bigr)
\qquad\qquad\ \,\text{if \ }n=2m,\\[5pt]
\sin\Bigl(s+(n-1)u+\displaystyle\sum_{\nu=1}^nx_\nu\Bigr)\cdot
\sin\Bigl(t+(n-1)u+\sum_{\nu=1}^nx_\nu\Bigr)
\qquad\text{if \ }n=2m+1.
\end{cases}\notag \end{align*} The direct proof of these identities using residue calculus is given by \cite{Oc}. It is interesting that similar identities of rational functions are given in \cite[Appendix]{Gl} which studies the systems of Schlesinger canonical form corresponding to Simpson's list (cf.~\S\ref{sec:rigidEx}).
\subsection{Rigid examples of order at most 4}\label{sec:4Ex} \index{tuple of partitions!rigid!order$\ \le4$} \index{000Delta1@$[\Delta(\mathbf m)]$} \subsubsection{order $1$} \underline{$1,1,1$} \begin{equation*}
u(x)=x^{\lambda_1}(1-x)^{\lambda_2}
\qquad
\begin{Bmatrix}-\lambda_1-\lambda_2&\lambda_1&\lambda_2\end{Bmatrix} \end{equation*} \subsubsection{order $2$} \underline{$11,11,11$} : $H_2$ (Gauss)\qquad$[\Delta(\mathbf m)]=1^4$ \begin{equation*}
u_{H_2} = \partial^{-\mu_1}u(x)
\qquad
\begin{Bmatrix}
-\mu_1+1& 0& 0\\
-\lambda_1-\lambda_2-\mu_1&\lambda_1+\mu_1&\lambda_2+\mu_1
\end{Bmatrix} \end{equation*} \subsubsection{order $3$} There are two types.
$\underline{111,21,111} : H_3\ ({}_3F_2)$ \qquad$[\Delta(\mathbf m)]=1^9$ \begin{align*} &u_{H_3} = \partial^{-\mu_2}x^{\lambda_3}u_{H_2} \\ & \begin{Bmatrix}
1-\mu_2 &0 & [0]_{(2)}\\
-\lambda_3-\mu_1-\mu_2+1& \lambda_3+\mu_2\\
-\lambda_1-\lambda_2-\lambda_3-\mu_1-\mu_2
&\lambda_1+\lambda_3+\mu_1+\mu_2&\lambda_2+\mu_1+\mu_2
\end{Bmatrix} \end{align*}
\underline{$21,21,21,21$} : $P_3$ (Jordan-Pochhammer) \qquad$[\Delta(\mathbf m)]=1^4\cdot2$ \begin{align*}
&u_{P_3} = \partial^{-\mu}x^{\lambda_0}(1-x)^{\lambda_1}(c_2-x)^{\lambda_2}\\
&
\begin{Bmatrix}
[1-\mu]_{(2)}& [0]_{(2)} & [0]_{(2)} & [0]_{(2)}\\
-\lambda_0-\lambda_1-\lambda_2-\mu &
\lambda_0+\mu &\lambda_1+\mu &\lambda_2+\mu
\end{Bmatrix} \end{align*} \subsubsection{order 4} There are 6 types.
\index{tuple of partitions!rigid!211,211,211} \underline{$211,211,211$}: $\alpha_2$ \qquad$[\Delta(\mathbf m)]=1^{10}\cdot2$ \begin{align*}
&\partial^{-\mu_2}x^{\lambda_3}(1-x)^{\lambda_4}u_{H_2}\\
&\begin{Bmatrix}
[-\mu_2+1]_{(2)} & [0]_{(2)}& [0]_{(2)}\\
-\mu_1-\lambda_3-\lambda_4-\mu_2+1& \lambda_3+\mu_2& \lambda_4+\mu_2\\
-\lambda_1-\lambda_2-\lambda_3-\lambda_4-\mu_1-\mu_2
&\lambda_1+\lambda_3+\mu_1+\mu_2
&\lambda_2+\lambda_4+\mu_1+\mu_2
\end{Bmatrix} \end{align*}
\underline{$1111,31,1111$} : $H_4\ ({}_4F_3)$ \qquad$[\Delta(\mathbf m)]=1^{16}$ {\small\begin{align*}
&\partial^{-\mu_3}x^{\lambda_4}u_{H_3}
\\
&\begin{Bmatrix}
-\mu_3+1&0& [0]_{(3)}\\
-\lambda_4-\mu_2-\mu_3+1 &\lambda_4 \\
-\lambda_3-\lambda_4-\mu_1-\mu_2-\mu_3+1
& \lambda_3+\lambda_4+\mu_2+\mu_3\\
-\lambda_1-\cdots-\lambda_4-\mu_1-\mu_2-\mu_3
&\lambda_1+\cdots+\lambda_4+\mu_1+\mu_2+\mu_3&\lambda_2+\mu_1+\mu_2+\mu_3
\end{Bmatrix} \end{align*}}
\underline{$211,22,1111$} : $EO_4$ \qquad$[\Delta(\mathbf m)]=1^{14}$ {\small\begin{align*}
&\partial^{-\mu_3}(1-x)^{-\lambda'}u_{H_3},\quad
\lambda'=\lambda_2+\mu_1+\mu_2\\
&
\begin{Bmatrix}
\lambda_2+\mu_1-\mu_2-\mu_3+1&[0]_{(2)}
& [-\lambda_2-\mu_1-\mu_2+\mu_3]_{(2)}\\
\lambda_2-\lambda_3-\mu_3+1& \lambda_3+\mu_2+\mu_3\\
-\lambda_1-\lambda_3-\mu_3
&\lambda_1+\lambda_3+\mu_1+\mu_2+\mu_3&[0]_{(2)}\\
-\mu_3+1\\
\end{Bmatrix} \end{align*}} We have the integral representation of the local solution corresponding to the exponent at 0: \[
\int_0^x\int_0^t\int_0^s (1-t)^{-\lambda_2-\mu_1-\mu_2}(x-t)^{\mu_3-1}
s^{\lambda_3}(t-s)^{\mu_2-1}
u^{\lambda_1}(1-u)^{\lambda_2}(s-u)^{\mu-1}du\,ds\,dt. \]
\underline{$211,22,31,31$}: $I_4$ \qquad$[\Delta(\mathbf m)]=1^6\cdot 2^2${\small \begin{align*}
&\partial^{-\mu_2}(c_2-x)^{\lambda_3}u_{H_2}\\
&\begin{Bmatrix}
[-\mu_2+1]_{(2)}&[0]_{(3)}&[0]_{(3)}&[0]_{(2)}\\
-\lambda_3-\mu_1-\mu_2+1& &
&[\lambda_3+\mu_2]_{(2)}\\
-\lambda_1-\lambda_2-\lambda_3-\mu_1-\mu_2
&\lambda_1+\mu_1+\mu_2&\lambda_2+\mu_1+\mu_2
\end{Bmatrix} \end{align*}}
\underline{$31,31,31,31,31$}: $P_4$ \qquad$[\Delta(\mathbf m)]=1^5\cdot3$ \begin{align*}
&u_{P_4}=\partial^{-\mu}x^{\lambda_0}(1-x)^{\lambda_1}(c_2-x)^{\lambda_2}(c_3-x)^{\lambda_3}
\\&
\begin{Bmatrix}
[-\mu+1]_{(3)} & [0]_{(3)} & [0]_{(3)} & [0]_{(3)} & [0]_{(3)}\\
-\lambda_0-\lambda_2-\lambda_3-\mu
&\lambda_0+\mu & \lambda_1+\mu & \lambda_2+\mu & \lambda_3+\mu
\end{Bmatrix} \end{align*}
\underline{$22,22,22,31$}: $P_{4,4}$ \qquad$[\Delta(\mathbf m)]=1^8\cdot2$ \begin{gather*}
\partial^{-\mu'}x^{-\lambda_0'}(1-x)^{-\lambda_1'}(c_2-x)^{-\lambda_2'}u_{P_3},
\quad
\lambda_j' = \lambda_j+\mu,\quad
\mu' = \lambda_0+\lambda_1+\lambda_2+2\mu\\
\begin{Bmatrix}
[1-\mu']_{(3)}& [\lambda_1+\lambda_2+\mu]_{(2)} & [\lambda_0+\lambda_2+\mu]_{(2)} & [\lambda_0+\lambda_1+\mu]_{(2)}\\
-\lambda_0-\lambda_1-\lambda_2& [0]_{(2)} &[0]_{(2)} &[0]_{(2)}
\end{Bmatrix} \end{gather*}
\subsubsection{Tuple of partitions $:\ 211,211,211$}\label{sec:eq211} \index{tuple of partitions!rigid!211,211,211} \qquad$[\Delta(\mathbf m)]=1^{10}\cdot 2$ \[211,211,211 = H_1\oplus H_3:6 = H_2\oplus H_2:4
= 2H_1\oplus H_2:1\] From the operations \begin{align*}
&\begin{Bmatrix}
x=\infty&0&1\\
1-\mu_1 & 0 & 0\\
-\alpha_1-\beta_1-\mu_1&\underline{\alpha_1+\mu_1}&\beta_1+\mu_1
\end{Bmatrix}\\
&\xrightarrow{x^{\alpha_2}(1-x)^{\beta_2}}{}
\begin{Bmatrix}
x=\infty&0&1\\
1-\alpha_2-\beta_2-\mu_1 & \alpha_2 & \beta_2\\
-\alpha_1-\alpha_2-\beta_1-\beta_2-\mu_1&
\underline{\alpha_1+\alpha_2+\mu_1}
&\beta_1+\beta_2+\mu_1
\end{Bmatrix}\\
&\xrightarrow{\partial^{-\mu_2}}{}
\begin{Bmatrix}
x=\infty&0&1\\
[-\mu_2+1]_{(2)} & [0]_{(2)} & [0]_{(2)}\\
1-\beta_2-\mu_1-\mu_2 &\alpha_2+\mu_2 &\beta_2+\mu_2\\
-\alpha_1-\beta_1-\beta_2-\mu_1-\mu_2
&\underline{\alpha_1+\mu_1+\mu_2}&\beta_1+\beta_2+\mu_1+\mu_2
\end{Bmatrix}\allowdisplaybreaks\\
&\longrightarrow
\begin{Bmatrix}
x=\infty&0&1\\
[\lambda_{2,1}]_{(2)} & [\lambda_{0,1}]_{(2)} & [\lambda_{1,1}]_{(2)}\\
\lambda_{2,2} & \lambda_{0,2} &\lambda_{1,2}\\
\lambda_{2,3} &\lambda_{0,3}&\lambda_{1,3}
\end{Bmatrix}\quad\text{with}\quad
\sum_{j=0}^2
(2\lambda_{j,1}+\lambda_{j,2}+\lambda_{j,3})=3, \end{align*} we have the integral representation of the solutions as in the case of other examples we have explained and so here we will not discuss them. The universal operator of type $11,11,11$ is \begin{align*}
Q=x^2(1-x)^2\partial^2 -(ax+b)x(1-x)\partial + (cx^2+dx+e). \end{align*} Here we have \begin{align*}
b&=\lambda_{0,1}'+\lambda_{0,2}'-1,&e&=\lambda_{0,1}'\lambda_{0,2}',\\
-a-b&=\lambda_{1,1}'+\lambda_{1,2}'-1,&c+d+e&=\lambda_{1,1}'\lambda_{1,2}',\\
&&c&=\lambda_{2,1}'\lambda_{2,2}',\\ \lambda_{0,1}'&=\alpha_2,&\lambda_{0,2}'&=\alpha_1+\alpha_2+\mu_1,\\
\lambda_{1,1}'&=\beta_2,&\lambda_{1,2}'&=\beta_1+\beta_2+\mu_2,\\
\lambda_{2,1}'&=1-\beta_2-\mu_1-\mu_2,&
\lambda_{2,2}'&=-\alpha_1-\beta_1-\beta_2-\mu_1-\mu_2 \end{align*} corresponding to the above second Riemann scheme. The operator corresponding to the tuple $211,211,211$ is \begin{align*} P&=\RAd(\partial^{-\mu_2}) Q\allowdisplaybreaks\\
&=\RAd(\partial^{-\mu_2})\Bigl(
(\vartheta-\lambda_{0,1}')(\vartheta-\lambda_{0,2}')\\ &\quad{}
+x\bigl(-2\vartheta^2+(2\lambda_{0,1}'+2\lambda_{0,2}'+\lambda_{1,1}'
+\lambda_{1,2}'-1)\vartheta+\lambda_{1,1}'\lambda_{1,2}'
-\lambda_{0,1}'\lambda_{0,2}'-\lambda_{2,1}'\lambda_{2,2}'\bigr)
\\
&\quad{}
+ x^2(\vartheta+\lambda_{2,1}')(\vartheta+\lambda_{2,2}')\Bigr)
\allowdisplaybreaks\\
&=\partial^2 (\vartheta-\lambda_{0,1}'-\mu_2)
(\vartheta-\lambda_{0,2}'-\mu_2)\\ &\quad{}
+\partial(\vartheta-\mu_2+1)
\bigl(-2(\vartheta-\mu_2)^2+(2\lambda_{0,1}'+2\lambda_{0,2}'+\lambda_{1,1}'
+\lambda_{1,2}'-1)(\vartheta-\mu_2)\\
&\quad{}+\lambda_{1,1}'\lambda_{1,2}'
-\lambda_{0,1}'\lambda_{0,2}'-\lambda_{2,1}'\lambda_{2,2}'\bigr)\\
&\quad{}
+ (\vartheta-\mu_2+1)(\vartheta-\mu_2+2)(\vartheta+\lambda_{2,1}'-\mu_2)(\vartheta+\lambda_{2,2}'-\mu_2). \end{align*}
\noindent The condition for the irreducibility: \begin{equation*}
\begin{cases}
\lambda_{0,1}+\lambda_{1,1}+\lambda_{2,1}\notin\mathbb Z,\\
\lambda_{0,\nu}+\lambda_{1,1}+\lambda_{2,1}\notin\mathbb Z,\
\lambda_{0,1}+\lambda_{1,\nu}+\lambda_{2,1}\notin\mathbb Z,\
\lambda_{0,1}+\lambda_{1,1}+\lambda_{2,\nu}\notin\mathbb Z\ \ (\nu=2,3),\\
\lambda_{0,1}+\lambda_{0,2}+\lambda_{1,1}+\lambda_{1,\nu}+\lambda_{2,1}
+\lambda_{2,\nu'}\notin\mathbb Z\ \ (\nu,\,\nu'\in\{2,3\}).
\end{cases} \end{equation*} There exist three types of direct decompositions of the tuple and there are 4 direct decompositions which give the connection coefficient $c(\lambda_{0,3}\!\rightsquigarrow\!\lambda_{1,3})$ by the formula \eqref{eq:connection} in Theorem~\ref{thm:c}:
\begin{align*}
21\overline{1},21\underline{1},211
&=00\overline{1},100,100\oplus210,11\underline{1},111\\
&=11\overline{1},210,111\oplus100,00\underline{1},100\\
&=10\overline{1},110,110\oplus110,10\underline{1},101\\
&=10\overline{1},110,101\oplus110,10\underline{1},110 \end{align*}
Thus we have \begin{align*}
&c(\lambda_{0,3}\!\rightsquigarrow\!\lambda_{1,3})
=\frac{\prod_{\nu=1}^2\Gamma(\lambda_{0,3}-\lambda_{0,\nu}+1)}
{\Gamma(\lambda_{0,3}+\lambda_{1,1}+\lambda_{2,1})\cdot
\Gamma(1-\lambda_{0,1}-\lambda_{1,3}-\lambda_{2,1})}\\
&\phantom{c(\lambda_{0,3}\!\rightsquigarrow\!\lambda_{1,3})=}
\quad\cdot
\frac{\prod_{\nu=1}^2\Gamma(\lambda_{1,\nu}-\Gamma_{1,3})}
{\prod_{\nu=2}^3
\Gamma(\lambda_{0,1}+\lambda_{0,3}+\lambda_{1,1}+\lambda_{1,2}
+\lambda_{2,1}+\lambda_{2,\nu}-1)}. \end{align*}
\index{Wronskian} We can also calculate generalized connection coefficient defined in Definition~\ref{def:GC}: \index{connection coefficient!generalized} \begin{align*}
&c([\lambda_{0,1}]_{(2)}\!\rightsquigarrow\![\lambda_{1,1}]_{(2)})
=\frac{
\prod_{\nu=2}^3\bigl(\Gamma(\lambda_{0,1}-\lambda_{0,\nu}+2)
\cdot\Gamma(\lambda_{1,\nu}-\lambda_{1,1}-1)\bigr)
}{
\prod_{\nu=2}^3\bigl(
\Gamma(\lambda_{0,1}+\lambda_{1,\nu}+\lambda_{2,1})
\cdot\Gamma(1-\lambda_{0,\nu}-\lambda_{1,1}-\lambda_{2,1})
\bigr)
}. \end{align*} This can be proved by the procedure given in Remark~\ref{rem:Cproc} as in the case of the formula \eqref{eq:e4dc2}. Note that the gamma functions in the numerator of this formula correspond to Remark~\ref{rem:Cproc} (2) and those in the denominator correspond to the rigid decompositions \[
\begin{split}
\underline{2}11,\overline{2}11,211
&=\underline{1}00,\overline{0}10,100\oplus
\underline{1}11,\overline{2}01,111
=\underline{1}00,\overline{0}01,100\oplus
\underline{1}11,\overline{2}10,111\\
&=\underline{2}10,\overline{1}11,111\oplus
\underline{0}01,\overline{1}00,100
=\underline{2}01,\overline{1}11,111\oplus
\underline{0}10,\overline{1}00,100.
\end{split} \] The equation $Pu=0$ with the Riemann scheme $\begin{Bmatrix} x=\infty & 0 & 1\\ [\lambda_{0,1}]_{(2)} & [0]_{(2)} & [0]_{(2)}\\ \lambda_{0,2} & \lambda_{1,2} & \lambda_{2,2}\\ \lambda_{0,3} & \lambda_{1,3} & \lambda_{2,3} \end{Bmatrix}$ is isomorphic to the system \begin{align*}
\tilde u'&=\frac Ax\tilde u+\frac B{x-1}\tilde u,\quad
\tilde u=\begin{pmatrix}u_1\\u_2\\u_3\\u_4\end{pmatrix},\ \ u_1=u, \allowdisplaybreaks\\
A&=\begin{pmatrix}
0 & 0 & c_1 & 0\\
0 & 0 & 0 & c_1\\
0 & 0 & a_1 & b_1-b_2-c_2\\
0 & 0 & 0 & a_2
\end{pmatrix}, \allowdisplaybreaks\\
B&=\begin{pmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
-a_1-b_2+c_1& -b_1+b_2+c_2& b_2 &0 \\
-a_1+a_2+c_2& -a_2-b_1+c_1& a_1-a_2-c_2& b_1
\end{pmatrix}, \allowdisplaybreaks\\
&\begin{cases}
a_1=\lambda_{1,2},\\
a_2=\lambda_{1,3},\\
b_1=\lambda_{2,2}-2,\\
b_2=\lambda_{2,3}-1,\\
c_1 = -\lambda_{0,1},\\
c_2=\lambda_{0,1}+\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,2}-1
\end{cases} \end{align*} when $\lambda_{0,1}(\lambda_{0,1}+\lambda_{2,2}) (\lambda_{0,1}+\lambda_{0,2}+\lambda_{1,2}+\lambda_{2,3}-2)\ne0$. Let $u(x)$ be a holomorphic solution of $Pu=0$ in a neighborhood of $x=0$. By a direct calculation we have \begin{align*}
&u_1(0)=\frac{(a_1-1)(a_2-1)}{(b_1-c_1+1)(b_1-b_2-c_2)c_1}u'(0)+\\ &\ \ \frac {(a_2+b_2+c_2-1)a_1-(c_1+c_2)a_2+(a_2-a_1+c_2)b_1-(c_2+1)b_2-c_2^2+c_1}{ (b_1-c_1+1)(b_1-b_2-c_2)}u(0). \end{align*} Since the shift described in Remark~\ref{rem:Cproc} (1) corresponds to the shift \[
(a_1,a_2,b_1,b_2,c_1,c_2)\mapsto
(a_1-k,a_2-k,b_1+k,b_2+k,c_1,c_2), \] it follows from Theorem~\ref{thm:paralimits} that \[
\lim_{k\to\infty}
c([\lambda_{0,1}]_{(2)}\!\rightsquigarrow\![\lambda_{1,1}]_{(2)})
\Bigl|_{\substack{\substack\lambda_{0,2}\mapsto \lambda_{0,2}-k,\
\substack\lambda_{0,3}\mapsto \lambda_{0,3}-k\\
\lambda_{1,2}\mapsto \lambda_{1,2}+k,\
\lambda_{1,3}\mapsto \lambda_{1,3}+k}}=1 \] as in the proof of \eqref{eq:e4dc2} because $u_1(0)\sim \frac k{(b_1-b_2-c_2)c_1}u'(0)+Cu(0)$ with $C\in \mathbb C$ when $k\to\infty$. Thus we can calculate this generalized connection coefficient by the procedure described in Remark~\ref{rem:Cproc}.
Using \eqref{eq:I0H}, we have the series expansion of the local solution at $x=0$ corresponding to the exponent $\alpha_1+\mu_1+\mu_2$ for the Riemann scheme parametrized by $\alpha_i$, $\beta_i$ and $\mu_i$ with $i=1,\,2$. \begin{align*}
&I_0^{\mu_2}x^{\alpha_2}(1-x)^{\beta_2}I_0^{\mu_1}x^{\alpha_1}(1-x)^{\beta_1}
\allowdisplaybreaks\\&
=I_0^{\mu_2} \frac{\Gamma(\alpha_1+1)}{\Gamma(\alpha_1+\mu+1)}
\sum_{n=0}^\infty
\frac{(\alpha_1+1)_n(-\beta_1)_n}{(\alpha_1+\mu+1)_nn!}
x^{\alpha_2}(1-x)^{\beta_2}x^{\alpha_1+\mu+n} \allowdisplaybreaks\\ &=\frac{\Gamma(\alpha_1+1)\Gamma(\alpha_1+\alpha_2+\mu_1+1)
x^{\alpha_1+\alpha_2+\mu_1+\mu_2}}
{\Gamma(\alpha_1+\mu_1+1) \Gamma(\alpha_1+\alpha_2+\mu_1+\mu_2+1)}\\ &\quad\cdot
\sum_{m,n=0}^\infty \frac{(\alpha_1+1)_n(\alpha_1+\alpha_2+\mu_1+1)_{m+n}
(-\beta_1)_n(-\beta_2)_m}
{(\alpha_1+\mu_1+1)_{n} (\alpha_1+\alpha_2+\mu_1+\mu_2+1)_{m+n}n!m!}x^{m+n} \allowdisplaybreaks\\
&=\frac{\Gamma(\alpha_1+1)\Gamma(\alpha_1+\alpha_2+\mu_1+1)
x^{\alpha_1+\alpha_2+\mu_1+\mu_2}(1-x)^{-\beta_2}}
{\Gamma(\alpha_1+\mu_1+1) \Gamma(\alpha_1+\alpha_2+\mu_1+\mu_2+1)}\\
&\quad \cdot\sum_{m,\,n=0}^\infty
\frac{(\alpha_1+1)_n(\alpha_1+\alpha_2+\mu_1+1)_n(\mu_2)_m(-\beta_1)_n
(-\beta_2)_m}
{(\alpha_1+\mu_1+1)_n(\alpha_1+\alpha_2+\mu_1+\mu_2+1)_{m+n}m!n!}
x^n\Bigl(\frac x{x-1}\Bigr)^m. \end{align*} Note that when $\beta_2=0$, the local solution is reduced to a local solution of the equation at $x=0$ satisfied by the hypergeometric series ${}_3F_2(\alpha'_1,\alpha'_2,\alpha'_3;\beta'_1,\beta'_2;x)$ and when $\alpha_2=0$, it is reduced to a local solution of the equation corresponding to the exponent at $x=1$ with free multiplicity.
Let $u_0(\alpha_1,\alpha_2,\beta_1,\beta_2,\mu_1,\mu_2;x)$ be the local solution normalized by \[
u_0(\alpha,\beta,\mu;x) - x^{\alpha_1+\alpha_2+\mu_1+\mu_2}
\in x^{\alpha_1+\alpha_2+\mu_1+\mu_2+1}\mathcal O_0 \] for generic $\alpha,\beta,\mu$. Then we have the recurrence relation \[ \begin{split}
u_0(\alpha,\beta_1-1,\beta_2,\mu;x) &= u_0(\alpha,\beta,\mu;x)
+\frac{(\alpha_1+1)(\alpha_1+\alpha_2+\mu_1+1)}
{(\alpha_1+\mu_1+1)(\alpha_1+\alpha_2+\mu_1+\mu_2+1)}\\
&\qquad\cdot u_0(\alpha_1+1,\alpha_2,\beta_ 1-1,\beta_2,\mu;x). \end{split} \]
\subsubsection{Tuple of partitions $:\ 211,22,31,31$} \index{tuple of partitions!rigid!211,22,31,31} \qquad$[\Delta(\mathbf m)]=1^6\cdot 2$ \begin{align*}
211,22,31,31
&=H_1\oplus P_3:4=H_2\oplus H_2:2=2H_1\oplus H_2:2\\
&=010,10,10,10\oplus 201,12,21,21
=010,01,10,10\oplus 201,21,21,21\\
&=001,10,10,10\oplus 210,12,21,21
=001,01,10,10\oplus 210,21,21,21\\
&=110,11,11,20\oplus 101,11,20,11
=110,11,20,11\oplus 101,11,11,20\\
&=200,20,20,20\oplus 011,02,11,11\\
&\xrightarrow{\partial_{max}}011,02,11,11 \end{align*} \begin{align*} &\begin{Bmatrix}
x=0 & \frac1{c_1} & \frac1{c_2} & \infty\\
[\lambda_{0,1}]_{(3)} & [\lambda_{1,1}]_{(3)} & [\lambda_{2,1}]_{(2)} &
[\lambda_{3,1}]_{(2)}\\
\lambda_{0,2} & \lambda_{1,2} & \lambda_{2,2} &
[\lambda_{3,2}]_{(2)}\\
& & \lambda_{2,3} \end{Bmatrix} \allowdisplaybreaks\\ &\xrightarrow{x^{-\lambda_{0,1}}
(1-c_1x)^{-\lambda_{1,1}}(1-c_2x)^{-\lambda_{2,1}}
}{}\\ & \begin{Bmatrix}
x=0 & \frac1{c_1} & \frac1{c_2} & \infty\\
[0]_{(3)} & [0]_{(3)} & [0]_{(2)} &
[\lambda_{3,1}+\lambda_{0,1}+\lambda_{1,1}+\lambda_{2,1}]_{(2)}\\
\lambda_{0,2}-\lambda_{0,1} & \lambda_{1,2}-\lambda_{1,1} &
\lambda_{2,2}-\lambda_{2,1} &
[\lambda_{3,2}+\lambda_{0,1}+\lambda_{1,1}+\lambda_{2,1}]_{(2)}\\
& & \lambda_{2,3}-\lambda_{2,1} \end{Bmatrix}\\ &\xrightarrow{\partial^{-\lambda_1'}}{} \\ & \begin{Bmatrix}
x=0 & \frac1{c_1} & \frac1{c_2} & \infty\\
0 & 0 \\
\lambda_{0,2}+\lambda_1'-\lambda_{0,1}
& \lambda_{1,2}+\lambda_1'-\lambda_{1,1}&
\lambda_{2,2}+\lambda_1'- \lambda_{2,1}&
[\lambda_{3,2}-\lambda_{3,1}+1]_{(2)}\\
& & \lambda_{2,3}+\lambda_1'-\lambda_{2,1} \end{Bmatrix} \end{align*} The condition for the irreducibility: \begin{equation*} \begin{cases}
\lambda_{0,1}+\lambda_{1,1}+\lambda_{2,\nu}+\lambda_{3,\nu'}\notin\mathbb Z
\ \ (\nu\in\{1,2,3\},\ \nu'\in\{1,2\}),\\
\lambda_{0,1}+\lambda_{0,2}+2\lambda_{1,1}+\lambda_{2,1}+\lambda_{2,\nu}
+\lambda_{3,1}+\lambda_{3,2} \notin\mathbb Z \ \ (\nu\in\{2,3\}), \end{cases} \end{equation*} \begin{align*}
c(\lambda_{0,2}\!\rightsquigarrow\!\lambda_{1,2})
&=\frac{\Gamma(\lambda_{0,2}-\lambda_{0,1}+1)
\Gamma(\lambda_{1,2}-\lambda_{1,1})
(1-\frac{c_2}{c_1})^{\lambda_{2,1}}}
{\prod_{\nu=2}^3
\Gamma(\lambda_{0,1}+\lambda_{0,2}+2\lambda_{1,1}+\lambda_{2,1}
+\lambda_{2,\nu}+\lambda_{3,1}+\lambda_{3,2}-1)}, \allowdisplaybreaks\\
c(\lambda_{0,2}\!\rightsquigarrow\!\lambda_{2,3})
&=\prod_{\nu=1}^2
\frac{\Gamma(\lambda_{2,3}-\lambda_{2,\nu})}
{\Gamma(1-\lambda_{0,1}-\lambda_{1,1}-\lambda_{2,3}-\lambda_{3,\nu})}\\
&\quad\cdot \frac{\Gamma(\lambda_{0,2}-\lambda_{0,1}+1)
(1-\frac{c_1}{c_2})^{\lambda_{1,1}}}
{
\Gamma(\lambda_{0,1}+\lambda_{0,2}+2\lambda_{1,1}+\lambda_{2,1}
+\lambda_{2,2}+\lambda_{3,1}+\lambda_{3,2}-1)}. \end{align*}
\subsubsection{Tuple of partitions $:\ 22,22,22,31$} \index{tuple of partitions!rigid!22,22,22,31} \qquad$[\Delta(\mathbf m)]=1^8\cdot2$ \begin{align*}
22,22,22,31
&=H_1\oplus P_3:8=2(11,11,11,20)\oplus00,00,00,(-1)1\\
&=10,10,10,10\oplus12,12,12,21
=10,10,01,10\oplus12,12,21,21\\
&=10,01,10,10\oplus12,21,12,21
=10,01,01,10\oplus12,21,21,21\\
&=01,10,10,10\oplus21,12,12,21
=01,10,01,10\oplus21,12,21,21\\
&=01,01,10,10\oplus21,21,12,21
=01,01,01,10\oplus21,21,21,21\\
&\overset2\to12,12,12,21 \end{align*} The condition for the irreducibility: \begin{equation*}
\begin{cases}
\lambda_{0,i}+\lambda_{1,j}+\lambda_{2,k}+\lambda_{3,1}\notin\mathbb Z
\quad(i,\,j,\,k\in\{1,2\}),\\
\lambda_{0,1}+\lambda_{0,2}+\lambda_{1,1}+\lambda_{1,2}+
\lambda_{2,1}+\lambda_{2,2}+\lambda_{3,1}+\lambda_{3,2}\notin\mathbb Z.
\end{cases} \end{equation*}
\subsection{Other rigid examples with a small order}\label{sec:ord6Ex} First we give an example which is not of Okubo type.
\subsubsection{$221,221,221$} \index{tuple of partitions!rigid!221,221,221} The Riemann Scheme and the direct decompositions are \begin{gather*}
\begin{Bmatrix}
x=0 & 1 & \infty\\
[\lambda_{0,1}]_{(2)} & [\lambda_{1,1}]_{(2)} & [\lambda_{2,1}]_{(2)}\\
[\lambda_{0,2}]_{(2)} & [\lambda_{1,2}]_{(2)} & [\lambda_{2,2}]_{(2)}\\
\lambda_{0,3} & \lambda_{1,3} & \lambda_{2,3}
\end{Bmatrix},\qquad
\sum_{j=0}^2(2\lambda_{j,1}+2\lambda_{j,2}+\lambda_{j,3})=4,
\allowdisplaybreaks\\ \begin{aligned}
\phantom{a}
[\Delta(\mathbf m)]&=1^{14}\cdot2\\
22\overline1,22\underline1,221
&=H_1\oplus 211,211,211:8\qquad\ 6=|2,2,2|\\
&=H_2\oplus H_3:6\qquad\qquad\qquad 11=|21,22,22|\\
&=2H_2\oplus H_1:1\\
&=10\overline1,110,110\oplus120,11\underline1,111
=01\overline1,110,110\oplus210,11\underline1,111\\
&=11\overline1,120,111\oplus110,10\underline1,110
=11\overline1,210,111\oplus110,01\underline1,110\\
&\to121,121,121 \end{aligned} \end{gather*} and a connection coefficient is give by \begin{align*}
c(\lambda_{0,3}\!\rightsquigarrow\!\lambda_{1,3})
&=\prod_{\nu=1}^{2}\biggl(
\frac{\Gamma(\lambda_{0,3}-\lambda_{0,\nu}+1)}
{\Gamma(\lambda_{0,\nu}+\lambda_{0,3}+\lambda_{1,1}+\lambda_{1,2}+\lambda_{2,1}+\lambda_{2,2}-1) }\\
&\qquad\cdot
\frac{\Gamma(\lambda_{1,\nu}-\lambda_{1,3})}
{\Gamma(2-\lambda_{0,1}-\lambda_{0,2}-\lambda_{1,\nu}
-\lambda_{1,3}-\lambda_{2,1}-\lambda_{2,2})}\biggr). \end{align*}
Using this example we explain an idea to get all the rigid decompositions $\mathbf m=\mathbf m'\oplus\mathbf m''$. Here we note that $\idx(\mathbf m,\mathbf m')=1$. Put $\mathbf m=221,221,221$. We may assume $\ord\mathbf m'\le\ord\mathbf m''$.
Suppose $\ord\mathbf m'=1$. Then $\mathbf m'$ is isomorphic to $1,1,1$ and there exists tuples of indices $(\ell_0,\ell_1,\ell_2)$ such that $m'_{j,\nu}=\delta_{j,\ell_j}$. Then $\idx(\mathbf m,\mathbf m')=m_{0,\ell_0}+m_{1,\ell_1}+m_{1,\ell_2}
- (3-2)\ord\mathbf m\cdot\ord\mathbf m'$ and we have
$m_{0,\ell_0}+m_{1,\ell_1}+m_{1,\ell_2}=6$. Hence $(m_{0,\ell_0},m_{1,\ell_1},m_{1,\ell_2})=(2,2,2)$, which is expressed by $6=|2,2,2|$ in the above. Since $\ell_j=1$ or $2$ for $0\le j\le 2$, it is clear that there exist 8 rigid decompositions with $\ord\mathbf m'=1$.
Suppose $\ord\mathbf m'=2$. Then $\mathbf m'$ is isomorphic to $11,11,11$ and there exists tuples of indices $(\ell_{0,1},\ell_{0,2},\ell_{1,1},\ell_{1,2},\ell_{2,1},\ell_{2,2})$ which satisfies $\sum_{j=0}^2\sum_{\nu=1}^2 m_{j,\ell_\nu}= (3-2)\ord\mathbf m\cdot\ord\mathbf m'+1=11$. Hence we may assume $(\ell_{0,1},\ell_{0,2},\ell_{1,1},\ell_{1,2},\ell_{2,1},\ell_{2,2}) =(2,1,2,2,2,2)$ modulo obvious symmetries, which is expressed by
$11=|21,22,22|$. There exist 6 rigid decompositions with $\ord\mathbf m'=2$.
In general, this method to get all the rigid decompositions of $\mathbf m$ is useful when $\ord\mathbf m$ is not big. For example if $\ord\mathbf m\le 7$, $\mathbf m'$ is isomorphic to $1,1,1$ or $11,11,11$ or $21,111,111$.
The condition for the irreducibility is given by Theorem~\ref{thm:irred} and it is \[
\begin{cases}
\lambda_{0,i}+\lambda_{1,j}+\lambda_{2,k}\notin\mathbb Z&
(i,\,j,\,k\in\{1,2\}),\\
\sum_{j=0}^2\sum_{\nu=1}^2\lambda_{j,\nu}
+(\lambda_{i,3}-\lambda_{i,k})\notin\mathbb Z&
(i\in\{0,1,2\},\ k\in\{1,2\}).
\end{cases} \] \subsubsection{Other examples} Theorem~\ref{thm:c} shows that the connection coefficients between local solutions of rigid differential equations which correspond to the eigenvalues of local monodromies with free multiplicities are given by direct decompositions of the tuples of partitions $\mathbf m$ describing their spectral types.
We list the rigid decompositions $\mathbf m=\mathbf m'\oplus \mathbf m''$ of rigid indivisible $\mathbf m$ in $\mathcal P^{(5)}\cup\mathcal P^{(6)}_3$ satisfying $m_{0,n_0}=m_{1,n_1}=m'_{0,n_0}=m''_{1,n_1}=1$. The positions of $m_{0,n_0}$ and $m_{1,n_1}$ in $\mathbf m$ to which Theorem~\ref{thm:c} applies are indicated by an overline and an underline, respectively. The number of decompositions in each case equals $n_0+n_1-2$ and therefore the validity of the following list is easily verified.
We show the tuple $\partial_{max}\mathbf m$ after $\to$. The type $[\Delta(\mathbf m)]$ of $\Delta(\mathbf m)$ is calculated by \eqref{eq:DmInd}, which is also indicated in the following with this calculation. For example, when $\mathbf m=311,221,2111$, we have $d(\mathbf m)=2$, $\mathbf m'=\partial\mathbf m=111,021,0111$, $[\Delta(s(111,021,0111))]=1^9$, $\{m'_{j,\nu}-m'_{j,1}\in\mathbb Z_{>0}\}\cup\{2\}=\{1,1,1,1,2,2\}$ and hence $[\Delta(\mathbf m)]=1^9\times 1^4\cdot 2^2=1^{13}\cdot2^2$, which is a partition of $h(\mathbf m)-1=17$. Here we note that $h(\mathbf m)$ is the sum of the numbers attached the Dynkin diagram \ \begin{xy}
\ar@{-} *+!D{1} *{\circ} ;
(5,0) *+!D{2} *{\circ}="A"
\ar@{-} "A";(10,0) *+!LD{5} *{\circ}="B"
\ar@{-} "B";(15,0) *+!D{3} *{\circ}="C"
\ar@{-} "C";(20,0) *+!D{2} *{\circ}="C"
\ar@{-} "C";(25,0) *+!D{1} *{\circ}="C"
\ar@{-} "B";(10,5) *+!LD{3} *{\circ}="D"
\ar@{-} "D";(10,10) *+!LD{1} *{\circ} \end{xy} \ corresponding to $\alpha_{\mathbf m}\in\Delta_+$.
All the decompositions of the tuple $\mathbf m$ corresponding to the elements in $\Delta(\mathbf m)$ are given, by which we easily get the necessary and sufficient condition for the irreducibility (cf.~Theorem~\ref{thm:irrKac} and \S\ref{sec:RobEx}). \index{000Delta1@$[\Delta(\mathbf m)]$}
{\small\begin{align*} &\text{\large\qquad$\ord\mathbf m=5$}\\
311,221,2111&=100,010,0001\oplus211,211,2110\qquad\qquad\quad\ \, 6=|3,2,1|\\
&=100,001,1000\oplus211,220,1111\qquad\qquad\quad\ \, 6=|3,1,2|\\
&=101,110,1001\oplus210,111,1110\qquad\qquad 11=|31,22,21|\\
&=2(100,100,1000)\oplus111,021,0111\\
&\overset{2}\to111,021,0111 \allowdisplaybreaks\\ [\Delta(\mathbf m)]&=1^9\times1^4\cdot2^2=1^{13}\cdot2^2\\ \mathbf m&=H_1\oplus211,211,211:6=H_1\oplus EO_4:1=H_2\oplus H_3:6
=2H_1\oplus H_3:2 \allowdisplaybreaks\\
311,22\overline{1},211\underline{1}
&=211,211,2110\oplus100,010,0001
=211,121,2110\oplus100,100,0001\\
&=100,001,1000\oplus211,220,1111\\
&=210,111,1110\oplus101,110,1001
=201,111,1110\oplus110,110,1001 \allowdisplaybreaks\\
31\overline{1},221,211\underline{1}
&=211,211,2110\oplus100,010,0001
=211,121,2110\oplus100,100,0001\\
&=201,111,1110\oplus110,110,1001\\
&=101,110,1010\oplus210,111,1101
=101,110,1100\oplus210,111,1011 \allowdisplaybreaks\\[5pt]
32,211\overline{1},211\underline{1}
&=22,1111,2110\oplus10,1000,0001
=10,0001,1000\oplus22,2110,1111\\
&=11,1001,1010\oplus21,1110,1101
=11,1001,1100\oplus21,1110,1011\\
&=21,1101,1110\oplus11,1010,1001
=21,1011,1110\oplus11,1100,1001\\
&\overset2\to12,0111,0111\\ [\Delta(\mathbf m)]&=1^9\times 1^7\cdot 2=1^{16}\cdot2\\ \mathbf m&=H_1\oplus H_4:1=H_1\oplus EO_4:6=H_2\oplus H_3:9
=2H_1\oplus H_3:1 \allowdisplaybreaks\\[5pt] 22\overline{1},22\underline{1},41,41
&=001,100,10,10\oplus220,121,31,31=001,010,10,10\oplus220,211,31,31\\
&=211,220,31,31\oplus010,001,10,10=121,220,31,31\oplus100,001,10,10\\
&\overset2\to021,021,21,21\\ [\Delta(\mathbf m)]&=1^4\cdot2\times1^4\cdot2^3=1^{6}\cdot2^4\\ \mathbf m&=H_1\oplus22,211,31,31:4=H_2\oplus H_3:2=2H_1\oplus P_3:4 \allowdisplaybreaks\\ 22\overline{1},221,4\underline{1},41
&=001,100,10,10\oplus220,121,31,31=001,010,10,10\oplus220,211,31,31\\
&=111,111,30,21\oplus110,110,11,20 \allowdisplaybreaks\\[5pt] 22\overline{1},32,32,4\underline{1}
&=101,11,11,20\oplus120,21,21,21=011,11,11,20\oplus210,21,21,21\\
&=001,10,10,10\oplus 220,22,22,31\\
&\overset2\to021,12,12,21 \allowdisplaybreaks\\ [\Delta(\mathbf m)]&=1^4\cdot2\times1^3\cdot2^2=1^{7}\cdot2^3 \allowdisplaybreaks\\ \mathbf m&=H_1\oplus22,22,22,31:1=H_1\oplus211,22,31,31:4=
H_2\oplus P_3:2\\
&=2H_1\oplus P_3:2 \allowdisplaybreaks\\[5pt] 31\overline{1},31\underline{1},32,41
&=001,100,10,10\oplus310,211,22,31
=211,301,22,31\oplus100,001,10,10\\
&=101,110,11,20\oplus210,201,21,21
=201,210,21,21\oplus110,101,11,20\\
&\overset3\to011,011,02,11\\ [\Delta(\mathbf m)]&=1^4\times1^4\cdot2\cdot3=1^8\cdot2\cdot3\\ \mathbf m&=H_1\oplus211,31,22,31:4=H_2\oplus P_3:4\\
&=2H_1\oplus H_3:1=3H_1\oplus H_2:1 \allowdisplaybreaks\\ 31\overline{1},311,32,4\underline{1}
&=001,100,10,10\oplus301,211,22,31\\
&=101,110,11,20\oplus210,201,21,21
=101,101,11,20\oplus210,210,21,21 \allowdisplaybreaks\\[5pt] 32,32,4\overline{1},4\underline{1},41
&=11,11,11,20,20\oplus21,21,30,21,21\\
&=21,21,21,30,21\oplus11,11,20,11,20\\
&\overset3\to02,02,11,11,11\\ [\Delta(\mathbf m)]&=1^4\times2^2\cdot 3=1^4\cdot2^2\cdot3\\ \mathbf m&=H_1\oplus P_4:1=H_2\oplus P_3:3=2H_1\oplus P_3:2=3H_1\oplus H_2:1 \allowdisplaybreaks\\[10pt]
&\text{\qquad\large$\ord\mathbf m=6$ and $\mathbf m\in\mathcal P_3$}\\
321,3111,222&=311,2111,221\oplus010,1000,001\qquad\qquad\quad\ \,7=|2,3,2|\\
&=211,2110,211\oplus110,1001,011\qquad\qquad13=|32,31,22|\\
&=210,1110,111\oplus111,2001,111\\
&\overset2\to121,1111,022\to111,0111,012\\ [\Delta(\mathbf m)]&=1^{14}\times1\cdot2^3=1^{15}\cdot2^3\\ \mathbf m&=H_1\oplus311,2111,221:3=
H_2\oplus211,211,211:6=H_3\oplus H_3:6\\
&= 2H_1\oplus EO_4:3 \allowdisplaybreaks\\ 32\overline{1},311\underline{1},222
&=211,2110,211\oplus110,1001,011
=211,2110,121\oplus110,1001,101\\
&=211,2110,112\oplus110,1001,110\\
&=111,2100,111\oplus210,1011,111
=111,2010,111\oplus210,1101,111 \allowdisplaybreaks\\ 321,311\overline{1},311\underline{1}
&=221,2111,3110\oplus100,1000,0001
=100,0001,1000\oplus221,3110,2111\\
&=211,2101,2110\oplus110,1010,1001
=211,2011,2110\oplus110,1101,1001\\
&=110,1001,1100\oplus211,2110,2011
=110,1001,1010\oplus211,2110,2101\\
&\overset3\to021,0111,0111\\ [\Delta(\mathbf m)]&=1^9\times1^7\cdot 2\cdot3=1^{16}\cdot2\cdot3\\ \mathbf m&=H_1\oplus221,2111,311:6=H_1\oplus32,2111,2111:1\\ &=H_2\oplus211,211,211:9=2H_1\oplus H_4:1= 3H_1\oplus H_3:1\\ \allowdisplaybreaks\\ 32\overline{1},311\underline{1},3111
&=221,3110,2111\oplus100,0001,1000
=001,1000,1000\oplus320,2111,2111\\
&=211,2110,2110\oplus110,1001,1001
=211,2110,2011\oplus110,1001,1100\\
&=211,2110,2011\oplus110,1001,1100 \allowdisplaybreaks\\[5pt] 32\overline{1},32\underline{1},2211
&=211,220,1111\oplus110,101,1100
=101,110,1100\oplus220,211,1111\\
&=111,210,1110\oplus210,111,1101
=111,210,1101\oplus210,111,1110\\
&\overset2\to121,121,0211\to101,101,0011\\ [\Delta(\mathbf m)]&=1^{10}\cdot2\times1^4\cdot2^2=1^{14}\cdot2^3\\ \mathbf m&=H_1\oplus311,221,2111:4=H_1\oplus 221,221,221:2\\
&=H_2\oplus EO_4:2= H_2\oplus211,211,211:4=H_3\oplus H_3:2\\
&=2H_1\oplus211,211,211:2=2(110,110,1100)\oplus101,101,0011:1 \allowdisplaybreaks\\ 321,32\overline{1},221\underline{1}
&=221,221,2210\oplus100,100,0001
=110,101,1100\oplus211,220,1111\\
&=211,211,2110\oplus110,110,0101
=211,211,1210\oplus110,110,1001\\
&=210,111,1110\oplus111,210,1101 \allowdisplaybreaks\\[5pt] 41\overline{1},221\underline{1},2211
&=311,2210,2111\oplus100,0001,0100
=311,2210,1211\oplus100,0001,1000\\
&=101,1100,1100\oplus310,1111,1111
=201,1110,1110\oplus210,1101,1101\\
&=201,1110,1101\oplus210,1101,1110\\
&\overset2\to211,0211,0211\to011,001,0011\\ [\Delta(\mathbf m)]&=1^{10}\cdot2\times1^4\cdot 2^3=1^{14}\cdot2^4\\ \mathbf m&=H_1\oplus311,221,2211:8=H_2\oplus H_4:2=H_3\oplus H_3:4\\ &=2H_1\oplus211,211,211:4 \allowdisplaybreaks\\ 411,221\overline{1},221\underline{1}
&=311,2111,2210\oplus100,0100,0001
=311,1211,2210\oplus100,1000,0001\\
&=100,0001,0100\oplus311,2210,2111
=100,0001,1000\oplus311,2210,1211\\
&=201,1101,1110\oplus210,1110,1101
=210,1101,1110\oplus201,1110,1101 \allowdisplaybreaks\\[5pt] 41\overline{1},222,2111\underline{1}
&=311,221,21110\oplus100,001,00001
=311,212,21110\oplus100,010,00001\\
&=311,122,21110\oplus100,100,00001
=201,111,11100\oplus210,111,10011\\
&=201,111,11010\oplus210,111,10101
=201,111,10110\oplus210,111,11001\\
&\overset2\to211,022,01111\to111,012,00111\\ [\Delta(\mathbf m)]&=1^{14}\times1^4\cdot 2^3=1^{18}\cdot2^3\\ \mathbf m&=H_1\oplus311,221,2111:12=H_3\oplus H_3:6=2H_1\oplus EO_4:3 \allowdisplaybreaks\\[5pt] 42,221\overline{1},2111\underline{1}
&=32,2111,21110\oplus10,0100,00001
=32,1211,21110\oplus10,1000,00001\\
&=10,0001,10000\oplus32,2210,11111
=31,1111,11110\oplus11,1100,10001\\
&=21,1101,11100\oplus21,1110,10011
=21,1101,11010\oplus21,1110,10101\\
&=21,1101,10110\oplus21,1110,11001\\
&\overset2\to22,0211,01111\to12,0111,00111\\ [\Delta(\mathbf m)]&=1^{14}\times1^6\cdot 2^2=1^{20}\cdot2^2\\ \mathbf m&=H_1\oplus32,2111,2111:8=H_1\oplus EO_4:2
=H_2\oplus H_4:4\\ &=H_3\oplus H_3:6=2H_1\oplus EO_4:2 \allowdisplaybreaks\\[5pt] 33,311\overline{1},2111\underline{1}
&=32,2111,21110\oplus01,1000,00001
=23,2111,21110\oplus10,1000,00001\\
&=22,2101,11110\oplus11,1010,10001
=22,2011,11110\oplus11,1100,10001\\
&=11,1001,11000\oplus22,2110,10111
=11,1001,10100\oplus22,2110,11011 \allowdisplaybreaks\\
&=11,1001,10010\oplus22,2110,11101\\
&\overset2\to13,1111,01111 \allowdisplaybreaks\\ [\Delta(\mathbf m)]&=1^{16}\times1^4\cdot2^2=1^{20}\cdot2^2\\ \mathbf m&=H_1\oplus32,2111,2111:8=H_2\oplus EO_4:12 =2H_1\oplus H_4:2 \allowdisplaybreaks\\[5pt] 32\overline{1},311\underline{1},3111
&=221,3110,2111\oplus100,0001,1000
=001,1000,1000\oplus320,2111,2111\\
&=211,2110,2110\oplus110,1001,1001
=211,2110,2101\oplus110,1001,1010\\
&=211,2110,2011\oplus110,1001,1100\\
&\overset3\to021,0111,0111\\ [\Delta(\mathbf m)]&=1^{9}\times1^7\cdot2\cdot3=1^{16}\cdot2\cdot3\\ \mathbf m&=H_1\oplus221,2111,311:6=H_1\oplus32,2111,2111:1\\ &=H_2\oplus 211,211,211:9=2H_1\oplus H_4:1=3H_1\oplus H_3:1 \allowdisplaybreaks\\ 321,311\overline{1},311\underline{1}
&=100,0001,1000\oplus221,3110,2111
=221,2111,3110\oplus100,1000,0001\\
&=211,2101,2110\oplus110,1010,1001
=211,2011,2110\oplus110,1100,1001\\
&=110,1001,1100\oplus211,2110,2011
=110,1001,1010\oplus211,2110,2101 \allowdisplaybreaks\\[5pt] 33,221\overline{1},221\underline{1}
&=22,1111,2110\oplus11,1100,1001
=22,1111,1210\oplus11,1100,0101\\
&=21,1101,1110\oplus12,1110,1011
=12,1101,1110\oplus21,1110,1011\\
&=11,1001,1100\oplus22,1210,1111
=11,0101,1100\oplus22,2110,1111\\
&\overset1\to23,1211,1211\to21,1011,1011\\ [\Delta(\mathbf m)]&=1^{16}\cdot2\times1^4=1^{20}\cdot2\\ \mathbf m&=H_1\oplus32,2111,2111:8=H_2\oplus EO_4:8= H_3\oplus H_3:4\\ &=2(11,1100,1100)\oplus11,0011,0011:1 \end{align*}}
We show all the rigid decompositions of the following simply reducible partitions of order 6, which also correspond to the reducibility of the universal models.\index{tuple of partitions!simply reducible} {\begin{align*}
42,222,111111
&=32,122,011111\oplus 10,100,100000\\
&=21,111,111000\oplus 21,111,000111\\
&\overset1\to32,122,011111\to22,112,001111\to12,111,000111\\ [\Delta(\mathbf m)]&=1^{28}\\ \mathbf m&=H_1\oplus EO_5:18=H_3\oplus H_3:10 \allowdisplaybreaks\\[5pt]
33,222,21111
&=23,122,11111\oplus10,100,10000\\
&=22,112,10111\oplus11,110,11000\\
&=21,111,11100\oplus12,111,10011\\
&\overset1\to23,122,11111\to22,112,01111\to12,111,00111\\ [\Delta(\mathbf m)]&=1^{24}\\ \mathbf m&=H_1\oplus EO_5:6=H_2\oplus EO_4:12=H_3\oplus H_3:6 \end{align*}}
\subsection{Submaximal series and minimal series }\label{sec:Rob} \index{tuple of partitions!rigid!(sub)maximal series} The rigid tuples $\mathbf m=\{m_{j,\nu}\}$ satisfying \begin{equation}
\#\{m_{j,\nu}\,;\,0<m_{j,\nu}<\ord\mathbf m\}\ge\ord\mathbf m +5 \end{equation} are classified by Roberts \cite{Ro}. They are the tuples of type $H_n$ and $P_n$ which satisfy \begin{equation}
\#\{m_{j,\nu}\,;\,0<m_{j,\nu}<\ord\mathbf m\}=2\ord\mathbf m+2 \end{equation} and those of 13 series $A_n=EO_n$, $B_n$, $C_n$, $D_n$, $E_n$, $F_n$, $G_{2m}$, $I_n$, $J_n$, $K_n$, $L_{2m+1}$, $M_n$, $N_n$ called \textsl{submaximal series} \index{tuple of partitions!rigid!(sub)maximal series} which satisfy \begin{equation}
\#\{m_{j,\nu}\,;\,0<m_{j,\nu}<\ord \mathbf m\}=\ord\mathbf m+5. \end{equation} The series $H_n$ and $P_n$ are called \textsl{maximal series}.
We examine these rigid series and give enough information to analyze the series, which will be sufficient to construct differential equations including their confluences, integral representation and series expansion of solutions and get connection coefficients and the condition of their reducibility.
In fact from the following list we easily get all the direct decompositions and Katz's operations decreasing the order. The number over an arrow indicates the difference of the orders. We also indicate Yokoyama's reduction for systems of Okubo normal form using extension and restriction, which are denoted $E_i$ and $R_i$ $(i=0,1,2)$, respectively (cf.~\cite{Yo2}). Note that the inverse operations of $E_i$ are $R_i$, respectively. In the following we put \begin{equation} \begin{split}
u_{P_m}&=\partial^{-\mu}x^{\lambda_0}(1-x)^{\lambda_1}(c_2-x)^{\lambda_2}
\cdots(c_{m-1}-x)^{\lambda_{m-1}},\\
u_{H_2}&=u_{P_2},\\
u_{H_{m+1}}&=\partial^{-\mu^{(m)}}x^{\lambda^{(m)}_0}u_{H_m}. \end{split} \end{equation} We give all the decompositions \begin{equation}\label{eq:allDec}
\mathbf m=\bigl(\idx(\mathbf m',\mathbf m)\cdot\mathbf m'\bigr)\oplus\mathbf m'' \end{equation} for $\alpha_{\mathbf m'}\in\Delta(\mathbf m)$. Here we will not distinguish between $\mathbf m'\oplus \mathbf m''$ and $\mathbf m''\oplus \mathbf m'$ when $\idx(\mathbf m',\mathbf m)=1$. Moreover note that the inequality assumed for the formula $[\Delta(\mathbf m)]$ below assures that the given tuple of partition is monotone.
\subsubsection{$B_n$}\label{sec:Bn} (${B_{2m+1}}=\text{III}_m$, $B_{2m}=\text{II}_m$, $B_3=H_3$, $B_2=H_2$) \index{000Delta1@$[\Delta(\mathbf m)]$} \begin{align*}
u_{B_{2m+1}}&=\partial^{-\mu'}(1-x)^{\lambda'}u_{H_{m+1}}\\
m^21,m+11^m,m1^{m+1} &=10,10,01\oplus mm-11,m1^m,m1^m\\
&=01,10,10\oplus m^2,m1^m,m-11^{m+1}\\
&=1^20,11,11\oplus (m-1)^21,m1^{m-1},m-11^m \allowdisplaybreaks\\
[\Delta(B_{2m+1})]&=1^{(m+1)^2}\times1^{m+2}\cdot m^2
=1^{m^2+3m+3}\cdot m^2\\ &\hspace{-1.08cm} \begin{aligned}
B_{2m+1}&=H_1\oplus B_{2m}&&:2(m+1) \\
&=H_1\oplus C_{2m}&&:1\\
&=H_2\oplus B_{2m-1}&&:m(m+1)\\
&=mH_1\oplus H_{m+1}&&:2 \end{aligned} \allowdisplaybreaks\\[5pt]
u_{B_{2m}}&=\partial^{-\mu'}x^{\lambda'}(1-x)^{\lambda''}u_{H_m}\\
mm-11,m1^m,m1^m&=100,01,10\oplus (m-1)^21,m1^{m-1},m-11^m\\
&=001,10,10\oplus mm-10,m-11^m,m-11^m\\
&=110,11,11\oplus m-1m-21,m-11^{m-1},m-11^{m-1} \allowdisplaybreaks\\ [\Delta(B_{2m})]&=1^{m^2}\times1^{2m+1}\cdot(m-1)=1^{(m+1)^2}\cdot(m-1)\cdot m \\ &\hspace{-.72cm} \begin{aligned} B_{2m}&=H_1\oplus B_{2m-1}&&:2m \\
&= H_1\oplus C_{2m-2}&&:1 \\
&= H_2\oplus B_{2m-2}&&:m^2\\
&= (m-1)H_1\oplus H_{m+1}&&:1\\
&= mH_1\oplus H_m&&:1 \end{aligned} \allowdisplaybreaks\\
B_{2m+1} &\overset{m}{\underset{R2E0}\longrightarrow} H_{m+1},\quad
B_n\overset{1}\longrightarrow B_{n-1},\quad B_n\overset{1}\longrightarrow C_{n-1} \allowdisplaybreaks\\
B_{2m} &\overset{m}{\underset{R1E0}\longrightarrow} H_{m},\quad
B_{2m} \overset{m-1}\longrightarrow H_{m+1} \end{align*}
\subsubsection{An example}\label{sec:RobEx} Using the example of type $B_{2m+1}$, we explain how we get explicit results from the data written in \S\ref{sec:Bn}.
The Riemann scheme of type $B_{2m+1}$ is \begin{gather*}
\begin{Bmatrix}
\infty & 0 & 1\\
[\lambda_{0,1}]_{(m)} & [\lambda_{1,1}]_{(m+1)} & [\lambda_{2,1}]_{(m)}\\
[\lambda_{0,2}]_{(m)} & \lambda_{1,2} & \lambda_{2,2}\\
\lambda_{0,3} & \vdots & \vdots\\
& \lambda_{1,m+1} & \lambda_{2,m+2}
\end{Bmatrix},
\allowdisplaybreaks\\
\sum_{j=0}^p\sum_{\nu=1}^{n_j} m_{j,\nu}\lambda_{j,\nu}=2m
\qquad(\text{Fuchs relation}). \end{gather*}
Theorem~\ref{thm:irrKac} says that the corresponding equation is irreducible if and only if any value of the following linear functions is not an integer. \begin{align*}
L^{(1)}_{i,\nu}&:=
\lambda_{0,i}+\lambda_{1,1}+\lambda_{2,\nu}\qquad(i=1,2,\ \ \nu=2,\dots,m+2),\\
L^{(2)}&:=
\lambda_{0,3}+\lambda_{1,1}+\lambda_{2,1},\\
L^{(3)}_{\mu,\nu}&:=
\lambda_{0,1}+\lambda_{0,2}+\lambda_{1,1}+\lambda_{1,\mu}
+\lambda_{2,1}+\lambda_{2,\nu}-1\\
&\qquad\qquad\qquad\quad(\mu=2,\dots,m+1,\ \ \nu=2,\dots,m+2),\\
L^{(4)}_i&:=
\lambda_{0,i}+\lambda_{1,1}+\lambda_{2,1}\qquad(i=1,2). \end{align*} Here $L^{(1)}_{i,\nu}$ (resp.~$L^{(2)}$ etc.) correspond to the terms $10,01,01$ and $H_1\oplus B_{2m}:2(m+1)$ (resp.~$01,10,10$ and $H_1\oplus C_{2m}:1$ etc.) in \S\ref{sec:Bn}.
It follows from Theorem~\ref{thm:univmodel} and Theorem~\ref{thm:irrKac} that the Fuchsian differential equation with the above Riemann scheme belongs to the universal equation $P_{B_{2m+1}}(\lambda)u=0$ if \[
L^{(4)}_i\notin\{-1,-2,\ldots,1-m\}\quad(i=1,\ 2). \]
Theorem~\ref{thm:c} says that the connection coefficient $c(\lambda_{1,m+1}\rightsquigarrow\lambda_{2,m+2})$ equals \[
\frac
{\prod_{\mu=1}^m\Gamma(\lambda_{1,m+1}-\lambda_{1,\mu}+1)\cdot
\prod_{\mu=1}^{m+1}\Gamma(\lambda_{2,\nu}-\lambda_{1,m+2})}
{\prod_{i=1}^2\Gamma(1-L^{(1)}_{i,m+2})\cdot
\prod_{\nu=2}^{m+1}\Gamma(L^{(3)}_{m+1,\nu})\cdot
\prod_{\mu=2}^m\Gamma(1-L^{(3)}_{\mu,m+2})} \] and \begin{align*}
c(\lambda_{1,m+1}\rightsquigarrow\lambda_{0,3})&
=\frac
{\prod_{\mu=1}^m\Gamma(\lambda_{1,m+1}-\lambda_{1,\mu}+1)\cdot
\prod_{i=1}^{2}\Gamma(\lambda_{0,i}-\lambda_{0,3})}
{\Gamma(1-L^{(2)})\cdot
\prod_{\nu=2}^{m+1}\Gamma(L^{(3)}_{m+1,\nu})},\\
c(\lambda_{2,m+2}\rightsquigarrow\lambda_{0,3})&
= \frac
{\prod_{\nu=1}^{m+1}\Gamma(\lambda_{2,m+2}-\lambda_{1,\nu}+1)\cdot
\prod_{i=1}^{2}\Gamma(\lambda_{0,i}-\lambda_{0,3})}
{\prod_{i=1}^2 \Gamma(L^{(1)}_{m+2})\cdot
\prod_{\nu=2}^{m+1}\Gamma(L^{(3)}_{m+1,\nu})}. \end{align*}
It follows from Theorem~\ref{thm:sftUniv} that the universal operators \index{shift operator} \index{Fuchsian differential equation/operator!universal operator} \[ \begin{aligned}
&P_{H_1}^{0}(\lambda) && P_{H_1}^{2}(\lambda)
&&P_{B_{2m}}^{0}(\lambda) && P_{B_{2m}}^{1}(\lambda)
&&P_{B_{2m}}^{2}(\lambda)
&&P_{C_{2m}}^{1}(\lambda)
&& P_{C_{2m}}^{2}(\lambda)\\
&P_{H_2}^{1}(\lambda) && P_{H_2}^{2}(\lambda)
&&P_{B_{2m-1}}^{0}(\lambda) && P_{B_{2m-1}}^{1}(\lambda)
&&P_{B_{2m-1}}^{2}(\lambda) \end{aligned} \] define shift operators $R_{B_{2m+1}}(\epsilon,\lambda)$ under the notation in the theorem.
We also explain how we get the data in \S\ref{sec:Bn}. Since $\partial_{max}:B_{2m+1}=\mathbf m:=mm1,m+11^m,m1^{m+1}\to H_{m+1}=\mathbf m':=0m1,11^m,01^{m+1}$, the equality \eqref{eq:DmInd} shows \begin{align*}
[\Delta(B_{2m+1})]&=[\Delta(H_{m+1})]
\cup\{d_{1,1,1}(\mathbf m)\}\cup\{m'_{j,\nu}-m'_{j,1}>0\}\\
&=1^{(m+1)^2}\times m^1\times 1^{m+2}\cdot m^1=1^{(m+1)^2}\times 1^{m+2}\cdot m^2
= 1^{m^2+3m+3}\cdot m^2. \end{align*} Here we note that $\{m'_{j,\nu}-m'_{j,1}>0\}=\{m,1,1^{m+1}\}=1^{m+2}\cdot m^1$ and $[\Delta(H_{m+1})]$ is given in \S\ref{sec:GHG}.
We check \eqref{eq:sumDelta} for $\mathbf m$ as follows:\\ $h(\mathbf m)= 2(1+\cdots+m)+(2m+1)+2(m+1)+1$\\
$\phantom{h(\mathbf m)}=m^2+5m+4,$\\ $\sum_{i\in[\Delta(\mathbf m)]}i =
(m^2+3m+3)+2m=m^2+5m+3$.
\hspace{7.6cm} {\small \begin{xy}
\ar@{-} *+!D{1} *{\circ} ;
(5,0) *+!D{2} *{\circ}="A"
\ar@{-} "A";(8,0)
\ar@{.} (8,0);(13,0)
\ar@{-} (13,0);(16,0) *+!D{m} *{\circ}="A"
\ar@{-} "A";(21,0) *+!U{2m+1} *{\circ}="A"
\ar@{-} "A";(26,0) *+!D{\ \ m+1} *{\circ}="C"
\ar@{-} "C";(29,0)
\ar@{.} (29,0);(34,0)
\ar@{-} (34,0);(37,0) *+!D{2} *{\circ}="C"
\ar@{-} "C";(42,0) *+!D{1} *{\circ}="C"
\ar@{-} "A";(21,5) *+!LD{m+1} *{\circ}="A"
\ar@{-} "A";(21,10) *+!LD{1} *{\circ} \end{xy}}
The decompositions $mH_1\oplus H_{m+1}$ and $H_1\oplus B_{2m}$ etc.\ in \S\ref{sec:Bn} are easily obtained and we should show that they are all the decompositions \eqref{eq:allDec}, whose number is given by $[\Delta(B_{2m+1})]$. There are 2 decompositions of type $mH_1\oplus H_{m+1}$, namely, $B_{2m+1}=mm1,m+11^m,m1^{m+1}=m(100,10,10)\oplus\cdots=m(010,10,10)\oplus\cdots$, which correspond to $L^{(4)}_i$ for $i=1$ and $2$. Then the other decompositions are of type $\mathbf m'\oplus\mathbf m''$ with rigid tuples $\mathbf m'$ and $\mathbf m''$ whose number equals $m^2+3m+3$. The numbers of decompositions $H_1\oplus B_{2m}$ etc.\ given in \S\ref{sec:Bn} are easily calculated which correspond to $L^{(1)}_{i,\nu}$ etc.\ and we can check that they give the required number of the decompositions.
\subsubsection{$C_n$} ($C_4=EO_4$, $C_3=H_3$, $C_2=H_2$) \begin{align*}
u_{C_{2m+1}}&=\partial^{-\mu'}x^{\lambda'}u_{H_{m+1}}\\
m+1m,m1^{m+1},m1^{m+1}&=10,01,10\oplus
m^2,m1^m,m-11^{m+1}\\
&=11,11,11\oplus m(m-1),m-11^mm-11^m \allowdisplaybreaks\\ [\Delta(C_{2m+1})]&=1^{(m+1)^2}\times1^{2m+2}\cdot m\cdot(m-1)\\ &=1^{(m+1)(m+3)}\cdot m\cdot(m-1)\\ &\hspace{-1.08cm} \begin{aligned}
C_{2m+1}&=H_1\oplus C_{2m}&&:2m+2\\
&=H_2\oplus C_{2m-2}&&:(m+1)^2\\
&=mH_1\oplus H_{m+1}&&:1\\
&=(m-1)H_1\oplus H_{m+2}&&:1 \end{aligned} \allowdisplaybreaks\\[5pt]
u_{C_{2m}}&=\partial^{-\mu'}x^{\lambda'}
(1-x)^{-\lambda_1-\mu-\mu^{(2)}-\cdots-\mu^{(m)}}
u_{H_{m+1}} \allowdisplaybreaks\\
m^2,m1^m,m-11^{m+1}&=1,10,01\oplus mm-1,m-11^{m-1},m-11^{m-1}\\
&=1^2,11,11\oplus (m-1)^2,m-11^{m-1},m-21^m \allowdisplaybreaks\\ [\Delta(C_{2m})]&=1^{(m+1)^2}\times1^{m+1}\cdot (m-1)^2=1^{m^2+3m+2}\cdot(m-1)^2\\ &\hspace{-.725cm} \begin{aligned}
C_{2m}&=H_1\oplus C_{2m-1}&&:2m+2\\
&=H_2\oplus C_{2m-2}&&:m(m+1)\\
&=(m-1)H_1\oplus H_{m+1}&&:2 \end{aligned} \allowdisplaybreaks\\
C_{2m+1}&\overset{m}{\underset{R2E0R0E0}\longrightarrow} H_{m+1},\quad
C_{2m+1}\overset{m-1}\longrightarrow H_{m+2}\\
C_{2m}&\overset{m-1}{\underset{R1E0R0E0}\longrightarrow} H_{m+1},\quad
C_n\overset{1}\longrightarrow C_{n-1} \end{align*}
\subsubsection{$D_n$} ($D_6=X_6$ : Extra case, $D_5=EO_5$) \begin{align*}
u_{D_5}&=\partial^{-\mu_5}(1-x)^{-\lambda_3-\mu_3-\mu_4}u_{E_4}\\
u_{D_6}&=\partial^{-\mu_6}(1-x)^{-\lambda_1-\mu-\mu_5}u_{D_5}\\
u_{D_n}&=\partial^{-\mu_n}(1-x)^{-\lambda'_n}u_{D_{n-2}}\quad(n\ge7) \allowdisplaybreaks\\
(2m-1)2,2^m1,2^{m-2}1^5&=10,01,10\oplus(2m-2)2,2^m,2^{m-3}1^6\\
&=10,10,01\oplus(2m-2)2,2^{m-1}1^2,2^{m-3}1^4\\
&=(m-1)1,1^m0,1^{m-2}1^2\oplus m1,1^m1,1^{m-2}1^3 \allowdisplaybreaks\\ m\ge2\ \Rightarrow\ [\Delta(D_{2m+1})]&=1^{6m+2}\cdot 2^{(m-1)(m-3)}\times
1^6\cdot 2^{2m-3}=1^{6m+8}\cdot 2^{m(m-2)}\\ &\hspace{-1.12cm} \begin{aligned}
D_{2m+1}&=H_1\oplus D_{2m}&&:m-2\\
&=H_1\oplus E_{2m}&&:5m\\
&=H_m\oplus H_{m+1}&&:10\\
&=2H_1\oplus D_{2m-1}&&:m(m-2) \end{aligned} \allowdisplaybreaks\\[5pt]
(2m-2)2,2^m,2^{m-3}1^6&=10,1,01\oplus
(2m-3)2,2^{m-1}1,2^{m-3}1^5\\
&=(m-1)1,1^m,1^{m-3}1^3\oplus(m-1)1,1^m,1^{m-3}1^3 \allowdisplaybreaks\\ m\ge3\ \Rightarrow\ [\Delta(D_{2m})]&=1^{6m+6}\cdot 2^{(m-1)(m-4)}\times
1^6\cdot 2^{2m-4}=1^{6m+10}\cdot 2^{m(m-3)}\\ &\hspace{-.74cm} \begin{aligned}
D_{2m}&=H_1\oplus D_{2m-1}&&:6m\\
&=H_m\oplus H_m&&:10\\
&=2H_1\oplus D_{2m-2}&&:m(m-3) \end{aligned} \allowdisplaybreaks\\
D_n &\overset{2}{\underset{R2E0}\longrightarrow} D_{n-2},\quad
D_n\overset{1}\longrightarrow D_{n-1},\quad
D_{2m+1}\overset{1}\longrightarrow E_{2m} \end{align*}
\subsubsection{$E_n$} ($E_5=C_5$, $E_4=EO_4$, $E_3=H_3$) \begin{align*}
u_{E_3}&= x^{-\lambda_0-\mu-\mu_3}\partial^{-\mu_3}(1-x)^{\lambda'_3}u_{H_2}\\
u_{E_4}&= \partial^{-\mu_4}u_{E_3}\\
u_{E_n}&=\partial^{-\mu_n}(1-x)^{\lambda'_n}u_{E_{n-2}}\quad(n\ge 5) \allowdisplaybreaks\\
(2m-1)2,2^{m-1}1^3,2^{m-1}1^3&=10,01,10\oplus (2m-2)2,2^{m-1}1^2,2^{m-2}1^4\\
&=(m-1)1,1^{m-1}1,1^{m-1}1\oplus m1,1^{m-1}1^3,1^{m-1}1^2\\
&=(m-2)1,1^{m-1}0,1^{m-1}0\oplus (m+1)1,1^{m-1}1^2,1^{m-1}1^3 \allowdisplaybreaks\\ m\ge2\ \Rightarrow\ [\Delta(E_{2m+1})]&=1^{6m-2}\cdot 2^{(m-2)^2}\times
1^6\cdot 2^{2m-3}=1^{6m+4}\cdot 2^{(m-1)^2}\\ &\hspace{-1.08cm} \begin{aligned} E_{2m+1}&=H_1\oplus E_{2m}&&:6(m-1)\\
&=H_{m-1}\oplus H_{m+2}&&:1\\
&=H_m\oplus H_{m+1}&&:9\\
&=2H_1\oplus E_{2m-1}&&:(m-1)^2 \end{aligned} \allowdisplaybreaks\\[5pt]
(2m-2)2,2^{m-1}1^2,2^{m-2}1^4&=10,10,01\oplus (2m-3)2,2^{m-2}1^3,2^{m-2}1^3\\
&=10,01,10\oplus (2m-3)2,2^{m-1}1,2^{m-3}1^5\allowdisplaybreaks\\
&=(m-2)1,1^{m-1}0,1^{m-2}1\oplus m1,1^{m-1}1^2,1^{m-2}1^3\\
&=(m-1)1,1^{m-1}1,1^{m-2}1^2\oplus (m-1)1,1^{m-1}1,1^{m-2}1^2 \allowdisplaybreaks\\ m\ge2\ \Rightarrow\ [\Delta(E_{2m})]&=1^{6m-4}\cdot 2^{(m-2)(m-3)}\times
1^6\cdot 2^{2m-4}=1^{6m+2}\cdot 2^{(m-1)(m-2)}\\ &\hspace{-0.72cm} \begin{aligned}
E_{2m}&=H_1\oplus E_{2m-1}&&:4(m-1)\\
&=H_1\oplus D_{2m-1}&&:2(m-2)\\
&=H_{m-1}\oplus H_{m+1}&&:4\\
&=H_m\oplus H_m&&:6\\
&=2H_1\oplus E_{2m-2}&&:(m-1)(m-2) \end{aligned} \allowdisplaybreaks\\
E_n &\overset{2}{\underset{R2E0}\longrightarrow} E_{n-2},\quad
E_n\overset{1}\longrightarrow E_{n-1},\quad
E_{2m}\overset{1}\longrightarrow D_{2m-1} \end{align*}
\subsubsection{$F_n$} ($F_5=B_5$, $F_4=EO_4$, $F_3=H_3$) \begin{align*}
u_{F_3}&=u_{H_3}\\
u_{F_4}&=\partial^{-\mu_4}(1-x)^{-\lambda_1-\lambda_0^{(3)}-\mu^{(3)}}
u_{F_3}\\
u_{F_n}&=\partial^{-\mu_n}(1-x)^{\lambda'_n}u_{F_{n-2}}\quad(n\ge 5) \allowdisplaybreaks\\
(2m-1)1^2,2^m1,2^{m-1}1^3&=10,10,01\oplus (2m-2)1^2,2^{m-1}1^2,2^{m-1}1^2\\
&=10,01,10\oplus (2m-2)1^2,2^m,2^{m-2}1^4\\
&=(m-1)1,1^m0,1^{m-1}1\oplus m1,1^m1,1^{m-1}1^2 \allowdisplaybreaks\\ m\ge1\ \Rightarrow\ [\Delta(F_{2m+1})]&=1^{4m+1}\cdot 2^{(m-1)(m-2)}\times
1^4\cdot 2^{2m-2}=1^{4m+5}\cdot 2^{m(m-1)}\\ &\hspace{-1.05cm} \begin{aligned}
F_{2m+1}&=H_1\oplus G_{2m}&&:3m\\
&=H_1\oplus F_{2m}&&:m-1\\
&=H_m\oplus H_{m+1}&&:6\\
&=2H_1\oplus F_{2m-1}&&:m(m-1) \end{aligned} \allowdisplaybreaks\\[5pt]
(2m-2)1^2,2^m,2^{m-2}1^4&=10,1,01\oplus (2m-3)1^2,2^{m-1}1,2^{m-2}1^3\\
&=(m-1)1,1^m,1^{m-2}1^2\oplus(m-1)1,1^m,1^{m-2}1^2 \allowdisplaybreaks\\ m\ge2\ \Rightarrow\ [\Delta(F_{2m})]&=1^{4m+2}\cdot 2^{(m-1)(m-3)}\times
1^4\cdot 2^{2m-3}=1^{4m+6}\cdot 2^{m(m-2)}\\ &\hspace{-0.7cm} \begin{aligned}
F_{2m}&=H_1\oplus F_{2m-1}&&:4m\\
&=H_m\oplus H_m&&:6\\
&=2H_1\oplus F_{2m-2}&&:m(m-2) \end{aligned} \allowdisplaybreaks\\
F_n &\overset{2}{\underset{R2E0}\longrightarrow} F_{n-2},\quad
F_n\overset{1}\longrightarrow F_{n-1},\quad
F_{2m+1}\overset{1}\longrightarrow G_{2m} \end{align*}
\subsubsection{$G_{2m}$} ($G_4=B_4$) \begin{align*}
u_{G_2}&=u_{H_2}\\
u_{G_{2m}}&=\partial^{-\mu_{2m}}(1-x)^{\lambda'_{2m}}u_{G_{2m-2}} \allowdisplaybreaks\\
(2m-2)1^2,2^{m-1}1^2,2^{m-1}1^2&=10,01,01\oplus
(2m-3)1^2,2^{m-1}1,2^{m-2}1^3\\
&=(m-2)1,1^{m-1}0,1^{m-1}0\oplus m1,1^{m-1}1^2,1^{m-1}1^2 \allowdisplaybreaks\\ m\ge2\ \Rightarrow\ [\Delta(G_{2m})]&=1^{4m-2}\cdot 2^{(m-2)^2}\times
1^4\cdot 2^{2m-3}=1^{4m+2}\cdot 2^{(m-1)^2}\\ &\hspace{-0.72cm} \begin{aligned}
G_{2m}&=H_1\oplus F_{2m-1}&&:4m\\
&=H_{m-1}\oplus H_{m+1}&&:2\\
&=2H_1\oplus G_{2m-2}&&:(m-1)^2 \end{aligned} \allowdisplaybreaks\\
G_{2m} &= H_1\oplus F_{2m-1} = H_{m-1}\oplus H_{m+1} \allowdisplaybreaks\\
G_{2m} &\overset{2}{\underset{R2E0}\longrightarrow} G_{2(m-1)},\quad
G_{2m}\overset{1}\longrightarrow F_{2m-1} \end{align*}
\subsubsection{$I_n$} ($I_{2m+1}=\text{III}^*_m$, $I_{2m}=\text{II}^*_m$, $I_3=P_3$) \begin{align*}
u_{I_{2m+1}}&=\partial^{-\mu'}x^{\lambda'}(c-x)^{\lambda''}u_{H_{m}} \allowdisplaybreaks\\
&\hspace{-1cm}(2m)1,m+1m,m+11^{m},m+11^{m}\\
&=10,10,10,01\oplus (2m-1)1,mm,m1^{m},m+11^{m-1}\\
&=20,11,11,11\oplus (2m-2)1,mm-1,m1^{m-1},m1^{m-1} \allowdisplaybreaks\\
[\Delta(I_{2m+1})]&=1^{m^2}\times
1^{2m}\cdot m\cdot(m+1)=1^{m^2+2m}\cdot m\cdot(m+1)\\ &\hspace{-0.9cm} \begin{aligned}
I_{2m+1}&=H_1\oplus I_{2m}&&:2m\\
&=H_2\oplus I_{2m-1}&&:m^2\\
&=mH_1\oplus H_{m+1}&&:1\\
&=(m+1)H_1\oplus H_m&&:1 \end{aligned} \allowdisplaybreaks\\[5pt]
u_{I_{2m}}&=\partial^{-\mu'}(1-cx)^{\lambda''}u_{H_{m}}\\
&\hspace{-1cm}(2m-1)1,mm,m1^{m},m+11^{m-1}\\
&=10,01,01,10\oplus (2m-2)1,mm-1,m1^{m-1},m1^{m-1}\\
&=20,11,11,11\oplus (2m-3)1,m-1m-1,m-11^{m-1},m1^{m-2} \allowdisplaybreaks\\
[\Delta(I_{2m})]&=1^{m^2}\times
1^{m}\cdot m^2=1^{m(m+1)}\cdot m^2\\ &\hspace{-0.6cm} \begin{aligned}
I_{2m}&=H_1\oplus I_{2m-1}&&:2m\\
&=H_2\oplus I_{2m-2}&&:m(m-1)\\
&=mH_1\oplus H_m&&:2 \end{aligned} \allowdisplaybreaks\\
I_{2m+1}&\overset{m+1}\longrightarrow H_m,\quad
I_{2m+1}\overset{m}\longrightarrow H_{m+1},\quad
I_{2m}\overset{m}\longrightarrow H_m,\quad
I_{n}\overset{1}\longrightarrow I_{n-1}\\
I_{2m+1}&\underset{R1E0}\longrightarrow I_{2m}
\underset{R2E0}\longrightarrow I_{2m-2} \end{align*}
\subsubsection{$J_n$} ($J_4=I_4$, $J_3=P_3$) \begin{align*}
u_{J_2}&=(c-x)^{\lambda'}u_{H_2}\\
u_{J_3}&=u_{P_3}\\
u_{J_n}&=\partial^{-\mu'_n}x^{\lambda'_n}u_{J_{n-2}}\quad(n\ge4) \allowdisplaybreaks\\
&\hspace{-1cm}(2m)1,(2m)1,2^{m}1,2^{m}1\\
&=10,10,01,10\oplus (2m-1)1,(2m-1)1,2^{m},2^{m-1}11\\
&=(m-1)1,m0,1^m0,1^m0\oplus(m+1),m1,1^m1,1^m1 \allowdisplaybreaks\\ [\Delta(J_{2m+1})]&=1^{2m}\cdot2^{(m-1)^2}\times
1^2\cdot 2^{2m-1}=1^{2m+2}\cdot2^{m^2}\\ &\hspace{-1cm} \begin{aligned}
J_{2m+1}&=H_1\oplus J_{2m}&&:2m\allowdisplaybreaks\\
&=H_m\oplus H_{m+1}&&:2\allowdisplaybreaks\\
&=2H_1\oplus J_{2m-2}&&:m^2 \end{aligned} \allowdisplaybreaks\\[5pt]
&\hspace{-1cm}(2m-1)1,(2m-1)1,2^m,2^{m-1}1^2\\
&=10,10,1,01\oplus (2m-2)1,(2m-2)1,2^{m-1}1,2^{m-1}1\\
&=(m-1)1,m0,1^m,1^{m-1}1\oplus m0,(m-1)1,1^m,1^{m-1}1 \allowdisplaybreaks\\
[\Delta(J_{2m})]&=1^{2m}\cdot2^{(m-1)(m-2)}\times
1^2\cdot2^{2m-2}=1^{2m+2}\cdot2^{m(m-1)}\\ &\hspace{-.65cm} \begin{aligned}
J_{2m}&=H_1\oplus J_{2m-1}&&:2m\allowdisplaybreaks\\
&=H_m\oplus H_m&&:2\allowdisplaybreaks\\
&=2H_1\oplus J_{2m-2}&&:m(m-1)\\ \end{aligned} \allowdisplaybreaks\\
J_{n}&\overset{2}{\underset{R2E0}\longrightarrow} J_{n-2}\ (n\ge6),\quad
J_{n}\overset{1}\longrightarrow J_{n-1} \end{align*}
\subsubsection{$K_n$} ($K_5=M_5$, $K_4=I_4$, $K_3=P_3$) \begin{align*}
&u_{K_{2m+1}}=\partial^{\mu+\lambda'+\lambda''}(c'-x)^{\lambda'}(c''-x)^{\lambda''}u_{P_m} \allowdisplaybreaks\\
&m+1m,m+1m,(2m)1,(2m)1,(2m)1,\ldots\in \mathcal P_{m+3}^{(2m+1)}\\
&\quad= 11,11,11,20,20,\ldots\oplus mm-1,mm-1,(2m-1)0,(2m-2)1,(2m-2)1,\ldots \allowdisplaybreaks\\
&[\Delta(K_{2m+1})]=1^{m+1}\cdot(m-1)\times
m^2\cdot(m+1)=1^{m+1}\cdot(m-1)\cdot m^2\cdot(m+1)\\
& \qquad\begin{aligned}
K_{2m+1}&=H_2\oplus K_{2m-1}&&:m+1\\
&=(m-1)H_1\oplus P_{m+2}&&:1\\
&=mH_1\oplus P_{m+1}&&:2\\
&=(m+1)H_1\oplus P_m&&:1 \end{aligned} \allowdisplaybreaks\\[5pt]
&u_{K_{2m}}=\partial^{-\mu'}(c'-x)^{\lambda'}u_{P_m}\\
&mm,mm-11,(2m-1)1,(2m-1)1,\ldots\in\mathcal P_{m+2}^{(2m)}\\
&\quad=01,001,10,10,10,\ldots\oplus mm-1,mm-10,(2m-2)1,(2m-2)1,\ldots \allowdisplaybreaks\\
&\quad=11,110,11,20,20,\ldots\oplus m-1m-1,m-1m-21,(2m-2)0,(2m-3)1,\ldots \allowdisplaybreaks\\
&[\Delta(K_{2m})]=1^{m+1}\cdot(m-1)\times
1\cdot(m-1)\cdot m^2=1^{m+2}\cdot(m-1)^2\cdot m^2 \allowdisplaybreaks\\ &\qquad\begin{aligned}
K_{2m}&=H_1\oplus K_{2m-1}&&:2\\
&=H_2\oplus K_{2m-2}&&:m\\
&=(m-1)H_1\oplus P_{m+1}&&:2\\
&=mH_1\oplus P_m&&:2 \end{aligned} \allowdisplaybreaks\\
&\ \ K_{2m+1}\overset{m+1}\longrightarrow P_{m},\quad
K_{2m+1}\overset{m}{\underset{R1}\longrightarrow} P_{m+1},\quad
K_{2m+1}\overset{m-1}\longrightarrow P_{m+2}\\
&\ \ K_{2m}\overset{m}{\underset{R1}\longrightarrow} P_{m},\quad
K_{2m}\overset{m-1}\longrightarrow P_{m+1},\quad
K_{2m}\overset{1}\longrightarrow K_{2m-1} \end{align*}
\subsubsection{$L_{2m+1}$} ($L_5=J_5$, $L_3=H_3$) \begin{align*}
&u_{L_{2m+1}}=\partial^{-\mu'}x^{\lambda'}u_{P_{m+1}} \allowdisplaybreaks\\
&mm1,mm1,(2m)1,(2m)1,\ldots\in\mathcal P_{m+2}^{(2m+1)}\\
&\quad=001,010,10,10,\ldots\oplus mm0,mm-11,(2m-1)1,(2m-1)1,\ldots\\
&\quad=110,110,11,20,\ldots\oplus m-1m-10,m-1m-11,(2m-1)0,(2m-2)1,\ldots \allowdisplaybreaks\\
&[\Delta(L_{2m+1})]=1^{m+2}\cdot m\times
1^{2}\cdot m^3=1^{m+4}\cdot m^4 \\&\quad\ \ \ \begin{aligned}
L_{2m+1}&=H_1\oplus K_{2m}&&:4\\
&=H_2\oplus L_{2m-1}&&:m\\
&=mH_1\oplus P_{m+1}&&:4 \end{aligned} \allowdisplaybreaks\\
&\ \ L_{2m+1}=H_1\oplus K_{2m},\quad L_{2m+1}=H_2\oplus L_{2m-1}\\
&\ \ L_{2m+1}\overset{m}{\underset{R2E0}\longrightarrow} P_{m+1},\quad
L_{2m+1}\overset{1}\longrightarrow K_{2m} \end{align*}
\subsubsection{$M_n$} ($M_5=K_5$, $M_4=I_4$, $M_3=P_3$) \begin{align*}
&u_{M_{2m+1}}=\partial^{\mu+\lambda'_3+\cdots+\lambda'_{m+2}}
(c_3-x)^{\lambda'_3}\cdots(c_{m+2} -x)^{\lambda'_{m+2}}u_{H_2} \allowdisplaybreaks\\
&(2m)1,(2m)1,(2m)1,(2m-1)2,(2m-1)2,\ldots\in \mathcal P^{(2m+1)}_{m+3}\\
&\quad=m-11,m0,m0,m-11,m-11,\ldots
\oplus m+10,m1,m1,m1,m1,\ldots\\
&\quad
=m-10,m-10,m-10,m-21,m-21,\ldots\\
&\quad\quad
\oplus m+11,m+11,m+11,m+11,m+11,\ldots \allowdisplaybreaks\\
&[\Delta(M_{2m+1})]=1^{4}\times
2^{m}\cdot(2m-1)=1^4\cdot2^m\cdot(2m-1) \allowdisplaybreaks\\ &\qquad\begin{aligned}
M_{2m+1}&=P_{m-1}\oplus P_{m+2} &&:1\\
&=P_{m}\oplus P_{m+1} &&:3\\
&=2H_1\oplus M_{2m-1} &&:m\\
&=(2m-1)H_1\oplus H_2 &&:1
\end{aligned} \allowdisplaybreaks\\[5pt]
&u_{M_{2m}}=\partial^{-\mu'}(c_3-x)^{\lambda'_3}\cdots(c_{m+1} -x)^{\lambda'_{m+1}}
u_{H_2}\\
&(2m-2)1^2,(2m-1)1,(2m-1)1,(2m-2)2,\ldots\in\mathcal P^{(2m)}_{m+2}\\
&\quad=01,10,10,10,\ldots\oplus (2m-2)1,(2m-2)1,(2m-2)1,(2m-3)2,\ldots \allowdisplaybreaks\\
&\quad=m-21,m-10,m-10,m-21,\ldots\oplus m1,m1,m1,m1,\ldots\\
&\quad=m-11,m-11,m0,m-11,\ldots\oplus m-11,m0,m-11,m-11,\ldots \allowdisplaybreaks\\
&[\Delta(M_{2m})]=1^{4}\times
1^{2}\cdot2^{m-1}\cdot(2m-2)=1^6\cdot2^{m-1}\cdot(2m-2)\\ &\qquad\begin{aligned}
M_{2m}&=H_1\oplus M_{2m-1} &&:2 \\
&=P_{m-1}\oplus P_{m+1} &&:2\\
&=P_m\oplus P_m&&:2\\
&=2H_1\oplus M_{2m-2} &&:m-1\\
&=(2m-2)H_1\oplus H_2 &&:1\\
\end{aligned} \allowdisplaybreaks\\
&\ \ M_{n}\overset{n-2}\longrightarrow H_2,\quad
M_{n}\overset{2}\longrightarrow M_{n-2},\quad
M_{2m}\overset{1}{\underset{R1E0}\longrightarrow} M_{2m-1}
\underset{R1}\longrightarrow M_{2m-3} \end{align*}
\subsubsection{$N_n$}\label{sec:minseries} ($N_6=\text{IV}^*$, $N_5=I_5$, $N_4=G_4$, $N_3=H_3$) \index{tuple of partitions!rigid!minimal series} \begin{align*}
&u_{N_{2m+1}}=\partial^{-\mu'}x^{\lambda'}(c_3-x)^{\lambda'_3}\cdots(c_{m+1}-x)^{\lambda'_{m+1}}u_{H_2} \allowdisplaybreaks\\
&(2m-1)1^2,(2m-1)1^2,(2m)1,(2m-1)2,(2m-1)2,\ldots\in \mathcal P^{(2m+1)}_{m+2}\\
&\quad =10,01,10,10,10\ldots\\
&\quad\quad\oplus
(2m-2)1^2,(2m-1)1,(2m-1)1,(2m-2)2,(2m-2)2,\ldots\\
&\quad =m-11,m-11,m0,m-11,m-11,\ldots
\oplus
m1,m1,m1,m1,m1,\ldots \allowdisplaybreaks\\
&[\Delta(N_{2m+1})]=1^{4}\times
1^{4}\cdot2^{m-1}\cdot(2m-1)=1^{8}\cdot2^{m-1}\cdot(2m-1)\\ &\qquad\begin{aligned}
N_{2m+1}&=H_1\oplus M_{2m}&&:4\\
&=P_m\oplus P_{m+1}&&:4\\
&=2H_1\oplus N_{2m-1}&&:m-1\\
&=(2m-1)H_1\oplus H_2&&:1\\ \end{aligned} \allowdisplaybreaks\\[5pt]
&u_{N_{2m}}=\partial^{-\mu'}x^{\lambda'_0}(1-x)^{\lambda'_1}
(c_3-x)^{\lambda'_3}\cdots(c_{m}-x)^{\lambda'_{m}}u_{H_2}
\quad(m\ge2)\\
&(2m-2)1^2,(2m-2)1^2,(2m-2)1^2,(2m-2)2,(2m-2)2,\ldots\in \mathcal P^{(2m)}_{m+1}
\\
&\quad =01,10,10,10,10\ldots\\
&\quad\quad
\oplus (2m-2)1,(2m-3)1^2,(2m-3)1^2,(2m-3)2,(2m-3)2,\ldots\allowdisplaybreaks\\
&\quad =m-11,m-11,m-11,m-11,m-11,\ldots\\
&\quad\quad
\oplus m-11,m-11,m-11,m-11,m-11,\ldots \allowdisplaybreaks\\
&[\Delta(N_{2m})]=1^{4}\times
1^{6}\cdot 2^{m-2}\cdot(2m-2)=1^{10}\cdot2^{m-2}\cdot(2m-2) \allowdisplaybreaks\\ &\qquad\begin{aligned}
N_{2m}&=H_1\oplus N_{2m-1}&&:6\\
&=P_m\oplus P_m&&:4\\
&=2H_1\oplus N_{2m-2}&&:m-2\\
&=(2m-2)H_1\oplus H_2&&:1 \end{aligned} \allowdisplaybreaks\\
&N_{n}\overset{n-2}\longrightarrow H_2,\quad
N_{n}\overset{2}\longrightarrow N_{n-2},\quad
N_{2m+1}\overset{1}{\underset{R1E0}\longrightarrow} M_{2m},\quad
N_{2m}\overset{1}{\underset{R1E0}\longrightarrow} N_{2m-1} \end{align*}
\subsubsection{minimal series}\label{sec:minS} \index{tuple of partitions!rigid!minimal series} The tuple $11,11,11$ corresponds to Gauss hypergeometric series, which has three parameters. Since the action of additions is easily analyzed, we consider the number of parameters of the equation corresponding to a rigid tuple $\mathbf m=\bigl(m_{j,\nu}\bigr)_{\substack{0\le j\le p\\1\le \nu\le n_j}} \in\mathcal P_{p+1}^{(n)}$ modulo additions and the Fuchs condition equals \begin{equation}\label{eq:NSpara}
n_0+n_1+\cdots+n_p - (p+1). \end{equation} Here we assume that $0<m_{j,\nu}<n$ for $1\le \nu\le n_j$ and $j=0,\dots,p$.
We call the number given by \eqref{eq:NSpara} the \textsl{effective length}\/ of $\mathbf m$. The tuple $11,11,11$ is the unique rigid tuple of partitions whose effective length equals 3. Since the reduction $\partial_{max}$ never increase the effective length and the tuple $\mathbf m\in\mathcal P_3$ satisfying $\partial_{max}=11,11,11$ is $21,111,111$ or $211,211,211$, it is easy to see that the non-trivial rigid tuple $\mathbf m\in\mathcal P_3$ whose effective length is smaller than $6$ is $H_2$ or $H_3$.
The rigid tuple of partitions with the effective length 4 is also uniquely determined by its order, which is \index{00P@$P_{p+1,n},\ P_n$} \begin{equation}
\begin{split}
P_{4,2m+1}&: m+1m,m+1m,m+1m,m+1m\\
P_{4,2m}&: m+1m-1,mm,mm,mm
\end{split} \end{equation} with $m\in\mathbb Z_{>0}$. Here $P_{4,2m+1}$ is a generalized Jordan-Pochhammer tuple in Example~\ref{ex:JPH} i).\index{Jordan-Pochhammer!generalized}
In fact, if $\mathbf m\in\mathcal P$ is rigid with the effective length $4$, the argument above shows $\mathbf m\in\mathcal P_4$ and $n_j=2$ for $j=0,\dots,3$. Then $2 = \sum_{j=0}^3 m_{j,1}^2+\sum_{j=0}^3(n-m_{j,1})^2-2n^2$ and $\sum_{j=0}^3(n-2m_{j,1})^2=4$ and therefore $\mathbf m=P_{4,2m+1}$ or $P_{4,2m}$.
We give decompositions of $P_{4,n}$: \begin{align*}
&
m+1,m;m+1,m;m+1,m;m+1,m\\
&\quad=k,k+1;k+1,k;k+1,k;k+1,k\\
&\qquad\oplus m-k+1,m-k-1;m-k,m-k;m-k,m-k;m-k,m-k\\
&\quad=2(k+1,k;k+1,k;k+1,k;\ldots)\\
&\qquad\oplus m-2k-1,m-2k;m-2k-1,m-2k;m-2k-1,m-2k;\ldots\\
&[\Delta(P_{4,2m+1})]=
1^{4m-4}\cdot2^{m-1}\times 1^4\cdot 2=1^{4m}\cdot2^m\\
&\qquad\begin{aligned}
P_{4,2m+1}&=P_{4,2k+1}\oplus P_{4,2(m-k)}&&:4\quad(k=0,\ldots,m-1)\\
&=2P_{4,2k+1}\oplus P_{4,2m-4k-1}&&:1\quad(k=0,\ldots,m-1)
\end{aligned} \end{align*} Here $P_{k,-n}=-P_{k,n}$ and in the above decompositions there appear ``tuples of partitions" with negative entries corresponding formally to elements in $\Delta^{re}$ with \eqref{eq:Kazpart} (cf.~Remark~\ref{rem:length}~i)).
It follows from the above decompositions that the Fuchsian equation with the Riemann scheme \begin{gather*}
\begin{Bmatrix}
\infty & 0 & 1 & c_3\\
[\lambda_{0,1}]_{(m+1)} & [\lambda_{1,1}]_{(m+1)}
& [\lambda_{2,1}]_{(m+1)} & [\lambda_{3,1}]_{(m+1)}\\
[\lambda_{0,2}]_{(m)} & [\lambda_{1,2}]_{(m)}
& [\lambda_{2,1}]_{(m)} & [\lambda_{3,2}]_{(m)}
\end{Bmatrix}\\
\sum_{j=0}^4\bigl((m+1)\lambda_{j,1}+m\lambda_{j,2}\bigr)=2m
\qquad(\text{Fuchs relation}). \end{gather*} is irreducible if and only if \[
\sum_{j=0}^4\sum_{\nu=1}^2
\bigl(k+\delta_{\nu,1}+(1-2\delta_{\nu,1})\delta_{j,i}\bigr)
\lambda_{j,\nu}\notin\mathbb Z\qquad(i=0,1,\ldots,5,\ k=0,1,\ldots,m). \]
When $\mathbf m=P_{4,2m}$, we have the following. \begin{align*}
&
m+1,m-1;m,m;m,m;m,m\\
&\quad=k+1,k;k+1,k;k+1,k;k+1,k\\
&\qquad\oplus m-k,m-k-1;m-k-1,m-k;m-k-1,m-k;m-k-1,m-k\\
&\quad=2(k+1,k-1;k,k;k,k;k,k)\\
&\qquad\oplus m-2k-1,m-2k+1;m-2k,m-2k;m-2k,m-2k;m-2k;m-2k\\
&[\Delta(P_{4,2m})]=
1^{4m-4}\cdot 2^{m-1}\times 1^4=1^{4m}\cdot 2^{m-1}\\
&\ \ \begin{aligned}
P_{4,2m}&=P_{4,2k+1}\text{\small$(=k+1,k;k+1,k;\ldots)$}
\oplus P_{4,2m-2k+1}&&\!:4\ \ (k=0,\ldots,m-1)\\
&=2P_{4,2k}\oplus P_{4,2m-4k}&&\!:1\ \ (k=1,\ldots,m-1)
\end{aligned} \end{align*} \[
P_{4,n}\xrightarrow1 P_{4,n-1},\ P_{4,2m+1}\xrightarrow2 P_{4,2m-1} \]
Roberts \cite{Ro} classifies the rigid tuples $\mathbf m\in\mathcal P_{p+1}$ so that \begin{equation}\index{tuple of partitions!rigid!minimal series}
\frac1{n_0}+\cdots+\frac1{n_p}\ge p-1. \end{equation} They are tuples $\mathbf m$ in 4 series $\alpha$, $\beta$, $\gamma$, $\delta$, which are close to the tuples $r\tilde E_6$, $r\tilde E_7$, $r\tilde E_8$ and $r\tilde D_4$, namely, $(n_0,\dots,n_p)=(3,3,3)$, $(2,2,4)$, $(2,3,6)$ and $(2,2,2,2)$, respectively (cf.~\eqref{eq:R2ineq}), and the series are called \textsl{minimal series}. Then $\delta_n=P_{4,n}$ and the tuples in the other three series belong to $\mathcal P_3$. For example, the tuples $\mathbf m$ of type $\alpha$ are \begin{equation}
\begin{aligned}
\alpha_{3m}&=m+1mm-1,m^3,m^3,&\alpha_3&=H_3,\\
\alpha_{3m\pm1}&=m^2m\pm1,m^2m\pm1,m^2m\pm1,&\alpha_4&=B_4,
\end{aligned} \end{equation} which are characterized by the fact that their effective lengths equal 6 when $n\ge 4$. As in other series, we have the following: \begin{align*}
\alpha_n&\xrightarrow 1\alpha_{n-1},\ \ \alpha_{3m+1}\xrightarrow2 \alpha_{3m-1}\\
[\Delta(\alpha_{3m})]&=[\Delta(\alpha_{3m-1})]\times 1^5,\
[\Delta(\alpha_{3m-1})]=[\Delta(\alpha_{3m-2})]\times 1^4,\\
[\Delta(\alpha_{3m-2})]&=[\Delta(\alpha_{3m-4})]\times 1^6\cdot 2\\
[\Delta(\alpha_{3m-1})]&=[\Delta(\alpha_2)]\times 1^{10(m-1)}\cdot 2^{m-1}
=1^{10m-6}\cdot 2^{m-1}\\
[\Delta(\alpha_{3m})]&=1^{10m-1}\cdot2^{m-1}\\
[\Delta(\alpha_{3m-2})]&=1^{10m-10}\cdot2^{m-1} \end{align*} \begin{align*}
&\alpha_{3m}=m+1mm-1,m^3,m^3\\
&\ \ =kkk-1,k^2k-1,k^2k-1\\
&\quad\oplus (m-k+1)(m-k)(m-k),(m-k)^2(m-k+1),(m-k)^2(m-k+1)\\
&\ \ =k+1k-1k,k^3,k^3\\
&\quad\oplus(m-k+1)(m-k)(m-k-1),(m-k)^3,(m-k)^3\\
&\ \ =2(k+1kk-1,k^3,k^3)\\
&\quad\oplus(m-2k-1)(m-2k)(m-2k+1),(m-2k)^3,(m-2k)^3 \allowdisplaybreaks\\
&\begin{aligned}
\alpha_{3m}&=\alpha_{3k-1}\oplus\alpha_{3(m-k)+1}&&:9\quad(k=1,\dots,m)\\
&=\alpha_{3k}\oplus\alpha_{3(m-k)}&&:1\quad(k=1,\ldots,m-1)\\
&=2\alpha_{3k}\oplus\alpha_{3(m-2k)}&&:1\quad(k=1,\ldots,m-1)
\end{aligned}\allowdisplaybreaks\\
&\alpha_{3m-1}=mmm-1,mmm-1,mmm-1\\
&\ \ =kk-1k-1,kk-1k-1,kk-1k-1\\
&\quad\oplus (m-k)(m-k+1)(m-k),(m-k)(m-k+1)(m-k),\cdots\\
&\ \ =k+1kk-1,k^3,k^3\\
&\quad\oplus(m-k-1)(m-k)(m-k),(m-k)(m-k)(m-k-1),\cdots\\
&\ \ =2(kkk-1,kkk-1,kkk-1)\\
&\quad\oplus(m-2k)(m-2k)(m-2k+1),(m-2k)(m-2k)(m-2k+1),\cdots \allowdisplaybreaks\\
&\begin{aligned}
\alpha_{3m-1}
&=\alpha_{3k-2}(=k,k-1,k-1;\cdots)\oplus\alpha_{3(m-k)+1}&&:4\quad(k=1,\dots,m)\\
&=\alpha_{3k}\oplus\alpha_{3(m-k)-1}&&:6\quad(k=1,\ldots,m-1)\\
&=2\alpha_{3k-1}\oplus\alpha_{3(m-2k)+1}&&:1\quad(k=1,\ldots,m-1)
\end{aligned} \allowdisplaybreaks\\
&\alpha_{3m-2}=mm-1m-1,mm-1m-1,mm-1m-1\\
&\ \ =kkk-1,kkk-1,kkk-1\\
&\quad\oplus (m-k)(m-k-1)(m-k),(m-k)(m-k-1)(m-k),\cdots\\
&\ \ =k+1kk-1,k^3,k^3\\
&\quad\oplus(m-k-1)(m-k-1)(m-k),(m-k)(m-k-1)(m-k-1),\cdots\\
&\ \ =2(kk-1k-1,kk-1k-1,kk-1k-1)\\
&\quad\oplus(m-2k)(m-2k+1)(m-2k+1),(m-2k)(m-2k+1)(m-2k+1),\cdots \allowdisplaybreaks\\
&\begin{aligned}
\alpha_{3m-2}
&=\alpha_{3k-1}(=k,k-1,k-1;\cdots)\oplus\alpha_{3(m-k)-1}&&:4\quad(k=1,\dots,m-1)
\\
&=\alpha_{3k}\oplus\alpha_{3(m-k)-2}&&:6\quad(k=1,\ldots,m-1)\\
&=2\alpha_{3k-2}\oplus\alpha_{3(m-2k)+2}&&:1\quad(k=1,\ldots,m-1)
\end{aligned} \end{align*}
The analysis of the other minimal series \begin{align*}
\beta_{4m,2}&=(2m+1)(2m-1),m^4,m^4&\beta_{4,2}&=H_4\\
\beta_{4m,4}&=(2m)^2,m^4,(m+1)m^2(m-1)&\beta_{4,4}&=EO_4\\
\beta_{4m\pm1}&=(2m)(2m\pm1),(m\pm1)m^3,(m\pm1)m^3&\beta_5&=C_5,\ \beta_3=H_3\\
\beta_{4m+2}&=(2m+1)^2,(m+1)^2m^2,(m+1)^2m^2 \allowdisplaybreaks\\[4pt]
\gamma_{6m,2}&=(3m+1)(3m-1),(2m)^3,m^6&\gamma_{6,2}&=D_6=X_6\\
\gamma_{6m,3}&=(3m)^2,(2m+1)(2m)(2m-1),m^6&\gamma_{6,3}&=EO_6\\
\gamma_{6m,6}&=(3m)^2,(2m)^3,(m+1)m^4(m-1) \allowdisplaybreaks\\
\gamma_{6m\pm1}&=(3m)(3m\pm1),(2m)^2(2m\pm1),m^5(m\pm1)&\gamma_5&=EO_5 \allowdisplaybreaks\\
\gamma_{6m\pm2}&=(3m\pm1)(3m\pm1),(2m)(2m\pm1)^2,m^4(m\pm1)^2&\gamma_4&=EO_4 \allowdisplaybreaks\\
\gamma_{6m+3}&=(3m+2)(3m+1),(2m+1)^3,(m+1)^3m^3&\gamma_3&=H_3 \end{align*} and general $P_{p+1,n}$ will be left to the reader as an exercise.
\subsubsection{Relation between series}\label{sec:Bseries} We have studied the following sets of families of spectral types of Fuchsian differential equations which are closed under the irreducible subquotients in the Grothendieck group. \begin{align*}
&\{H_n\}&&\text{(hypergeometric family)}\allowdisplaybreaks\\
&\{P_n\}&&\text{(Jordan-Pochhammer series)}\allowdisplaybreaks\\
&\{A_n=EO_n\}&&\text{(even/odd family)}\allowdisplaybreaks\\
&\{B_n,\ C_n,\ H_n\}&&\text{(3 singular points)}\allowdisplaybreaks\\
&\{C_n,\ H_n\}&&\text{(3 singular points)}\allowdisplaybreaks\\
&\{D_n,\ E_n,\ H_n\}&&\text{(3 singular points)}\allowdisplaybreaks\\
&\{F_n,\ G_{2m},\ H_n\}&&\text{(3 singular points)}\allowdisplaybreaks\\
&\{I_n,\ H_n\}&&\text{(4 singular points)}\allowdisplaybreaks\\
&\{J_n,\ H_n\}&&\text{(4 singular points)}\allowdisplaybreaks\\
&\{K_n,\ P_n\}&&\text{($[\tfrac{n+5}{2}]$ singular points)}\allowdisplaybreaks\\
&\{L_{2m+1},\ K_n,\ P_n\}&&\text{($m+2$ singular points)}\allowdisplaybreaks\\
&\{M_n,\ P_n\}&&\text{($[\tfrac{n+5}{2}]$ singular points)}
\qquad\supset\{M_{2m+1},\ P_n\}\allowdisplaybreaks\\
&\{N_n,\ M_n,\ P_n\}&&\text{($[\tfrac{n+3}{2}]$ singular points)}
\qquad\supset\{N_{2m+1},\ M_n.\ P_n\}
\allowdisplaybreaks\\
&\{P_{4,n}=\delta_n\}&&\text{(4 effective parameters)}\allowdisplaybreaks\\
&\{\alpha_n\}&&\text{(6 effective parameters and 3 singular points)} \end{align*}
\index{tuple of partitions!rigid!Yokoyama's list} Yokoyama classified $\mathbf m= \bigl(m_{j,\nu}\bigr)_{\substack{0\le j\le p\\1\le \nu\le n_j}} \in\mathcal P_{p+1}$ such that \begin{align} &\text{$\mathbf m$ is irreducibly realizable}, \allowdisplaybreaks\\ &m_{0,1}+\cdots+m_{p-1,1}=(p-1)\ord\mathbf m\quad (\text{$\mathbf m$ is of Okubo type}), \allowdisplaybreaks\\ &m_{j,\nu}=1\quad(0\le j\le p-1,\ 2\le\nu\le n_j). \end{align} \index{Okubo type} The tuple $\mathbf m$ satisfying the above conditions is in the following list given by \cite[Theorem 2]{Yo} (cf.~\cite{Ro}).
\begin{tabular}{|c|c|c|c|c|}\hline Yokoyama&type&order& p+1 &tuple of partitions\\ \hline\hline $\text{I}_n$&$H_n$&$n$ & $3$ & $1^n,n-11,1^n$\\ \hline $\text{I}^*_n$&$P_n$ & $n$ & $n+1$ & $n-11,n-11,\ldots,n-11$ \\ \hline $\text{II}_n$&$B_{2n}$&$2n$&$3$ & $n1^n, n1^n,nn-11$\\ \hline $\text{II}^*_n$ &$I_{2n}$ &$2n$&$4$ & $n1^n,n+11^{n-1},2n-11,nn$\\ \hline $\text{III}_n$&$B_{2n+1}$&$2n+1$ &$3$&$ n1^{n+1},n+11^n,nn1$\\ \hline $\text{III}^*_n$&$I_{2n+1}$&$2n+1$& $4$& $ n+11^n,n+11^n,(2n)1,n+1n$\\ \hline $\text{IV}\phantom{^*}$&$F_6$&$6$& $3$ &$21111,411,222$\\ \hline $\text{IV}^*$&$N_6$&$6$& $4$ &$411,411,411,42$\\ \hline \end{tabular}
\subsection{Appell's hypergeometric functions}\label{sec:ApEx} First we recall the Appell hypergeometric functions. \index{hypergeometric equation/function!Appell} \begin{align}
F_1(\alpha;\beta,\beta';\gamma;x,y)&=\sum_{m,n=0}^\infty
\frac{(\alpha)_{m+n}(\beta)_m(\beta')_n}
{(\gamma)_{m+n}m!n!}x^my^n,\label{eq:ApF1} \allowdisplaybreaks\\
F_2(\alpha;\beta,\beta';\gamma,\gamma';x,y)&=\sum_{m,n=0}^\infty
\frac{(\alpha)_{m+n}(\beta)_m(\beta')_n}
{(\gamma)_m(\gamma')_nm!n!}x^my^n, \allowdisplaybreaks\\
F_3(\alpha,\alpha';\beta,\beta';\gamma;x,y)&=\sum_{m,n=0}^\infty
\frac{(\alpha)_m(\alpha')_n(\beta)_m(\beta')_n}
{(\gamma)_{m+n}m!n!}x^my^n,\label{eq:F3} \allowdisplaybreaks\\
F_4(\alpha;\beta;\gamma,\gamma';x,y)&=\sum_{m,n=0}^\infty
\frac{(\alpha)_{m+n}(\beta)_{m+n}}
{(\gamma)_m(\gamma')_nm!n!}x^my^n. \end{align} They satisfy the following equations \begin{align} \iffalse
&x(1-x)\frac{\partial^2F_1}{\partial x^2}
+y(1-x)\frac{\partial^2F_1}{\partial x\partial y}
+\bigl(\gamma-(\alpha+\beta+1)x\bigr)\frac{\partial F_1}{\partial x}
-\beta y\frac{\partial F_1}{\partial y} -\alpha\beta F_1=0, \allowdisplaybreaks\\
&x(1-x)\frac{\partial^2F_2}{\partial x^2}
->xy\frac{\partial^2F_2}{\partial x\partial y}
+>\bigl(\gamma-(\alpha+\beta+1)x\bigr)\frac{\partial F_2}{\partial x}
-\beta y\frac{\partial F_2}{\partial y} -\alpha\beta F_2=0, \allowdisplaybreaks\\
&x(1-x)\frac{\partial^2F_3}{\partial x^2}
+y\frac{\partial^2F_3}{\partial x\partial y}
+\bigl(\gamma-(\alpha+\beta+1)x\bigr)\frac{\partial F_3}{\partial x}
-\alpha\beta F_3=0, \allowdisplaybreaks\\
&x(1-x)\frac{\partial^2F_4}{\partial x^2}
-2xy\frac{\partial^2F_4}{\partial x\partial y} - y^2\frac{\partial^2 F_4}{\partial y^2}
+\bigl(\gamma-(\alpha+\beta+1)x\bigr)\frac{\partial F_4}{\partial x}
- (\alpha+\beta+1)y\frac{\partial F_4}{\partial y} \allowdisplaybreaks\\
&\qquad
-\alpha\beta F_4=0.\\ \fi
\Bigl((\vartheta_x+\vartheta_y+\alpha)(\vartheta_x+\beta)-\partial_x(\vartheta_x+\vartheta_y+\gamma-1)\Bigr)F_1&=0,\\
\Bigl((\vartheta_x+\vartheta_y+\alpha)(\vartheta_x+\beta)-\partial_x(\vartheta_x+\gamma-1)\Bigr)F_2&=0, \allowdisplaybreaks\\
\Bigl((\vartheta_x+\alpha)(\vartheta_x+\beta)-\partial_x(\vartheta_x+\vartheta_y+\gamma-1)\Bigr)F_3&=0,\\
\Bigl((\vartheta_x+\vartheta_y+\alpha)(\vartheta_x+\vartheta_y+\beta)
-\partial_x(\vartheta_x+\gamma-1)\Bigr)F_4&=0. \end{align} Similar equations hold under the symmetry $x\leftrightarrow y$ with $(\alpha,\beta,\gamma)\leftrightarrow (\alpha',\beta',\gamma')$.
\subsubsection{Appell's $F_1$} \index{hypergeometric equation/function!Appell!$F_1$} First we examine $F_1$. Put \begin{align*} u(x,y)&:=\int_0^xt^\alpha(1-t)^\beta(y-t)^{\gamma-1}
(x-t)^{\lambda-1} dt\qquad(t=xs)\\
&=\int_0^1 x^{\alpha+\lambda+1}s^\alpha(1-xs)^\beta(y-xs)^{\gamma-1}
(1-s)^{\lambda-1} ds
\\
&=x^{\alpha+\lambda}y^{\gamma-1}\int_0^1
s^\alpha(1-s)^{\lambda-1}(1-xs)^\beta\Bigl(1-\frac yxs\Bigr)^{\gamma-1} ds,\\ h_x&:= x^\alpha(x-1)^\beta(x-y)^{\gamma-1}. \end{align*} Since the left ideal of $\overline W[x,y]$ is not necessarily generated by a single element, we want to have good generators of $\RAd(\partial_x^{-\lambda})\circ \RAd(h_x)\bigl(W[x,y]\partial_x+W[x,y]\partial_y\bigr)$ and we have \begin{align*} P&:=\Ad(h_x)\partial_x=\partial_x-\frac{\alpha}{x}-\frac{\beta}{x-1}-\frac{\gamma-1}{x-y},\\ Q&:=\Ad(h_x)\partial_y=\partial_y+\frac{\gamma-1}{x-y},\\ R&:=xP+yQ=x\partial_x+y\partial_y-(\alpha+\gamma-1)-\frac{\beta x}{x-1}, \allowdisplaybreaks\\ S&:=\partial_x(x-1)R= (\vartheta_x+1)(\vartheta_x+\vartheta_y-\alpha-\beta-\gamma+1) -\partial_x(\vartheta_x+\vartheta_y-\alpha-\gamma+1) \allowdisplaybreaks\\ T&:=\partial_x^{-\lambda}\circ S\circ\partial_x^{\lambda}\\
&=(\vartheta_x-\lambda+1)(\vartheta_x+\vartheta_y-\alpha-\beta-\gamma-\lambda+1) -\partial_x(\vartheta_x+\vartheta_y-\alpha-\gamma-\lambda+1) \end{align*} with \begin{align*} a=-\alpha-\beta-\gamma-\lambda+1,\ b=1-\lambda,\
c=2-\alpha-\gamma-\lambda. \end{align*} This calculation shows the equation $Tu(x,y)=0$ and we have a similar equation by changing $(x,y,\gamma,\lambda)\mapsto(y,x,\lambda,\gamma)$. Note that $TF_1(a;b,b';c;x,y)=0$ with $b'=1-\gamma$.
Putting \begin{align*} v(x,z)&=I_{0,x}^\mu(x^\alpha(1-x)^\beta(1-zx)^{\gamma-1})\\
&=\int_0^xt^\alpha(1-t)^\beta(1-zt)^{\gamma-1}
(x-t)^{\mu-1} dt\\
&=x^{\alpha+\mu}\int_0^1s^\alpha(1-xs)^\beta(1-xzs)^{\gamma-1}
(1-s)^{\mu-1} ds,\allowdisplaybreaks \intertext{we have}
&\qquad u(x,y)=y^{\gamma-1}v(x,\tfrac1y),\\ t^\alpha(1-t)^\beta&(1-zt)^{\gamma-1}=
\sum_{m,n=0}^\infty
\frac{(-\beta)_m(1-\gamma)_n}{m!n!}
t^{\alpha+m+n}z^n,\allowdisplaybreaks\\ v(x,z)&=\sum_{m,n=0}^\infty
\frac{\Gamma(\alpha+m+n+1)(-\beta)_m(1-\gamma)_n}
{\Gamma(\alpha+\mu+m+n+1)m!n!}
x^{\alpha+\gamma+m+n}z^n\\
&=x^{\alpha+\mu}\frac{\Gamma(\alpha+1)}{\Gamma(\alpha+\mu+1)}
\sum_{m,n=0}^\infty
\frac{(\alpha+1)_{m+n}(-\beta)_m(1-\gamma)_n}
{(\alpha+\mu+1)_{m+n}m!n!}x^{m+n}z^n\\
&=x^{\alpha+\mu}\frac{\Gamma(\alpha+1)}{\Gamma(\alpha+\mu+1)}
F_1(\alpha+1;-\beta,1-\gamma;\alpha+\mu+1;x,xz). \end{align*}
Using a versal addition to get the Kummer equation, we introduce the functions \index{hypergeometric equation/function!Appell!confluence} \begin{align*}
v_c(x,y)&:=\int_0^xt^\alpha(1-ct)^{\frac \beta c}(y-t)^{\gamma-1}
(x-t)^{\lambda-1},\\
h_{c,x}&:=x^\alpha(1-cx)^{\frac\beta c}(x-y)^{\gamma-1}. \end{align*} Then we have \begin{align*}
&R:=\Ad(h_{c,x})(\vartheta_x+\vartheta_y)
=\vartheta_x+\vartheta_y-(\alpha+\gamma-1)+\frac{\beta x}{1-cx},
\allowdisplaybreaks\\
&S:=\partial_x(1-cx)R\\
&\ \ \ =(\vartheta_x+1)
\bigl(\beta-c(\vartheta_x+\vartheta_y-\alpha-\gamma+1)
\bigr) + \partial_x(\vartheta_x+\vartheta_y-\alpha-\gamma+1),
\allowdisplaybreaks\\
&T:=\Ad(\partial^{-\lambda})R\\
&\ \ \ =(\vartheta_x-\lambda+1)
\bigl(\beta-c(\vartheta_x+\vartheta_y-\lambda-\alpha-\gamma+1)
\bigr) + \partial_x(\vartheta_x+\vartheta_y-\lambda-\alpha-\gamma+1) \end{align*} and hence $u_c(x,y)$ satisfies the differential equation \begin{align*}
&\Bigl(x(1-cx)\partial_x^2+y(1-cx)\partial_x\partial_y\\
&\quad{}+\bigl(2-\alpha-\gamma-\lambda
+(\beta+\lambda-2+c(\alpha+\gamma+\lambda-1))x\bigr)\partial_x
+(\lambda-1)\partial_y\\
&\quad{}-
(\lambda-1)(\beta+c(\alpha+\gamma+\lambda-1))\Bigr)u=0. \end{align*}
\subsubsection{Appell's $F_4$} \index{hypergeometric equation/function!Appell!$F_4$} To examine $F_4$ we consider the function \[
v(x,y):=\int_\Delta s^{\lambda_1}t^{\lambda_2}(st-s-t)^{\lambda_3}
(1-sx-ty)^\mu ds\,dt \] and the transformation \begin{equation}\label{eq:1-tx}
J^\mu_x(u)(x)
:=\int_\Delta u(t_1,\dots,t_n)(1-t_1x_1-\dots-t_nx_n)^\mu
dt_1\cdots dt_n \end{equation} for function $u(x_1,\dots,u_n)$. For example the region $\Delta$ is given by \begin{align*}
v(x,y)&=
\int_{s\le0,\ t\le0} s^{\lambda_1}t^{\lambda_2}(st-s-t)^{\lambda_3}
(1-sx-ty)^\mu ds\,dt. \end{align*}
Putting $s\mapsto s^{-1}$, $t\mapsto t^{-1}$ and $|x|+|y|<c<\frac{1}{2}$, Aomoto \cite{Ao} shows \begin{equation}
\begin{split}
&\int_{c-\infty i}^{c+\infty i}
\int_{c-\infty i}^{c+\infty i} s^{-\gamma}t^{-\gamma^\prime}
(1-s-t)^{\gamma+\gamma^\prime-\alpha-2}
\left(1-\frac{x}{s}-\frac{y}{t}\right)^{-\beta}ds\,dt\\
&\quad=-\frac{4\pi^2\Gamma(\alpha)}{\Gamma(\gamma)\Gamma(\gamma^\prime)
\Gamma(\alpha-\gamma-\gamma^\prime+2)}
F_4(\alpha;\beta;\gamma,\gamma^\prime;x,y),
\end{split} \end{equation} which follows from the integral formula \begin{equation}
\begin{split}
&\frac1{(2\pi i)^n}\int_{\frac1{n+1}-\infty i}^{\frac1{n+1}+\infty i}\cdots
\int_{\frac1{n+1}-\infty i}^{\frac1{n+1}+\infty i}
\prod_{j=1}^{n}t_j^{-\alpha_j}
\Bigl(1-\sum_{j=1}^nt_j\Bigr)^{-\alpha_{n+1}}dt_1\cdots dt_n\\
&\qquad=\frac{\Gamma\bigl(\sum_{j=1}^{n+1}\alpha_j-n\bigr)}
{\prod_{j=1}^{n+1}\Gamma\bigl(\alpha_j\bigr)}.
\end{split} \end{equation}
Since \begin{align*} J^\mu_x(u)&=J^{\mu-1}_x(u) - \sum x_\nu J^{\mu-1}_x(x_\nu u) \intertext{and} \frac{d}{dt_i}\bigl(u(t)(1-\sum t_\nu x_\nu)^{\mu}\bigr)\\ &\hspace{-2cm}=\frac{d u}{dt_i}(t)(1-\sum t_\nu x_\nu)^{\mu}
- \mu u(t)x_i(1-\sum t_\nu x_\nu)^{\mu-1}, \end{align*} we have \begin{align}
J^\mu_x(\partial_i u)(x)
&=\mu x_i J^{\mu-1}_x(u)(x)\notag\\
&=-x_i\int t_i^{-1}u(t)\frac{d}{dx_i}
\bigl(1-\textstyle\sum x_\nu t_\nu\bigr)^{\mu}dt\notag\\
&=-x_i\frac{d}{d x_i} J^\mu_x\bigl(\frac u{x_i}\bigr)(x),\notag\\
J^\mu_x\bigl(\partial_i(x_iu)\bigr)&=-x_i\partial_i J^\mu_x(u)
,\notag\\
J^\mu_x(\partial_i u)&=\mu x_iJ^{\mu-1}_x(u)\notag\\
&=\mu x_iJ^\mu_x(u)+\mu x_i\textstyle\sum x_\nu J^{\mu-1}_x(x_\nu u)\notag\\
&=\mu x_iJ^\mu_x(u)+x_i\textstyle\sum J^\mu_x\bigl(\partial_\nu(x_\nu u)\bigr) \notag\\
&=\mu x_iJ^\mu_x(u) - x_i\textstyle\sum x_\nu \partial_\nu J^\mu_x(u)\notag \intertext{and therefore}
J^\mu_x(x_i\partial_i u)&=(-1-x_i\partial_i)J^\mu_x(u),\\
J^\mu_x(\partial_i u)
&=x_i\bigl(\mu-\textstyle\sum x_\nu\partial_\nu\bigr)J^\mu_x(u). \end{align} Thus we have \begin{prop}\label{prop:1-tx} For a differential operator \begin{equation}
P =\sum_{\substack{\alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb Z^n_{\ge0}\\
\beta=(\beta_1,\dots,\beta_n)\in\mathbb Z^n_{\ge0}}}
c_{\alpha,\beta}
\partial_{1}^{\alpha_1}\cdots\partial_{n}^{\alpha_n}
\vartheta_{1}^{\beta_1}\cdots\vartheta_{n}^{\beta_n}, \end{equation} we have \begin{equation}
\begin{split}
J^\mu_x\bigl(Pu(x)\bigr)&=J^\mu_x(P)J^\mu_x\bigl(u(x)\bigr),\\
J^\mu_x(P) :\!&= \sum_{\alpha,\,\beta}
c_{\alpha,\beta}
\prod_{k=1}^n\bigl(x_k(\mu-\sum_{\nu=1}^n\vartheta_{\nu})\bigr)^{\alpha_k}
\prod_{k=1}^n\bigl(-\vartheta_{k}-1\bigr)^{\beta_k}. \end{split} \end{equation} \end{prop} Using this proposition, we obtain the system of differential equations satisfied by $J^\mu_x(u)$ from that satisfied by $u(x)$. Denoting the Laplace transform of the variable $x=(x_1,\dots,x_n)$ by $\Lap_x$ (cf.\ Definition~\ref{def:Lap}), we have \begin{equation} J_x^\mu L_x^{-1}(\vartheta_i)=\vartheta_i,\quad J_x^\mu L_x^{-1}(x_i)=x_i\bigl(\mu-\sum_{\nu=1}^n\vartheta_\nu\bigr). \end{equation}
We have \begin{align}
\Ad\bigl(x^{\lambda_1} y^{\lambda_2} (xy - x - y)^{\lambda_3}\bigr)\partial_x &=\partial_x -\frac{\lambda_1}x - \frac{\lambda_3(y-1)}{xy-x-y}, \allowdisplaybreaks\notag\\
\Ad\bigl(x^{\lambda_1} y^{\lambda_2} (xy - x - y)^{\lambda_3}\bigr)\partial_y &=\partial_y -\frac{\lambda_2}y - \frac{\lambda_3(x-1)}{xy-x-y}, \allowdisplaybreaks\notag\\
\Ad\bigl(x^{\lambda_1} y^{\lambda_2} (xy - x - y)^{\lambda_3}\bigr)
&\bigl(x(x-1)\partial_x\bigr)\notag\\
&\hspace{-2cm}=x(x-1)\partial_x-\lambda_1(x-1)
- \frac{\lambda_3(x-1)(xy-x)}{xy-x-y}, \allowdisplaybreaks\notag\\
\Ad\bigl(x^{\lambda_1} y^{\lambda_2} (xy - x - y)^{\lambda_3}\bigr)
&\bigl(x(x-1)\partial_x-y\partial_y\bigr)\notag\\
&\hspace{-2cm}=x(x-1)\partial_x-y\partial_y-\lambda_1(x-1)
-\lambda_2 - \lambda_3(x-1)\notag\\
&\hspace{-2cm}=x\vartheta_x-\vartheta_x-\vartheta_y-(\lambda_1+\lambda_3)x
+\lambda_1-\lambda_2+\lambda_3, \allowdisplaybreaks\notag\\
\begin{split}
\partial_x\Ad\bigl(x^{\lambda_1} y^{\lambda_2} (xy - x - y)^{\lambda_3}\bigr)
&\bigl(x(x-1)\partial_x-y\partial_y\bigr) \\
&\hspace{-2cm}=\partial_xx(\vartheta_x-\lambda_1-\lambda_3)
-\partial_x(\vartheta_x+\vartheta_y-\lambda_1+\lambda_2-\lambda_3)
\end{split}\label{eq:AP40} \end{align} and \begin{align*}
&J^\mu_{x,y}\bigl
(\partial_xx(\vartheta_x-\lambda_1-\lambda_3)
-\partial_x(\vartheta_x+\vartheta_y-\lambda_1+\lambda_2-\lambda_3)\bigr)\\
&\quad=\vartheta_x(1+\vartheta_x+\lambda_1+\lambda_3)
-x(-\mu+\vartheta_x+\vartheta_y)(2+\vartheta_x+\vartheta_y+\lambda_1-\lambda_2+\lambda_3). \end{align*} Putting \begin{align*}
T&:=(\vartheta_x+\vartheta_y-\mu)
(\vartheta_x+\vartheta_y+\lambda_1-\lambda_2+\lambda_3+2)
-\partial_x(\vartheta_x+\lambda_1+\lambda_3+1) \end{align*} with \begin{align*} \quad\alpha=-\mu,\quad \beta=\lambda_1-\lambda_2+\lambda_3+2,\quad \gamma=\lambda_1+\lambda_3+2, \end{align*} we have $Tv(x,y)=0$ and moreover it satisfies a similar equation by replacing $(x,y,\lambda_1,\lambda_3,\gamma)$ by $(y,x,\lambda_3,\lambda_1,\gamma')$. Hence $v(x,y)$ is a solution of the system of differential equations satisfied by $F_4(\alpha;\beta;\gamma,\gamma';x,y)$.
In the same way we have \begin{gather}
\Ad\bigl(x^{\beta-1}y^{\beta'-1}(1-x-y)^{\gamma-\beta-\beta'-1}\bigr)\vartheta_x
=\vartheta_x-\beta+1+\frac{(\gamma-\beta-\beta'-1) x}{1-x-y},\notag
\allowdisplaybreaks\\
\begin{split}
&\Ad\bigl(x^{\beta-1}y^{\beta'-1}(1-x-y)^{\gamma-\beta-\beta'-1}\bigr)
(\vartheta_x-x(\vartheta_x+\vartheta_y))\\
&\quad=\vartheta_x-x(\vartheta_x+\vartheta_y)
-\beta+1+(\gamma-3)x\\
&\quad=(\vartheta_x-\beta+1)-x(\vartheta_x+\vartheta_y-\gamma+3),
\end{split}\label{eq:F120}\allowdisplaybreaks\\
\begin{split}
&J^\mu_{x,y}\bigl(\partial_x(\vartheta_x-\beta+1)-\partial_xx(\vartheta_x+\vartheta_y-\gamma+3)\bigr)
\notag\\
&\quad=x(-\vartheta_x-\vartheta_y+\mu)(-\vartheta_x-\beta)
+\vartheta_x(-2-\vartheta_x-\vartheta_y-\gamma+3)
\notag\\
&\quad=x\Bigl((\vartheta_x+\vartheta_y-\mu)(\vartheta_x+\beta)
-\partial_x(\vartheta_x+\vartheta_y+\gamma-1)\Bigr).\notag
\end{split} \end{gather} which is a differential operator killing $F_1(\alpha;\beta,\beta';\gamma;x,y)$ by putting $\mu=-\alpha$ and in fact we have \begin{align*}
&\iint_{\substack{s\ge 0,\ t\ge 0\\1-s-t\ge0}}
s^{\beta-1}t^{\beta'-1}(1-s-t)^{\gamma-\beta-\beta'-1}
(1-sx-ty)^{-\alpha}ds\,dt\allowdisplaybreaks\\
&\quad=\iint_{\substack{s\ge 0,\ t\ge 0\\1-s-t\ge0}}\sum_{m,\,n=0}^\infty
s^{\beta+m-1}t^{\beta'+n-1}(1-s-t)^{\gamma-\beta-\beta'-1}
\frac{(\alpha)_{m+n}x^my^n}{m!n!}ds\,dt\allowdisplaybreaks\\
&\quad=\sum_{m,\,n=0}^\infty
\frac{\Gamma(\beta+m)\Gamma(\beta'+n)\Gamma(\gamma-\beta-\beta')}
{\Gamma(\gamma+m+n)}\cdot\frac{(\alpha)_{m+n}}{m!n!}x^my^n\allowdisplaybreaks\\
&\quad=\frac{\Gamma(\beta)\Gamma(\beta')\Gamma(\gamma-\beta-\beta')}
{\Gamma(\gamma)} F_1(\alpha;\beta,\beta';\gamma;x,y). \end{align*} Here we use the formula \begin{equation}
\iint_{\substack{s\ge 0,\ t\ge 0\\1-s-t\ge0}}
s^{\lambda_1-1}t^{\lambda_2-1}(1-s-t)^{\lambda_3-1}ds\,dt
=\frac{\Gamma(\lambda_1)\Gamma(\lambda_2)\Gamma(\lambda_3)}
{\Gamma(\lambda_1+\lambda_2+\lambda_3)}. \end{equation} \subsubsection{Appell's $F_3$} \index{hypergeometric equation/function!Appell!$F_3$} Since \begin{align*}
T_3:\!&=
J^{-\alpha'}_yx^{-1} J^{-\alpha}_x
\bigl(\partial_x(\vartheta_x-\beta+1)-\partial_xx(\vartheta_x+\vartheta_y-\gamma+3)\bigr)\\
&=J^{-\alpha'}_y\bigl((-\vartheta_x-\alpha)(-\vartheta_x-\beta)
+\partial_x(-\vartheta_x+\vartheta_y-\gamma+2)\bigr)\\
&=(\vartheta_x+\alpha)(\vartheta_x+\beta)
-\partial_x(\vartheta_x+\vartheta_y+\gamma-1) \end{align*} with \eqref{eq:F120}, the operator $T_3$ kills the function \begin{align*}
&\iint_{\substack{s\ge 0,\ t\ge 0\\ 1-s-t\ge 0}}
s^{\beta-1}t^{\beta'-1}(1-s-t)^{\gamma-\beta-\beta'-1}(1-xs)^{-\alpha}
(1-yt)^{-\alpha'}ds\,dt\\
&\quad=\iint_{\substack{s\ge 0,\ t\ge 0\\ 1-s-t\ge 0}}\sum_{m,\,n=0}^\infty
s^{\beta+m-1}t^{\beta'+n-1}(1-s-t)^{\gamma-\beta-\beta'-1}
\frac{(\alpha)_m(\alpha')_nx^my^n}{m!n!}ds\,dt\\
&\quad =\sum_{m,\,n=0}^\infty
\frac{\Gamma(\beta+m)\Gamma(\beta'+n)\Gamma(\gamma-\beta-\beta')(\alpha)_m(\alpha')_n}
{\Gamma(\gamma+m+n)m!n!}x^my^n\\
&\quad=\frac{\Gamma(\beta)\Gamma(\beta')\Gamma(\gamma-\beta-\beta')}
{\Gamma(\gamma)}
F_3(\alpha,\alpha';\beta,\beta';\gamma;x,y). \end{align*} Moreover since \begin{align*} T'_3:\!&=\Ad(\partial_x^{-\mu})\Ad(\partial_y^{-\mu'})
\bigl
((\vartheta_x+1)(\vartheta_x-\lambda_1-\lambda_3)
-\partial_x(\vartheta_x+\vartheta_y-\lambda_1+\lambda_2-\lambda_3)\bigr)\\
&=(\vartheta_x+1-\mu)(\vartheta_x-\lambda_1-\lambda_3-\mu)
-\partial_x(\vartheta_x+\vartheta_y-\lambda_1+\lambda_2-\lambda_3-\mu-\mu') \end{align*} with \eqref{eq:AP40} and \begin{align*} \alpha&=-\lambda_1-\lambda_3-\mu,\quad \beta=1-\mu,\quad \gamma=-\lambda_1+\lambda_2-\lambda_3-\mu-\mu'+1, \end{align*} the function \begin{equation}\label{eq:F3I2}
u_3(x,y)
:=\int_\infty^y\int_\infty^x s^{\lambda_1} t^{\lambda_2}(st-s-t)^{\lambda_3}
(x-s)^{\mu-1}(y-t)^{\mu'-1}ds\,dt \end{equation} satisfies $T'_3u_3(x,y)=0$. Hence $u_3(x,y)$ is a solution of the system of the equations that $F_3(\alpha,\alpha';\beta,\beta';\gamma;x,y)$ satisfies.
\subsubsection{Appell's $F_2$} \index{hypergeometric equation/function!Appell!$F_2$} Since \begin{align*}
&\partial_x\Ad\bigl(x^{\lambda_1-1}(1-x^{\lambda_2-1})\bigr)x(1-x)\partial_x\\
&\quad=
\partial_x x(1-x)\partial_x-(\lambda_1-1)\partial_x+\partial_x(\lambda_1+\lambda_2-2)x\\
&\quad=\partial_xx(-\vartheta_x+\lambda_1+\lambda_2-2)
+\partial_x(\vartheta-\lambda_1+1) \end{align*} and \begin{align*} T_2:\!&=J^\mu_{x,y}\bigl(\partial_xx(-\vartheta_x+\lambda_1+\lambda_2-2)
+\partial_x(\vartheta_x-\lambda_1+1)\bigr)\\ &
=-\vartheta_x(\vartheta_x+1+\lambda_1+\lambda_2-2)
+x(\mu - \vartheta_x-\vartheta_y)(-1-\vartheta_x-\lambda_1+1) \\&
=x\Bigl((\vartheta_x+\lambda_1)(\vartheta_x+\vartheta_y-\mu)
-\partial_x(\vartheta_x+\lambda_1+\lambda_2-1)\Bigr)
\end{align*} with \[ \alpha=-\mu,\quad\beta=\lambda_1,\quad\gamma=\lambda_1+\lambda_2, \] the function \begin{align*}
u_2(x,y)&:=\int_0^1\!\int_0^1
s^{\lambda_1-1}(1-s)^{\lambda_2-1}
t^{\lambda'_1-1}(1-t)^{\lambda'_2-1}
(1-xs-yt)^{\mu}ds\,dt\\
&=\int_0^1\!\int_0^1\sum_{m,\,n=0}^\infty s^{\lambda_1+m-1}(1-s)^{\lambda_2-1}
t^{\lambda_1'+n-1}(1-t)^{\lambda_2'-1}\frac{(-\mu)_{m+n}}{m!n!}x^my^n ds\,dt\\
&=\sum_{m,\,n=0}^\infty
\frac{\Gamma(\lambda_1+m)\Gamma(\lambda_2)}
{\Gamma(\lambda_1+\lambda_2+m)}
\frac{\Gamma(\lambda_1'+n)\Gamma(\lambda_2')}
{\Gamma(\lambda_1'+\lambda_2'+m)}
\frac{(-\mu)_{m+n}}{m!n!}x^my^n\\
&=\frac{\Gamma(\lambda_1)\Gamma(\lambda_2)
\Gamma(\lambda_1')\Gamma(\lambda_2')}
{\Gamma(\lambda_1+\lambda_2)\Gamma(\lambda_1'+\lambda_2')}
\sum_{m,\,n=0}^\infty\frac{(\lambda_1)_m(\lambda_1')_n(-\mu)_{m+n}}
{(\lambda_1+\lambda_2)_m(\lambda_1'+\lambda_2')_nm!n!}x^my^n \end{align*} is a solution of the equation $T_2u=0$ that $F_2(\alpha;\beta,\beta';\gamma,\gamma';x,y)$ satisfies.
Note that the operator $\tilde T_3$ transformed from $T'_3$ by the coordinate transformation $(x,y)\mapsto (\frac1x,\frac1y)$ equals \begin{align*}
\tilde T_3&=
(-\vartheta_x+\alpha)(-\vartheta_x+\beta)
-x(-\vartheta_x)(-\vartheta_x-\vartheta_y+\gamma-1)\\
&=(\vartheta_x-\alpha)(\vartheta_x-\beta)
-x\vartheta_x(\vartheta_x+\vartheta_y-\gamma+1) \end{align*} and the operator \begin{align*}
\Ad(x^{-\alpha} y^{-\alpha'})\tilde T_3&=
\vartheta_x(\vartheta_x+\alpha-\beta)
-x(\vartheta_x+\alpha)(\vartheta_x+\vartheta_y+\alpha+\alpha'-\gamma+1) \end{align*} together with the operator obtained by the transpositions $x\leftrightarrow y$, $\alpha\leftrightarrow\alpha'$ and $\beta\leftrightarrow \beta'$ defines the system of the equations satisfied by the functions \begin{equation}
\begin{cases}
F_2(\alpha+\alpha'-\gamma+1;\alpha,\alpha';
\alpha-\beta+1,\alpha'-\beta'+1;x,y),\\
x^{-\alpha'}y^{-\alpha'}F_3(\alpha,\alpha';\beta,\beta';\gamma;\frac1x,\frac1y),
\end{cases} \end{equation} which also follows from the integral representation \eqref{eq:F3I2} with the transformation $(x,y,s,t)\mapsto(\frac1x,\frac1y,\frac1s,\frac1t)$.
\subsection{\texttt{Okubo} and \texttt{Risa/Asir}}\label{sec:okubo} Most of our results in this paper are constructible and they can be explicitly calculated and implemented in computer programs.
The computer program \texttt{okubo} \cite{O5} written by the author handles combinatorial calculations in this paper related to tuples of partitions. It generates basic tuples (cf.~\S\ref{sec:Exbasic}) and rigid tuples (cf.~\S\ref{sec:rigidEx}), calculates the reductions originated by Katz and Yokoyama, the position of accessory parameters in the universal operator (cf.~Theorem~\ref{thm:univmodel} iv)) and direct decompositions etc.
The author presented Theorem~\ref{thm:c} in the case when $p=3$ as a conjecture in the fall of 2007, which was proved in May in 2008 by a completely different way from the proof given in \S\ref{sec:C1}, which is a generalization of the original proof of Gauss's summation formula of the hypergeometric series explained in \S\ref{sec:C2}. The original proof of Theorem~\ref{thm:c} in the case when $p=3$ was reduced to the combinatorial equality \eqref{eq:numdec}. The author verified \eqref{eq:numdec} by \texttt{okubo} and got the concrete connection coefficients for the rigid tuples $\mathbf m$ satisfying $\ord\mathbf m\le 40$. Under these conditions ($\ord\mathbf m\le 40,\, p=3,\, m_{0,n_0}=m_{1,n_1}=1$) there are 4,111,704 independent connection coefficients modulo obvious symmetries and it took about one day to got all of them by a personal computer with \texttt{okubo}.
Several operations on differential operators such as additions and middle convolutions defined in \S\ref{sec:frac} can be calculated by a computer algebra and the author wrote a program for their results under \texttt{Risa/Asir}, which gives a reduction procedure of the operators (cf.~Definition~\ref{def:redGRS}), integral representations and series expansions of the solutions (cf.~Theorem~\ref{thm:expsol}), connection formulas (cf.~Theorem~\ref{thm:conG}), differential operators (cf.~Theorem~\ref{thm:univmodel} iv)), the condition of their reducibility (cf.~Corollary~\ref{cor:irred} i)), recurrence relations (cf.~Theorem~\ref{thm:shifm1} ii)) etc.\ for any given spectral type or Riemann scheme \eqref{eq:IGRS} and displays the results using \TeX. This program for Risa/Asir written by the author contains many useful functions calculating rational functions, Weyl algebra and matrices. These programs can be obtained from
\quad\texttt{http://www.math.kobe-u.ac.jp/Asir/asir.html}
\quad\texttt{ftp://akagi.ms.u-tokyo.ac.jp/pub/math/muldif}
\quad\texttt{ftp://akagi.ms.u-tokyo.ac.jp/pub/math/okubo}.
\section{Further problems}\label{sec:prob} \subsection{Multiplicities of spectral parameters} Suppose a Fuchsian differential equation and its middle convolution are given. Then we can analyze the corresponding transformation of a global structure of its local solution associated with an eigenvalue of the monodromy generator at a singular point if the eigenvalue is free of multiplicity.
When the multiplicity of the eigenvalue is larger than one, we have not a satisfactory result for the transformation (cf.~Theorem~\ref{thm:conG}). The value of a generalized connection coefficient defined by Definition~\ref{def:GC} may be interesting. Is the procedure in Remark~\ref{rem:Cproc} always valid? In particular, is there a general result assuring Remark~\ref{rem:Cproc} (1) (cf.~Remark~\ref{rem:Cgamma})? Are the multiplicities of zeros of the generalized connection coefficients of a rigid Fuchsian differential equation free?
\subsection{Schlesinger canonical form}\label{prob:Sch} Can we define a natural \textsl{universal} Fuchsian system of Schlesinger canonical form \eqref{eq:SCF} with a given realizable spectral type? Here we recall Example~\ref{ex:univSch}.
Let $P_{\mathbf m}$ be the universal operator in Theorem~\ref{thm:univmodel}. Is there a natural system of Schlesinger canonical form which is isomorphic to the equation $P_{\mathbf m}u=0$ together with the explicit correspondence between them?
\subsection{Apparent singularities} Katz \cite{Kz} proved that any irreducible rigid local system is constructed from the trivial system by successive applications of middle convolutions and additions and it is proved in this paper that the system is realized by a single differential equation without an apparent singularity.
In general, an irreducible local system cannot be realized by a single differential equation without an apparent singularity but it is realized by that with apparent singularities. Hence it is expected that there exist some natural operations of single differential equations with apparent singularities which correspond to middle convolutions of local systems or systems of Schlesinger canonical form.
The Fuchsian ordinary differential equation satisfied by an important special function often hasn't an apparent singularity even if the spectral type of the equation is not rigid. Can we understand the condition that a $W(x)$-module has a generator so that it satisfies a differential equation without an apparent singularity? Moreover it may be interesting to study the existing of contiguous relations among differential equations with fundamental spectral types which have no apparent singularity.
\subsection{Irregular singularities} Our fractional operations defined in \S\ref{sec:frac} give transformations of ordinary differential operators with polynomial coefficients, which have irregular singularities in general. The reduction of ordinary differential equations under these operations is a problem to be studied. Note that versal additions and middle convolutions construct such differential operators from the trivial equation.
A similar result as in this paper is obtained for certain classes of ordinary differential equations with irregular singularities (cf.~\cite{Hi}).
A ``versal" path of integral in an integral representation of the solution and a ``versal" connection coefficient and Stokes multiplier should be studied. Here ``versal" means a natural expression corresponding to the versal addition.
We define a complete model with a given spectral type as follows. For simplicity we consider differential operators without singularities at the origin. For a realizable irreducible tuple of partitions $\mathbf m=\bigl(m_{j,\nu}\bigr)_{\substack{0\le j\le p\\1\le\nu\le n_j}}$ of a positive integer $n$ Theorem~\ref{thm:univmodel} constructs the universal differential operator \begin{equation}\label{eq:univcj}
P_{\mathbf m}=\prod_{j=1}^p(1-c_jx)^n \cdot \frac{d^n}{dx^n}+\sum_{k=0}^{n-1} a_k(x,c,\lambda,g)\frac{d^k}{dx^k} \end{equation} with the Riemann scheme \begin{align*}
&\begin{Bmatrix}
x=\infty & \frac1{c_1} & \cdots &\frac1{c_p}\\
[\lambda_{0,1}]_{(m_{0,1})} & [\lambda_{1,1}]_{(m_{1,1})}
& \cdots &[\lambda_{p,1}]_{(m_{p,1})} \\
\vdots & \vdots & \vdots & \vdots\\
[\lambda_{0,n_0}]_{(m_{0,n_0})} & [\lambda_{1,n_1}]_{(m_{1,n_1})} &
\cdots &[\lambda_{p,n_p}]_{(m_{p,n_p})}
\end{Bmatrix} \end{align*} and the Fuchs relation \begin{align*} \sum_{j=0}^p\sum_{\nu=1}^{n_j}m_{j,\nu}
\lambda_{j,\nu}=n-\frac{\idx\mathbf m}2. \end{align*} Here $c=(c_0,\dots,c_p)$, $\lambda=(\lambda_{j,\nu})$ and $g=(g_1,\dots,g_N)$ are parameters. We have $c_ic_j(c_i-c_j)\ne 0$ for $0\le i<j\le p$. The parameters $g_j$ are called accessory parameters and we have $\idx\mathbf m=2-2N$. We call the Zariski closure $\overline P_{\mathbf m}$ of $P_{\mathbf m}$ in $W[x]$ the \textsl{complete model} \index{differential equation/operator!complete model} of differential operators with the spectral type $\mathbf m$, whose dimension equals $p+\sum_{j=0}^p n_j + N-1$. It is an interesting problem to analyze the complete model $\overline P_{\mathbf m}$.
When $\mathbf m=11,11,11$, the complete model equals \[
(1-c_1x)^2(1-c_2x)^2\tfrac{d^2}{dx^2}
-(1-c_1x)(1-c_2x)(a_{1,1}x+a_{1,0})\tfrac d{dx}
+a_{0,2}x^2+a_{0,1}x+a_{0,0}, \] whose dimension equals 7. Any differential equation defined by the operator belonging to this complete model is transformed into a Gauss hypergeometric equation, a Kummer equation, an Hermite equation or an airy equation by a suitable gauge transformation and a coordinate transformation. A good understanding together with a certain completion of our operators is required even in this fundamental example. It is needless to say that the good understanding is important in the case when $\mathbf m$ is fundamental.
\subsection{Special parameters} Let $P_{\mathbf m}$ be the universal operator of the form \eqref{eq:univcj} for an irreducible tuple of partition $\mathbf m$. When a decomposition $\mathbf m=\mathbf m'+\mathbf m''$ with realizable tuples of partitions $\mathbf m'$ and $\mathbf m''$ is given, Theorem~\ref{thm:prod} gives the values of the parameters of $P_\mathbf m$ corresponding to the product $P_{\mathbf m'}P_{\mathbf m''}$. A $W(x,\xi)$-automorphism of $P_{\mathbf m}u=0$ gives a transformation of the parameters $(\lambda,g)$, which is a contiguous relation and called Schlesinger transformation in the case of systems of Schlesinger canonical form. How can we describe the values of the parameters obtained in this way and characterize their position in all the values of the parameters when the universal operator is reducible? In general, they are not all even in a rigid differential equation. A direct decomposition $32,32,32,32=12,12,12,12\oplus 2(10,10,10,10)$ of a rigid tuples $32,32,32,32$ gives this example (cf.~\eqref{eq:32idx}).
Analyse the reducible differential equation with an irreducibly realizable spectral type. This is interesting even when $\mathbf m$ is a rigid tuple. For example, describe the monodromy of its solutions.
Describe the characteristic exponents of the generalized Riemann scheme with an irreducibly realizable spectral type such that there exists a differential operator with the Riemann scheme which is outside the universal operator (cf.~Example \ref{ex:outuniv} and Remark~\ref{rem:inuniv}). In particular, when the spectral type is not fundamental nor simply reducible, does there exist such a differential operator?
The classification of rigid and simply reducible spectral types coincides with that of indecomposable objects described in \cite[Theorem~2.4]{MWZ}. Is there some meaning in this coincidence?
Has the condition \eqref{eq:CSR} a similar meaning in the case of Schlesinger canonical form? What is the condition on the local system or a (single) Fuchsian differential equation which has a realization of a system of Schlesinger canonical form?
Give the condition so that the monodromy group is finite. Give the condition so that the centralizer of the monodromy is the set of scalar multiplications.
Suppose $\mathbf m$ is fundamental. Study the condition so that the connection coefficients is a quotient of the products of gamma functions as in Theorem~\ref{thm:c} or the solution has an integral representation only by using elementary functions.
\subsection{Shift operators} Calculate the polynomial function $c_{\mathbf m}(\epsilon;\lambda)$ of $\lambda$ defined in Theorem~\ref{thm:shiftC}. Is it square free? See Conjecture~\ref{conj:shift}.
Is the shift operator $R_{\mathbf m}(\epsilon,\lambda)$ Fuchsian?
Study the shift operators given in Theorem~\ref{thm:sftUniv}.
Study the condition on the characteristic exponents and accessory parameters assuring the existence of a shift operator for a Fuchsian differential operator with a fundamental spectral type.
Study the shift operator or Schlesinger transformation of a system of Schlesinger canonical form with a fundamental spectral type. When is it not defined or when is it not bijective?
\subsection{Several variables} We have analyzed Appell hypergeometric equations in \S\ref{sec:ApEx}. What should be the geometric structure of singularities of more general system of equations when it has a good theory?
Describe or define operations of differential operators that are fundamental to analyze good systems of differential equations.
A series expansion of a local solution of a rigid ordinal differential equation indicates that it may be natural to think that the solution is a restriction of a solution of a system of differential equations with several variables (cf.~Theorem~\ref{thm:expsol} and \S\ref{sec:PoEx}--\ref{sec:GHG}). Study the system.
\subsection{Other problems} \begin{itemize} \item For a rigid decomposition $\mathbf m=\mathbf m'\oplus\mathbf m''$, can we determine whether $\alpha_{\mathbf m'}\in \Delta(\mathbf m)$ or $\alpha_{\mathbf m''}\in \Delta(\mathbf m)$ (cf.~Proposition~\ref{prop:wm} iv))?
\item Are there analyzable series $\mathcal L$ of rigid tuples of partitions different from the series given in \S\ref{sec:Rob}? Namely, $\mathcal L\subset \mathcal P$, the elements of $\mathcal L$ are rigid, the number of isomorphic classes of $\mathcal L\cap \mathcal P^{(n)}$ are bounded for $n\in\mathbb Z_{>0}$ and the following condition is valid.
Let $\mathbf m=k\mathbf m'+\mathbf m''$ with $k\in\mathbb Z_{>0}$ and rigid tuples of partitions $\mathbf m$, $\mathbf m'$ and $\mathbf m''$. If $\mathbf m\in\mathcal L$, then $\mathbf m'\in\mathcal L$ and $\mathbf m''\in\mathcal L$. Moreover for any $\mathbf m''\in\mathcal L$, this decomposition $\mathbf m=k\mathbf m'+\mathbf m''$ exists with $\mathbf m\in\mathcal L$, $\mathbf m'\in\mathcal L$ and $k\in\mathbb Z_{>0}$. Furthermore $\mathcal L$ is indecomposable. Namely, if $\mathcal L=\mathcal L'\cup\mathcal L''$ so that $\mathcal L'$ and $\mathcal L''$ satisfy these conditions, then $\mathcal L'=\mathcal L$ or $\mathcal L''=\mathcal L$.
\item Characterize the ring of automorphisms and that of endomorphisms of the localized Weyl algebra $W(x)$.
\item In general, different procedures of the reduction of the universal operator $P_{\mathbf m}u=0$ give different integral representations and series expansions of its solution (cf.~Example~\ref{ex:localGauss}, Remark~\ref{rem:Irep} and the last part of \S\ref{sec:PoEx}). Analyze the difference. \end{itemize}
\section{Appendix}\label{sec:appendix} In this section we give a theorem which is proved by K.~Nuida. The author greatly thanks to K.~Nuida for allowing the author to put the theorem with its proof in this section.
Let $(W,S)$ be a \textsl{Coxeter system}. \index{Coxeter system} Namely, $W$ is a group with the set $S$ of generators and under the notation $S=\{s_i\,;\,i\in I\}$, the fundamental relations among the generators are \begin{equation}
s_k^2=(s_is_j)^{m_{i,j}}=e\text{ \ and \ }m_{i,j}=m_{j,i}
\text{ \ for \ }\forall i,\ j,\ k\in I\text{ \ satisfying \ }i\ne j. \end{equation} Here $m_{i,j}\in\{2,3,4,\ldots\}\cup\{\infty\}$ and the condition $m_{i,j}=\infty$ means $(s_is_j)^m\ne e$ for any $m\in\mathbb Z_{>0}$. Let $E$ be a real vector space with the basis set $\Pi=\{\alpha_i\,;\,i\in I\}$
and define a symmetric bilinear form $(\ \ |\ \ )$ on $E$ by \begin{equation}
(\alpha_i|\alpha_i)=2\text{ \ and \ }
(\alpha_i|\alpha_j)=-2\cos\frac{\pi}{m_{i,j}}. \end{equation} Then the \textsl{Coxeter group} $W$ is naturally identified with the reflection group generated by the reflections $s_{\alpha_i}$ with respect to $\alpha_i$ $(i\in I)$. The set $\Delta_\Pi$ of the roots of $(W,S)$ equals $W\Pi$, which is a disjoint union of the set of positive roots $\Delta_\Pi^+:= \Delta_\Pi\cap \sum_{\alpha\in\Pi}\mathbb Z_{\ge 0}\alpha$ and the set of negative roots $\Delta_\Pi^-:=-\Delta_\Pi^+$. For $w\in W$ the length $L(w)$ is the minimal number $k$ with the expression $w=s_{i_1}s_{i_2}\cdots s_{i_k}$ ($i_1,\dots,i_k\in I$). Defining $\Delta_\Pi(w):=\Delta_\Pi^+\cap w^{-1}\Delta_\Pi^-$, we have $L(w)=\#\Delta_\Pi(w)$.
Fix $\beta$ and $\beta'\in\Delta_\Pi$ and put \begin{equation} W^\beta_{\!\beta'}:=\{w\in W\,;\,\beta'=w\beta\} \text{ \ and \ }W^\beta:=W^\beta_\beta. \end{equation}
\begin{thm}[K.~Nuida]\label{thm:Nuida} Retain the notation above. Suppose\/ $W^\beta_{\!\beta'}\ne\emptyset$ and \begin{equation}\label{eq:noloop} \begin{split}
&\text{there exist no sequence $s_{i_1},s_{i_2},\ldots s_{i_k}$ of
elements of $S$ such that}\\
&\begin{cases}
k\ge 3,\\
s_{i_\nu}\ne s_{i_\nu'}\quad(1\le \nu<\nu'\le k),\\
m_{i_\nu, i_{\nu+1}}\text{ and }m_{i_1,i_k}
\text{\ are odd integers}\quad(1\le\nu<k).
\end{cases} \end{split} \end{equation} Then an element $w\in W^\beta_{\!\beta'}$ is uniquely determined by the condition \begin{equation}\label{eq:minCox}
L(w)\le L(v)\quad(\forall v\in W^\beta_{\!\beta'}). \end{equation} \end{thm} \begin{proof}
Put $\Delta_\Pi^\beta:=\{\gamma\in\Delta_\Pi^+\,;\,(\beta|\gamma)=0\}$. First note that the following lemma. \begin{lem}\label{lem:Coxeter} If\/ $w\in W^\beta_{\!\beta'}$ satisfies \eqref{eq:minCox}, then\/ $w\Delta_\Pi^\beta\subset\Delta_\Pi^+$. \end{lem}
In fact, if $w\in W^\beta_{\beta'}$ satisfies \eqref{eq:minCox} and there exists $\gamma\in\Delta_\Pi^\beta$ satisfying $w\gamma\in\Delta_\Pi^-$, then there exists $j$ for a minimal expression $w=s_{i_1}\cdots s_{i_{L_\Pi(w)}}$ such that $s_{i_{j+1}}\cdots s_{j_{L_\Pi(w)}}\gamma=\alpha_{i_j}$, which implies $W^\beta_{\beta'}\ni v :=ws_\gamma=s_{i_1}\cdots s_{i_{j-1}}s_{i_{j+1}}\cdots s_{i_{L_\Pi(w)}}$ and contradicts to \eqref{eq:minCox}.
It follows from \cite{Br} that the assumption \eqref{eq:noloop} implies that $W^\beta$ is generated by $\{s_\gamma\,;\,\gamma\in \Delta_\Pi^\beta\}$. Putting $\Pi^\beta=\Delta_\Pi^\beta\setminus\{r_1\gamma_1+r_2\gamma_2\in \Delta_\Pi^\beta \,;\,\gamma_2\notin\mathbb R \gamma_1,\ \gamma_j\in \Delta_\Pi^\beta\text{ and }r_j>0\text{ for }j=1,2\}$ and $S^\beta=\{s_\gamma\,;\,\gamma\in \Pi^\beta\}$, the pair $(W^\beta,S^\beta)$ is a Coxeter system and moreover the minimal length of the expression of $w\in W^\beta$ by the product of the elements of $S^\beta$ equals $\#\bigl(\Delta_\Pi^\beta\cap w^{-1}\Delta_\Pi^-\bigr)$ (cf.~\cite[Theorem~2.3]{Nu}).
Suppose there exist two elements $w_1$ and $w_2\in W^\beta_{\!\beta'}$ satisfying $L(w_j)\le L(v)$ for any $v\in W^\beta_{\!\beta'}$ and $j=1$, $2$.
Since $e\ne w_1^{-1}w_2\in W^\beta$, there exists $\gamma\in \Delta_\Pi^\beta$ such that $w_1^{-1}w_2\gamma \in \Delta_\Pi^-$. Since $-w_1^{-1}w_2\gamma\in \Delta^\beta_\Pi$, Lemma~\ref{lem:Coxeter} assures $-w_2\gamma=w_1(-w_1^{-1}w_2\gamma)\in\Delta_\Pi^+$, which contradicts to Lemma~\ref{lem:Coxeter}. \end{proof} The above proof shows the following corollary. \begin{cor}\label{cor:Nuida} Retain the assumption in\/ {\rm Theorem~\ref{thm:Nuida}.} For an element $w\in W^\beta_{\beta'}$, the condition \eqref{eq:minCox} is equivalent to $w\Delta^\beta_\Pi\subset \Delta_\Pi^+$.
Let $w\in W^\beta_{\beta'}$ satisfying \eqref{eq:minCox}. Then \begin{equation}
W^\beta_{\beta'}=w\bigl\langle s_\gamma\,;\,(\gamma|\beta)=0,\ \gamma\in\Delta_\Pi^+\bigr\rangle. \end{equation} \end{cor}
\printindex
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Singularities of Fitzpatrick and convex functions}
\author{Dmitry Kramkov\footnote{Carnegie Mellon University, Department
of Mathematical Sciences, 5000 Forbes Avenue, Pittsburgh, PA,
15213-3890, USA,
\href{mailto:[email protected]}{[email protected]}. The author also
has a research position at the University of Oxford.} and Mihai
S\^{i}rbu\footnote{The University of Texas at Austin, Department of
Mathematics, 2515 Speedway Stop C1200, Austin, Texas 78712,
\href{mailto:[email protected]}{[email protected]}. The
research of this author was supported in part by the National
Science Foundation under Grant DMS 1908903.} }
\date{\today}
\maketitle
\begin{abstract}
In a pseudo-Euclidean space with scalar product $S(\cdot, \cdot)$,
we show that the singularities of projections on $S$-monotone sets
and of the associated Fitzpatrick functions are covered by countable
$c-c$ surfaces having positive normal vectors with respect to the
$S$-product. By~\citet{Zajicek:79}, the singularities of a convex
function $f$ can be covered by a countable collection of $c-c$
surfaces. We show that the normal vectors to these surfaces are
restricted to the cone generated by $F-F$, where
$F:= \cl \range \nabla f$, the closure of the range of the
gradient of $f$. \end{abstract}
\begin{description} \item[Keywords:] convexity, subdifferential, Fitzpatrick function,
projection, pseudo-Euclidean space, normal vector, singularity. \item[AMS Subject Classification (2020):] 26B25,
26B05,
47H05,
52A20.
\end{description}
\section{Introduction}
A locally Lipschitz function $f=f(x)$ on $\real{d}$ is differentiable almost everywhere, according to the Rademacher's Theorem. The set of its singularities \begin{displaymath}
\Sigma(f) := \descr{x\in \real{d}}{f \text{ is not differentiable at
} x} \end{displaymath} can be quite irregular. For instance, for $d=1$, \citet{Zahorski:46} (see \citet{FowlPreiss:09} for a simple proof) shows that \emph{any} $G_{\delta \sigma}$ set (countable union of countable intersections of open sets) of Lebesgue measure zero is the singular set of \emph{some} Lipschitz function.
By \citet{Zajicek:79}, the $\Sigma(f)$ of a convex function $f=f(x)$ on $\real{d}$ has $c-c$ structure: it can be covered by a countable collection of the graphs of the differences of two convex functions of dimension $d-1$. \citet{Alberti:94} proves that, except for sets with zero $\mathcal{H}^{d-1}$ measure, the Hausdorff measure of dimension $d-1$, the covering can be achieved with smooth surfaces. These results yield sharp conditions for the existence and uniqueness of optimal maps in $\mathcal{L}_2$ optimal transport, see~\citet{Bren:91} and \citet[Theorem~1.26]{AmbrGigli:13}.
Let $E$ be a closed subset of $\real{d}$ and \begin{displaymath}
d_E(x):=\inf _{y\in E}|x-y|, \quad
P_E(x):= \descr{y\in E}{|x-y|=d_E(x)}, \end{displaymath} be the Euclidean distance to $E$ and the projection on $E$, respectively. \citet{Erdos:46} proves that the singular set \begin{displaymath}
\Sigma (P_E)
:= \descr{x\in \real{d}}{P_E(x) \text{ contains at least two points}} \end{displaymath} can be covered by countable sets with finite $\mathcal {H}^{d-1}$ measure. \citet{Hajlasz:22} uses \cite{Zajicek:79} and the observation of \citet{Asplund:69} that the function \begin{displaymath}
\psi _E (x):= \frac 12 |x|^2-\frac 12 d_E^2(x), \quad x\in \real{d}, \end{displaymath} is convex, to conclude that $\Sigma (P_E)$ has the $c-c$ structure. \citet{AlbanCannar:99} obtain a lower bound on the size of the set $\Sigma (d_E)$, where $d_E$ is not differentiable.
Let $S$ be a $d\times d$ invertible symmetric matrix with $m \in \braces{0,1,\dots,d}$ positive eigenvalues. Motivated by applications to backward martingale transport in \cite{KramXu:22} and \cite{KramSirb:22a}, we investigate in this paper the singularities of projections on monotone sets in the pseudo-Euclidean space with the scalar product \begin{displaymath}
S(x,y) := \ip{x}{Sy} = \sum_{i,j=1}^d x^iS^{ij} y^j, \quad x,y\in
\real{d}. \end{displaymath}
Let $G\subset \real{d}$ be an $S$-monotone or $S$-positive set: \begin{displaymath}
S(x-y,x-y)\geq 0, \quad x,y \in G. \end{displaymath} For every $x\in \real{d}$, the scalar square to $G$ and the projection on $G$ are given by \begin{align*}
\phi _G(x) & := \inf _{y\in G}S(x-y,x-y), \\
P_G(x) & := \argmin_{y\in G}S(x-y,x-y). \end{align*} Note that $S$-monotonicity is equivalent to $x\in G\implies x\in P_G(x)$ and that \begin{displaymath}
\psi _G(x):= \frac 12 S(x,x)- \frac 12 \phi _G(x), \quad
x\in\real{d}, \end{displaymath} is the Fitzpatrick function studied in \cite{Fitzpatrick:88}, \cite{Simons:07}, and \cite{Penot:09}.
The singularities of the projection $P_G$ can be classified as \begin{align*}
\Sigma(P_G) & := \descr{x\in \real{d}}{P_G(x) \text{ contains at
least two points}} \\
&\; = {\Sigma}_0(P_G) \cup \Sigma_1(P_G), \\
{ \Sigma}_0(P_G) & := \descr{x\in \Sigma(P_G)}{S(y_1-y_2,y_1-y_2)=0
\text{~for \emph{all}~} y_1,y_2 \in P_G(x) }, \\
\Sigma_1(P_G) & := \descr{x\in \Sigma(P_G)}{S(y_1-y_2,y_1-y_2)>0
\text{~for \emph{some}~} y_1,y_2 \in P_G(x)}. \end{align*} By Theorem \ref{th:3}, $\Sigma _1(P_G)$ is contained in a countable union of $c-c$ surfaces having strictly positive normal vectors in the $S$-space. The structure of the zero-order singularities is described in Theorems~\ref{th:5} and~\ref{th:4}. If $m=1$, then ${\Sigma}_0(P_G)$ is covered by a countable number of hyperplanes having isotropic normal vectors in the $S$-space. If $m\geq 2$, then ${\Sigma}_0(P_G)$ is covered by a countable family of $c-c$ surfaces whose normal vectors are positive and almost isotropic in the $S$-space.
Using similar tools, in Theorem \ref{th:2}, we improve the $c-c$ description of singularities of general convex functions $f=f(x)$ from \citet{Zajicek:79} by showing that the covering surfaces have normal vectors belonging to the cone generated by $F-F$, where $F:= \cl \range \nabla f$, the closure of the range of the gradient of $f$.
\section{Parametrization of singularities} \label{sec:param-sing}
We say that a function $g = g(x)$ on $\real{d}$ has \emph{linear
growth} if there is a constant $K=K(g)>0$ such that \begin{displaymath}
\abs{g(x)} \leq K (1 + \abs{x}), \quad x\in \real{d}. \end{displaymath} We write $\dom{\nabla g}$ for the set of points where $g$ is differentiable.
Let $j\in \braces{1,\dots,d}$. We denote by $\mathcal{C}^j$ the collection of compact sets $C$ in $\real{d}$ such that \begin{displaymath}
y^j=1, \quad y\in C. \end{displaymath} Any compact set $C\subset \rplus{d}{j}$ can be rescaled as \begin{displaymath}
\theta^j(C):= \descr{ \frac{y}{y^j}}{y\in C}\in \mathcal{C}^j. \end{displaymath} For $x \in \real{d}$, we denote by $x^{-j}$ its sub-vector without the $j$th coordinate: \begin{displaymath}
x^{-j} :=
\cbraces{x^1,\dots,x^{j-1},x^{j+1},\dots,x^{d}}
\in \real{d-1}. \end{displaymath} For $C\in \mathcal{C}^j$, we write $\mathcal{H}^{j}_{C}$ for the family of functions $h=h(x)$ on $\real{d}$ having the decomposition: \begin{equation}
\label{eq:1}
h(x) = x^j + g_1(x^{-j}) - g_2(x^{-j}), \quad
x\in \real{d}, \end{equation} where the functions $g_1$ and $g_2$ on $\real{d-1}$ are convex, have linear growth, and \begin{equation}
\label{eq:2}
\nabla h(x)\in C, \quad x^{-j} \in \dom{\nabla g_1} \cap \dom{\nabla
g_2}. \end{equation}
The latter property has a clear geometric interpretation. Let $H$ be a closed set in $\real{d}$ and $x\in H$. Following \cite[Definition~6.3 on page~199]{RockWets:98}, we call a vector $w\in \real{d}$ \emph{regular normal to $H$ at $x$} if \begin{displaymath}
\limsup_{\substack{H\ni y\to x \\ y\not= x}}
\frac{\ip{w}{y-x}}{\abs{y-x}} \leq 0. \end{displaymath} A vector $w\in \real{d}$ is called \emph{normal to $H$ at $x$} if there exist $x_n\in H$ and a regular normal vector $w_n$ to $H$ at $x_n$ such that $x_n\rightarrow x$ and $w_n\rightarrow w$.
For a set $B$ in $\real{d}$, we denote by $\conv{B}$ its convex hull. \begin{Lemma}
\label{lem:1}
Let $j\in \braces{1,\dots,d}$, $C\in \mathcal{C}^j$, $h$ be given
by~\eqref{eq:1} for convex functions $g_1$ and $g_2$ with linear
growth, and $H$ be the zero-level set of $h$:
\begin{displaymath}
H := \descr{x\in \real{d}}{h(x) = 0}.
\end{displaymath}
Then $h\in \mathcal{H}^j_C$, that is, \eqref{eq:2} holds, if and
only if for every $x\in H$, there exists a normal vector $w\in C$ to
$H$ at $x$. \end{Lemma}
\begin{proof}
We can assume that $j=1$. Denote $f:= g_1-g_2$, so that
$h(x)=x^1+f(x^{-1})$. We have that
$\dom {\nabla{h}}=\real{}\times \dom{\nabla{f}}$,
$\nabla h(x)= \cbraces{1, \nabla f(x^{-1})}$, and
\begin{displaymath}
H=\descr{(-f(u), u)}{u\in \real{d-1}}.
\end{displaymath}
Denote also
$U:= \dom{\nabla g_1} \cap \dom{\nabla g_2} \subset \dom
{\nabla{f}}$.
$\implies$: Clearly, the gradient $\nabla h(x)$ is a regular normal
vector to $H$ at $x\in \dom{\nabla h}\cap H$. As $U$ is dense in
$\real{d-1}$, the result holds by standard compactness arguments.
$\impliedby$: Let $u\in U$, $v\in \real{d-1}$, and $w=(1,v)\in C$ be
a normal vector to $H$ at $x=(-f(u),u)$. Take a sequence
$(x_n,w_n)$, $n\geq 1$, that converges to $(x,w)$ and where $w_n$ is
a regular normal vector to $H$ at $x_n\in H$ with $w_n^1=1$. Such a
sequence exists by the definition of a normal vector. We can
represent $x_n=(-f(u_n), u_n)$ for $u_n\in \real{d-1}$ and
$w_n=(1, v_n)$ for $v_n\in \real{d-1}$.
We claim that $v_n$ belongs to the Clarke gradient of $f$ at $u_n$:
\begin{displaymath}
v_n \in \cpartial f(u_n) := \conv \descr{\lim_m \nabla
f(r_m)}{\dom \nabla f \ni r_m \to u_n}.
\end{displaymath}
The continuity of $\nabla f$ at $u\in U$ then yields that
$v_n \to \nabla f(u)$. Hence, $\nabla f(u)=v$, implying that
$\nabla h(x) =(1, \nabla f(u))= w \in C$.
In order to prove the claim, we write the definition of
$w_n=(1,v_n)$ being regular normal to $H$ at $x_n=(-f(u_n), u_n)$ as
\begin{displaymath}
\limsup _{\substack{r\rightarrow u_n \\ r\not= u_n}}
\frac{-\cbraces{f(r)-f(u_n)}+\ip{v_n}{r-u_n}}{|f(r)-f(u_n)|+|r-u_n|}
\leq 0.
\end{displaymath}
Using the Lipschitz property of $f$, we obtain that
\begin{displaymath}
\ip{v_n}{s}\leq \limsup_{\delta \downarrow 0}\frac{f(u_n+\delta
s)-f(u_n)}{\delta}, \quad s\in \real{d-1}.
\end{displaymath}
By~\cite[Corollary~1.10]{Clarke:75}, we have that
$v_n\in \cpartial f (u_n)$, as claimed. \end{proof}
We recall that the subdifferential $\smap{\partial f}{\real{d}}{\real{d}}$ of a closed convex function $\map{f}{\real{d}}{\mathbb{R} \cup \braces{+\infty}}$ is defined as \begin{displaymath}
\partial f(x):= \descr{y\in \real d}{\ip{z}{y}\leq f(x+z)-f(x), \,
z\in \real{d}}. \end{displaymath} Clearly, $\dom \partial f := \descr{x\in \real{d}}{\partial f(x)\not=
\emptyset} \subset \dom f := \descr{x\in \real{d}}{f(x)<\infty}$.
The following theorem is our main technical tool for the study of singularities of convex and Fitzpatrick functions in Sections~\ref{sec:sing-points-conv} and~\ref{sec:sing-points-fitzp}.
\begin{Theorem}
\label{th:1}
Let $\map{f}{\real{d}}{\mathbb{R} \cup \braces{+\infty}}$ be a closed convex function,
$j\in \braces{1,\dots, d}$, and $C_1$ and $C_2$ be compact sets in
$\real{d}$ such that
\begin{displaymath}
y^j> 0, \quad y \in C_2-C_1.
\end{displaymath}
There exists a function $h\in \mathcal {H}^j_{\theta^j(C_2-C_1)}$
such that
\begin{align*}
\Sigma_{C_1,C_2}(\partial f)
:= & \descr{x\in \dom{\partial f(x)}}{\partial f(x) \cap C_i \not
= \emptyset, \, i=1,2} \\
\subset & \descr{x\in \real{d}}{h(x)=0}.
\end{align*} \end{Theorem}
The proof of the theorem relies on Lemma~\ref{lem:2}. For a closed set $A\subset \real{d}$, we denote \begin{equation}
\label{eq:3}
f_A(x): = \sup_{y\in A}\cbraces{\ip{x}{y} - f^*(y)}, \quad x\in
\real{d}, \end{equation} where $f^*$ is the convex conjugate of $f$: \begin{displaymath}
f^*(y):= \sup_{x\in \real{d}}\cbraces{\ip{x}{y}-f(x)} \in
\mathbb{R} \cup \braces{+\infty}, \quad y \in
\real{d}. \end{displaymath} We have that $f_A$ is a closed convex function taking values in $\mathbb{R} \cup \braces{+\infty}$ if and only if \begin{displaymath}
A\cap \dom {f^*} = \descr{x\in A}{f^*(x)< \infty} \not= \emptyset; \end{displaymath} otherwise, $f_A=-\infty$. We recall that \begin{equation}
\label{eq:4}
f(x)=\ip{x}{y}-f^*(y)\iff y\in
\partial f(x) \iff x\in \partial f^*(y). \end{equation}
\begin{Lemma}
\label{lem:2}
Let $\map{f}{\real{d}}{\mathbb{R} \cup \braces{+\infty}}$ be a closed convex function and
$C$ be a compact set in $\real{d}$ such that
$C\cap \dom{f^*}\not=\emptyset$. Then $f_C$ has linear growth and
for every $x\in \real{d}$,
\begin{gather*}
\partial f_C(x) \cap C = \Arg_C(x) := \argmax_{y\in
C}\cbraces{\ip{x}{y} -
f^*(y)} \not=\emptyset, \\
\partial f_C(x) = \conv{\cbraces{\partial f_C(x) \cap C}}, \\
\partial f(x) \cap C \not= \emptyset \iff f(x) = f_C(x) \iff
\partial f(x) \cap C = \partial f_C(x) \cap C.
\end{gather*}
In particular, $f_C$ is differentiable at $x$ if and only if
$\partial f_C(x) \cap C$ is a singleton, in which case
\begin{displaymath}
\partial f_C(x) = \braces{\nabla f_C(x)} \in C.
\end{displaymath} \end{Lemma}
\begin{proof}
Since $C$ is a compact set, $f^*$ is a closed convex function, and
$C\cap\dom {f^*}\not= \emptyset$, we have that
\begin{displaymath}
\sup_{y\in C} \abs{y} < \infty, \quad \inf_{x\in C}
f^*(x) > -\infty,
\end{displaymath}
and that $\Arg_C(x)$ is a non-empty compact. Let
$y_0\in C\cap \dom {f^*}$. From the definition of $f_C$ we deduce
that
\begin{displaymath}
-\abs{x} \abs{y_0} -f^*(y_0) \leq f_C(x) \leq \abs{x}\sup_{y\in
C}\abs{y}-\inf_{y\in C}f^*(y), \quad x\in \real{d}.
\end{displaymath}
It follows that $f_C$ has linear growth.
The function
\begin{displaymath}
h(x,y) := \ip{x}{y}-f^*(y) \in \real{}\cup\braces{-\infty}, \quad
x,y\in \real{d},
\end{displaymath}
is linear in $x$ and concave and upper semi-continuous in $y$. Fix
$x\in \real{d}$. We can choose $K$ large enough such that for
\begin{displaymath}
E:= \descr{z\in C}{f^*(z)\leq K},
\end{displaymath}
we have that $f_C=f_{E}$ in a neighborhood of $x$ and
\begin{displaymath}
\Arg _C(x)=\Arg _E(x):=
\argmax_{y\in E}\cbraces{\ip{x}{y} -f^*(y)}.
\end{displaymath}
Since $E$ is compact and the function $h(\cdot, y)$ is finite for
$y\in E$, the classical envelope theorem
\cite[Theorem~4.4.2, p.~189]{HiriarLemar:01} yields that
\begin{align*}
\partial f_C(x) \
& = \partial f_E(x)= \partial \max_{y\in E}h(x,y) =
\conv\bigcup_{y\in \Arg_E(x)} \partial_x h(x,y) = \conv
\Arg_E(x) \\
& = \conv{\Arg _C(x)}.
\end{align*}
From the concavity of $h(x,\cdot)$ we deduce that
\begin{displaymath}
\partial f_C(x) \cap C= \cbraces{\conv{\Arg _C(x)}} \cap C
= \Arg _C(x).
\end{displaymath}
Clearly, $f_C\leq f$. If $y\in \partial f(x) \cap C$, then
$f(x) = \ip{x}{y} - f^*(y) \leq f_C(x)$. Hence, $f_C(x)= f(x)$ and
$y\in \Arg _C(x)\subset \partial f_C(x)$.
Conversely, let $f_C(x) = f(x)$. For every
$y\in \partial f_C(x) \cap C$, we have that
$f(x) = f_C(x) = \ip{x}{y} - f^*(y)$ and then that
$y\in \partial f(x)\cap C$.
Finally, being a convex function, $f_C$ is differentiable at $x$ if
and and only if $\partial f_C(x)$ is a singleton. In this case,
$\partial f_C(x) = \braces{\nabla f_C(x)}$. \end{proof}
\begin{proof}[Proof of Theorem~\ref{th:1}]
Hereafter, $i=1,2$. We assume that
$\Sigma_{C_1,C_2}(\partial f) \not= \emptyset$ as otherwise, there
is nothing to prove. This implies that
$C_i\cap \dom {f^*}\not=\emptyset$. Let $C:= C_1\cup C_2$.
Lemma~\ref{lem:2} yields that
\begin{equation}
\label{eq:5}
\Sigma_{C_1,C_2}(\partial f) = \Sigma_{C_1,C_2}(\partial
f_C) \cap \descr{x\in \real{d}}{f(x) = f_C(x)}.
\end{equation}
To simplify notations we assume that $j=1$. Denote by
\begin{displaymath}
a := \max_{y\in C_1} {y}^1 <\min_{y \in C_2} {y}^1 := b.
\end{displaymath}
We write $x \in \real{d}$ as $(t,u)$, where $t\in \real{}$ and
$u\in \real{d-1}$, and define the saddle function
\begin{displaymath}
g(t,u): = \inf_{s\in \real{}} \cbraces{f_C(s,u) - st},
\quad a<t<b, u\in \real{d-1}.
\end{displaymath}
Select $y_i =(q_i, z_i)\in C_i\cap \dom {f^*}$. We have that
\begin{align*}
f_C(s,u) & \geq \max_{i=1,2} \cbraces{sq_i+\ip{u}{z_i} - f^*(y_i)}.
\end{align*}
Since $q_1\leq a<b\leq q_2$, it follows from the definition of $g$
that
\begin{displaymath}
-\infty< g(t,u) \leq f_C(0,u), \quad a<t<b, u\in \real{d-1}.
\end{displaymath}
By Lemma~\ref{lem:2}, $f_C$ has linear growth. The classical results
on saddle functions, Theorems~33.1 and~37.5 in \cite{Rock:70}, imply
that
\begin{enumerate}[label = {\rm (\roman{*})}, ref={\rm (\roman{*})}]
\item \label{item:1} For every $a<t <b$, the function $g(t,\cdot)$
is convex and has linear growth on $\real{d-1}$.
\item \label{item:2} For every $u\in \real{d-1}$, the function
$g(\cdot, u)$ is concave and finite on $(a,b)$.
\item \label{item:3} For any $a<t<b$, we have that
$(t,v) \in \partial f_C(s,u)$ if and only if
$v \in \partial_u g(t,u)$ and $-s \in \partial_tg(t,u)$. In this
case, $g(t,u) = f_C(s,u) - st$.
\end{enumerate}
We take $a <r_1<r_2<b$ and denote $g_i := g(r_i, \cdot)$. We
verify the assertions of the theorem for the function
\begin{displaymath}
h(s,u):= s +
\frac1{r_2-r_1}\cbraces{g_2(u) - g_1(u)}, \quad s\in \real{}, u\in
\real{d}.
\end{displaymath}
More precisely, we show that
\begin{displaymath}
\Sigma_{C_1,C_2}(\partial f_C)= \descr{x\in \real{d}}{h(x)=0},
\end{displaymath}
which together with~\eqref{eq:5} implies the result.
Let $x=(s,u)$ and $(t_i,v_i) \in \partial f_C(x) \cap C_i$. As
$t_1 \leq a<r_i<b \leq t_2$, the convexity of subdifferentials
yields that $(r_i,w_i) \in \partial f_C(x)$, where
\begin{equation}
\label{eq:6}
w_i = \frac{t_2-r_i}{t_2-t_1} v_1 + \frac{r_i-t_1}{t_2-t_1} v_2.
\end{equation}
By~\ref{item:3}, $g_i(u) = f_C(x) - s r_i$ and then $h(x) = 0$.
Conversely, let $x=(s,u)$ be such that $h(x) = 0$ or, equivalently,
\begin{displaymath}
-s = \frac{g_2(u) - g_1(u)}{r_2-r_1} = \frac{g(r_2,u) - g(r_1,u)}{r_2-r_1}.
\end{displaymath}
The mean-value theorem yields $r\in [r_1,r_2]$ such that
$-s \in \partial_t g(r,u)$. Observe that $a<r<b$. Taking any
$w \in \partial_u g(r,u)$, we deduce from~\ref{item:3} that
$y:= (r,w) \in \partial f_C(x)$. As
$\partial f_C = \conv \cbraces{\partial f_C \cap C}$, the point $y$
is a convex combination of some $y_i \in \partial f_C(x) \cap C_i$,
$i=1,2$. In particular, $x\in \Sigma_{C_1,C_2}(\partial f_C)$.
Finally, let $x=(s,u)$ be such that $h(x) = 0$ and $g_1$ and $g_2$
are differentiable at $u$. As we have already shown, the gradients
$w_i := \nabla g_i(u)$ are given by~\eqref{eq:6} for some
$(t_i,v_i) \in \partial f_C(x) \cap C_i$. It follows that
\begin{displaymath}
\nabla h(x) = \cbraces{1, \frac{w_2-w_1}{r_2-r_1}} = \cbraces{1,
\frac{v_2-v_1}{t_2-t_1}} \in
\theta^1(C_2-C_1).
\end{displaymath}
Hence, $h \in \mathcal{H}^1_{\theta^1(C_2-C_1)}$. \end{proof}
\section{Singular points of convex functions} \label{sec:sing-points-conv}
For a multi-function $\smap{\Psi}{\real{d}}{\real{d}}$ taking values in closed subsets of $\real{d}$, we denote its domain by \begin{displaymath}
\dom \Psi := \descr{x\in \real{d}}{\Psi (x)\not= \emptyset}. \end{displaymath} Given an index $j\in \braces{1,\dots,d}$ and a closed set $A$ in $\real{d}$, the \emph{singular} set of $\Psi$ is defined as \begin{gather*}
\Sigma^j_{A}(\Psi) := \descr{x\in \dom \Psi}{\exists y_1,y_2 \in
\Psi(x) \cap A \text{ with } y_1^j\not=y_2^j}. \end{gather*} We also write $\Sigma^j(\Psi) = \Sigma ^j_{\real{d}}(\Psi)$ and $\Sigma(\Psi) = \cup_{j=1,\dots,d} \Sigma^j(\Psi)$.
Let $\map{f}{\real{d}}{\mathbb{R} \cup \braces{+\infty}}$ be a closed convex function such that its domain has a non-empty interior: \begin{displaymath}
D:= \interior\dom f \not= \emptyset. \end{displaymath} It is well-known that $\dom \nabla f := \descr{x\in D}{\nabla f(x) \text{~exists}}$ is dense in $D$ and \begin{displaymath}
D \setminus \dom \nabla f = \Sigma(\partial f) \cap D =
\descr{x\in D}{\partial f(x) \text{ is not a point}}. \end{displaymath} According to~\cite{Zajicek:79}, see also~\cite{Alberti:94} and~\cite{Hajlasz:22}, this set of \emph{interior} singularities can be covered by countable $c-c$ surfaces $H_n = \descr{x\in \real{d}}{h_n(x) = 0}$, $n\geq 1$, where \begin{displaymath}
h_n(x) = x^j + g_{n,1}(x^{-j}) - g_{n,2}(x^{-j}), \quad
x\in \real{d}, \end{displaymath} for some $j\in \braces{1,\dots,d}$ and finite convex functions $g_{n,1}$ and $g_{n,2}$ on $\real{d-1}$.
Theorem~\ref{th:2} and Lemma \ref{lem:1} describe the \emph{orientation} of the covering surface $H_n$ by showing that, at any point, it has a normal vector $w$ with $w^j=1$, that belongs to the cone $K$ generated by $F-F$, where \begin{displaymath}
F:= \cl \range \nabla f =
\descr{\lim_n \nabla f(x_n)}{x_n \in \dom
\nabla f}. \end{displaymath} Of course, this information has some value only if $K$ is distinctively smaller than $\real{d}$. A related example of Fitzpatrick functions is studied in Section~\ref{sec:sing-points-fitzp}. Proposition~\ref{prop:1} contains the geometric interpretation of the $\range \nabla f$ and Lemma~\ref{lem:4} explains the special feature of its closure $F$.
It turns out that the same surfaces $(H_n)$ also cover the singularities of the Clarke-type subdifferential $\cpartial f$ defined on the \emph{closure} of $D$: \begin{displaymath}
\cpartial f(x) := \cl \conv \descr{\lim_n \nabla
f(x_n)}{\dom
\nabla f \ni x_n \to x}, \quad x\in \cl D. \end{displaymath} By Theorem~25.6 in~\cite{Rock:70}, \begin{displaymath}
\partial f(x) = \cpartial f(x) + N_{\cl D}(x), \quad x\in \cl D, \end{displaymath} where $N_A(x)$ denotes the normal cone to a closed convex set $A\subset \real{d}$ at $x\in A$: \begin{align*}
N_{A}(x) := & \descr{s\in \real{d}}{\ip{s}{y-x}\leq 0 \text{ for
all } y\in A} \\
= & \descr{s\in \real{d}}{s \text{ is a regular normal vector to } A \text{ at }
x}. \end{align*} Recalling that $0\in N_{A}(x)$ for $x\in A$ and $N_{A}(x) =\braces{0}$ for $x\in \interior{A}$, we deduce that \begin{align}
\label{eq:7}
\dom{\cpartial f} & = \dom{\partial f}, \\
\label{eq:8}
\cpartial f(x) & \subset \partial f(x), \quad x \in \cl D, \\
\label{eq:9}
\cpartial f(x) & = \partial f(x), \quad x\in D. \end{align}
The diameter of a set $E$ is denoted by $\diam{E} := \sup_{x,y\in E} \abs{x-y}$.
\begin{Theorem}
\label{th:2}
Let $\map{f}{\real{d}}{\mathbb{R} \cup \braces{+\infty}}$ be a closed convex function with
$D:= \interior\dom f \not= \emptyset$. Let
$j\in \braces{1,\dots,d}$ and $A$ be a closed set in $\real{d}$
containing $\range \nabla f$. Then
\begin{gather}
\label{eq:10}
\Sigma^j(\partial f) \cap D = \Sigma^j(\cpartial f) \cap D =
\Sigma^j_A(\partial f)\cap
D, \\
\label{eq:11}
\Sigma^j(\cpartial f) \subset \Sigma^j_A(\partial f).
\end{gather}
If $y^j = z^j$ for all $y,z \in A$, then, clearly,
$\Sigma_A^j(\partial f)=\emptyset$. Otherwise, for every $n\geq 1$,
there exist a compact set $C_n \subset A-A$ with $y^j > 0$,
$y\in C_n$, and a function $h_n \in \mathcal{H}^j_{\theta^j(C_n)}$
such that
\begin{equation}
\label{eq:12}
\Sigma_A^j(\partial f) \subset
\bigcup_n \descr{x\in \real{d}}{h_n(x)=0}.
\end{equation}
For any $\epsilon>0$, all $C_n$ can be chosen so that
$\diam{\theta^j(C_n)}< \epsilon$. \end{Theorem}
Taking the unions over $j\in \braces{1,\dots,d}$, we obtain the descriptions of the \emph{full} singular sets $\Sigma(\partial f)$ on $D$, $\Sigma(\cpartial f)$, and $\Sigma_A(\partial f)$. Taking a smaller $\epsilon>0$ in the last sentence of the theorem, we make the directions of the normal vectors to the covering surface $H_n$ closer to each other. As a result, $H_n$ gets approximated by a hyperplane.
Theorem~\ref{th:2} uses a larger closed set $A$ instead of $F$ to allow for more flexibility in the treatment of singularities of $\partial f$ on the boundary of $D$. Lemma~\ref{lem:5} shows that \begin{displaymath}
\partial f(x) \cap A
= \Arg_A(x) := \argmax_{y\in A}\cbraces{\ip{x}{y} -
f^*(y)}, \quad x \in \cl D. \end{displaymath}
In the framework of Fitzpatrick functions in Section~\ref{sec:sing-points-fitzp}, $\Arg_A$ becomes a projection on the monotone set $A$ in a pseudo-Euclidean space.
The proof of Theorem~\ref{th:2} relies on some lemmas. We start with a simple fact from convex analysis.
\begin{Lemma}
\label{lem:3}
Let $\map{f}{\real{d}}{\mathbb{R} \cup \braces{+\infty}}$ be a closed convex function
attaining a strict minimum at a point $x_0$:
\begin{displaymath}
f(x_0)<f(x), \quad x\in \real{d}, \, x\not =x_0.
\end{displaymath}
Then
\begin{displaymath}
f(x_0) < \inf_{\abs{x-x_0} \geq \epsilon} f(x), \quad \epsilon >
0.
\end{displaymath} \end{Lemma}
\begin{proof}
If the conclusion is not true, then there exist $\epsilon > 0$ and a
sequence $(x_n)$ with $|x_n-x_0|\geq \epsilon$ such that
$f(x_n)\rightarrow f(x_0)$. As
\begin{displaymath}
z_n:= x_0+\frac{\epsilon}{\abs{x_n-x_0}}(x_n-x_0)
\end{displaymath}
is a convex combination of $x_0$ and $x_n$, we deduce that
\begin{displaymath}
f(z_n) \leq \max(f(x_0),f(x_n)) = f(x_n) \to f(x_0).
\end{displaymath}
By compactness, $z_n\rightarrow z_0$ over a subsequence. Clearly,
$\abs{z_0-x_0}=\epsilon$, while by the lower semi-continuity,
$f(z_0)\leq \liminf f(z_n)\leq f(x_0)$. We have arrived to a
contradiction. \end{proof}
The following result explains the special role played by $\cl\range \nabla f$. Recall the notation $f_A$ from~\eqref{eq:3}.
\begin{Lemma}
\label{lem:4}
Let $\map{f}{\real{d}}{\mathbb{R} \cup \braces{+\infty}}$ be a closed convex function with
$D:= \interior \dom f \not=\emptyset$ and $A$ be a closed set in
$\real{d}$. Then
\begin{displaymath}
f_A(x) = f(x), \, x\in \cl D \iff \range \nabla f \subset A.
\end{displaymath}
In other words, $F:= \cl\range \nabla f$ is the minimal closed set
such that $f_F=f$ on $\cl D$. \end{Lemma}
\begin{proof}
$\impliedby$: If $x \in \dom \nabla f$, then
$y:= \nabla f(x)\in A$ and~\eqref{eq:4} yields that
\begin{displaymath}
f(x) = \ip{x}{y} - f^*(y) = f_A(x).
\end{displaymath}
Since $\dom{\nabla f}$ is dense in $D$, the closed convex functions
$f_A$ and $f$ coincide on $\cl D$.
$\implies$: We fix $x_0\in \dom \nabla f$ and set
$y_0:= \nabla f(x_0)$. By the assumption, $f_A(x_0)=f(x_0)$. If
$y_0\notin A$, then the distance between $y_0$ and $A$ is at least
$\epsilon >0$. According to~\eqref{eq:4}, the concave upper
semi-continuous function
\begin{displaymath}
y\rightarrow \ip{x_0}{y}-f^*(y)
\end{displaymath}
attains a strict global maximum at $y_0$ and has the maximum value
$f(x_0)$. By Lemma~\ref{lem:3},
\begin{displaymath}
\sup_{\abs{y-y_0} \geq \epsilon}
\cbraces{\ip{x_0}{y} - f^*(y)} <f(x_0).
\end{displaymath}
As $A\subset \descr{y\in \real{d}}{|y-y_0|\geq \epsilon}$, we arrive
to a contradiction:
\begin{displaymath}
f_A(x_0)=\sup_{y\in A}\cbraces{\ip{x_0}{y} - f^*(y)} \leq
\sup_{\abs{y-y_0}\geq \epsilon}
\cbraces{\ip{x_0}{y} - f^*(y)} <f(x_0).
\end{displaymath}
Hence, $y_0 \in A$, as required. \end{proof}
\begin{Lemma}
\label{lem:5}
Let $\map{f}{\real{d}}{\mathbb{R} \cup \braces{+\infty}}$ be a closed convex function with
$D:= \interior \dom f \not=\emptyset$. Let $A$ be a closed set in
$\real{d}$ such that $f=f_A$ on $\cl D$. Then
\begin{gather*}
\partial f(x) \cap A = \Arg_A(x) := \argmax_{y\in
A}\cbraces{\ip{x}{y} -
f^*(y)}, \quad x \in \cl D, \\
\dom \Arg_A \cap \cl D
= \dom \partial f=\dom \cpartial f, \\
\cpartial f(x) = \partial f(x) = \conv{\cbraces{\partial f(x) \cap
A}}, \quad
x\in D, \\
\cpartial f(x) \subset \cl\conv\cbraces{\partial f(x) \cap A},
\quad x\in \dom \cpartial f.
\end{gather*} \end{Lemma} \begin{proof}
If $x \in \dom \Arg_A \cap \cl D$ and $y\in \Arg_A(x)$, then
\begin{equation}
\label{eq:13}
f(x) = f_A(x) = \ip{x}{y} - f^*(y).
\end{equation}
Hence, $y\in \partial f(x) \cap A$, by the properties of
subdifferentials in~\eqref{eq:4}.
Conversely, let $x\in \dom \partial f$. Lemma~\ref{lem:4} shows
that $F:= \cl\range \nabla f \subset A$. Accounting
for~\eqref{eq:7} and~\eqref{eq:8} and the definition of
$\cpartial f(x)$, we obtain that
\begin{displaymath}
\partial f(x) \cap A \supset \partial f(x) \cap F
\supset \cpartial f(x)\cap F\not=\emptyset.
\end{displaymath}
Let $y\in \partial f(x) \cap A$. From~\eqref{eq:4} we
deduce~\eqref{eq:13} and then that $y \in \Arg_A(x)$. We have proved
the first two assertions of the lemma.
The fact that $\cpartial f = \partial f$ on $D$ has been already
stated in~\eqref{eq:9}. To prove the second equality in the third
assertion, we fix $x_0\in D$ and choose $\epsilon>0$ such that
\begin{displaymath}
B(x_0,\epsilon) := \descr{x\in \real{d}}{\abs{x-x_0}\leq
\epsilon} \subset D.
\end{displaymath}
The uniform boundedness of $\partial f$ on compacts in $D$ implies
the existence of a constant $K>0$ such that
\begin{displaymath}
\abs{y} \leq K<\infty, \quad y\in \partial f(x), \, x\in
B(x_0,\epsilon).
\end{displaymath}
Denote by $A_K := A \cap \descr{x\in \real{d}}{\abs{x}\leq K}$.
As $\range \nabla f \subset A$, we deduce from~\eqref{eq:4} that
\begin{displaymath}
f(x)=f_{A_K}(x), \quad x\in B(x_0,\epsilon)\cap \dom \nabla f,
\end{displaymath}
and then, by the density of $\dom \nabla f$ in $D$, that $f=f_{A_K}$
on $B(x_0,\epsilon)$. Finally, Lemma~\ref{lem:2} shows that
\begin{displaymath}
\partial f(x_0) = \partial f_{A_K}(x_0) = \conv\cbraces{\partial
f_{A_K}(x_0) \cap A_K} = \conv\cbraces{\partial f(x_0) \cap A}.
\end{displaymath}
If $\dom \nabla f \ni x_n \to x$ and $\nabla f(x_n) \to y$, then,
clearly, $y\in F \subset A$. Moreover, $y \in \partial f(x)$, by the
continuity of subdifferentials. The last assertion of the lemma
readily follows. \end{proof}
We are ready to finish the proof of Theorem~\ref{th:2}.
\begin{proof}[Proof of Theorem~\ref{th:2}]
Lemma~\ref{lem:4} shows that $f=f_A$ on $\cl D$. This fact and the
last two assertions of Lemma~\ref{lem:5} readily yield the
relations~\eqref{eq:10} and~\eqref{eq:11}.
Fix $\epsilon >0$. Let $(x_n)$ be a dense sequence in $A$ and
$(r_n)$ be an enumeration of all positive rationals. Denote by
$\alpha:= (m, n,k,l)$ the indexes for which the compacts
\begin{displaymath}
C^{\alpha}_1 :=\descr{x\in A }{ |x-x_m|\leq r_k}, \quad
C^{\alpha}_2 :=\descr{x\in A}{|x-x_n|\leq r_l},
\end{displaymath}
satisfy the constraints:
\begin{displaymath}
\diam \theta^j(C^{\alpha}_2-C^{\alpha}_1) < \epsilon \text{~and~}
x^j > 0, \; x \in C_2^{\alpha} -
C_1^{\alpha}.
\end{displaymath}
We have that
\begin{displaymath}
\Sigma^j_A(\partial f) = \bigcup_{\alpha} \Sigma_{C^{\alpha}_1,
C^{\alpha}_2}(\partial f) = \bigcup_{\alpha} \descr{x\in
\real{d}}{\partial f(x) \cap C^{\alpha}_i \not
= \emptyset, \, i=1,2}.
\end{displaymath}
For every index $\alpha$, Theorem~\ref{th:1} yields a function
$h\in \mathcal H^j_{\theta^j(C^{\alpha}_2 - C^{\alpha}_1)}$ such
that
\begin{displaymath}
\Sigma_{C^{\alpha}_1, C^{\alpha}_2}(\partial f)\subset \descr{x\in
\real{d}}{h(x)=0}.
\end{displaymath}
We have proved~\eqref{eq:12} and with it the theorem. \end{proof}
We conclude the section with the geometric interpretation of $\range \nabla f$. For a closed convex function $\map{f}{\real{d}}{\mathbb{R} \cup \braces{+\infty}}$, we denote by $\epi f$ its epigraph: \begin{displaymath}
\epi f := \descr{(x,q) \in \real{d}\times \real{}}{f(x) \leq q}. \end{displaymath} Let $E$ be a closed convex set in $\real{d}$. A point $x_0\in E$ is called \emph{exposed} if there is a hyperplane intersecting $E$ only at $x_0$. In other words, there is $y_0 \in \real{d}$ such that \begin{displaymath}
\ip{x-x_0}{y_0} > 0, \quad x\in E \setminus \braces{x_0}. \end{displaymath}
\begin{Proposition}
\label{prop:1}
Let $\map{f}{\real{d}}{\mathbb{R} \cup \braces{+\infty}}$ be a closed convex function with
$\interior{\dom f} \not= \emptyset$. Then
\begin{displaymath}
\descr{\cbraces{y, f^*(y)} }{y\in \range \nabla f} = \text{
exposed points of } \epi f^*.
\end{displaymath} \end{Proposition}
\begin{proof}
By definition, $(y_0,r_0)$ is an exposed point of $\epi f^*$ if it
belongs to $\epi f^*$ and
\begin{displaymath}
\ip{y-y_0}{x_0} + (r-r_0) q_0 > 0, \quad (y,r) \in \epi f^*
\setminus \braces{(y_0,r_0)},
\end{displaymath}
for some $(x_0,q_0)\in \real{d}\times \real{}$. The definition of
$\epi f^*$ ensures that $q_0>0$ and $r_0=f^*(y_0)$. Rescaling
$(x_0,q_0)$ so that $q_0=1$, we deduce that the function
\begin{displaymath}
{y} \to {\ip{x_0}{y} +f^*(y)}
\end{displaymath}
has the unique minimizer $y_0$. This is equivalent to $y_0$ being
the only element of $\partial f(-x_0)$, which in turn is equivalent
to $-x_0\in \dom \nabla f$ and $y_0=\nabla f(-x_0)$. \end{proof}
\section{Singular points of projections in $S$-spaces} \label{sec:sing-points-fitzp}
We denote by $\msym{d}{m}$ the family of symmetric $d\times d$-matrices of full rank with $m \in \braces{0,1,\dots,d}$ positive eigenvalues. For $S \in \msym{d}{m}$, the bilinear form \begin{displaymath}
S(x,y) := \ip{x}{Sy} = \sum_{i,j=1}^d x^i S_{ij}y^j, \quad
x,y\in \real{d}, \end{displaymath} defines the scalar product on a pseudo-Euclidean space $\real{d}_m$ with dimension $d$ and index $m$, which we call the $S$-space. The quadratic form $S(x,x)$ is the \emph{scalar square} on the $S$-space; its value may be negative.
For a closed set $G\subset \real{d}$, we define the \emph{Fitzpatrick-type} function \begin{displaymath}
\psi_G (x):= \sup _{y\in G}\cbraces{S(x,y)-\frac 12S(y,y)} \in
\mathbb{R} \cup \braces{+\infty},
\quad x \in \real{d}, \end{displaymath} and the projection multi-function \begin{align*}
P_G (x):= & \argmin_{y\in G}S(x-y, x-y) = \argmax_{y\in G}\cbraces{S(x,y)-\frac 12S(y,y)}, \quad x \in \real{d}. \end{align*} Clearly, $\psi_G$ is a closed convex function and $P_G$ takes values in the closed (possibly empty) subsets of $G$.
A closed set $G\subset \real{d}$ is called $S$-\emph{monotone} or $S$-\emph{positive} if \begin{displaymath}
S(x-y,x-y) \geq 0, \quad x, y
\in G, \end{displaymath} or, equivalently, if its projection multi-function has the natural fixed-point property: \begin{displaymath}
x\in P_G(x), \quad x\in G. \end{displaymath} We denote by $\mathcal{M}(S)$ the family of closed non-empty $S$-monotone sets in $\real{d}$. We refer to \citet{Fitzpatrick:88}, \citet{Simons:07}, and~\citet{Penot:09} for the results on Fitzpatrick functions $\psi_G$ associated with $G \in \mathcal{M}(S)$.
\begin{Example}[Standard form]
\label{ex:1}
If $d=2m$ and
\begin{displaymath}
S(x,y) = \sum_{i=1}^m \cbraces{x^i y^{m+i} + x^{m+i}y^i}, \quad
x,y\in \real{2m},
\end{displaymath}
then $S \in \msym{2m}m$ and the $S$-monotonicity means the standard
monotonicity in $\real{2m}=\real{m}\times \real{m}$. For a
\emph{maximal} monotone set $G$, the function $\psi_G$ becomes the
classical Fitzpatrick function from~\cite{Fitzpatrick:88}. \end{Example}
\begin{Example}[Canonical form]
\label{ex:2}
If $\Lambda$ is the \emph{canonical} quadratic form in $\msym{d}m$:
\begin{displaymath}
\Lambda(x,y) = \sum_{i=1}^m x^i y^i - \sum_{i=m+1}^d x^i y^i, \quad
x,y\in \real{d},
\end{displaymath}
then a closed set $G$ is $\Lambda$-monotone if and only if
\begin{displaymath}
G = \graph{f} := \descr{(u,f(u))}{u\in D},
\end{displaymath}
where $D$ is a closed set in $\real{m}$ and $\map{f}{D}{\real{d-m}}$
is a $1$-Lipschitz function:
\begin{displaymath}
\abs{f(u)-f(v)} \leq \abs{u-v}, \quad u,v \in D.
\end{displaymath}
In view of the Kirszbraun Theorem, \cite[2.10.43]{Feder:69}, $G$ is
maximal $\Lambda$-monotone if and only if $D=\real{m}$. \end{Example}
While working on the $S$-space, it is convenient to use an appropriate version of subdifferential for a convex function $f$ : \begin{align*}
\partial^Sf(x) := & \descr{y\in \real d}{S(z,y)\leq f(x+z)-f(x), \,
z\in \real{d}} \\
= & \descr{S^{-1}y\in \real d}{\ip{z}{y} \leq f(x+z)-f(x), \,
z\in \real{d}} \\
= & \; S^{-1} \partial f(x). \end{align*}
\begin{Lemma}
\label{lem:6}
Let $S\in \msym{d}{m}$, $G\in \mathcal{M}(S)$, and assume that
$D:= \interior\dom{\psi_G} \not=\emptyset$. Then
$\range{S^{-1}\nabla \psi_G} \subset G$,
$\dom{P_G} = \dom{\partial \psi_G} = \dom{\partial^S \psi_G}$, and
\begin{gather*}
\psi_G(x) = \psi_G^*(Sx) = \frac12 S(x,x), \quad x\in G, \\
P_G(x) = \partial^S \psi_G(x) \cap G, \quad x\in \real{d}.
\end{gather*} \end{Lemma} \begin{proof}
We write $\psi_G$ as
\begin{displaymath}
\psi_G(x) = g^*(x) := \sup_{y\in \real{d}}(\ip{x}{y} - g(y)), \quad
x\in \real{d},
\end{displaymath}
where $g(y)=\frac{1}{2}S^{-1}(y,y)$ for $y\in SG$ and
$g(y) = +\infty$ for $y\not\in SG$. From the definition of
$\psi _G$ and the $S$-monotonicity of $G$ we deduce that
\begin{displaymath}
\psi _G(x)=\frac12 S(x,x) - \frac12 \inf_{y\in
G} S(x-y,x-y) = \frac12 S(x,x), \quad x\in G,
\end{displaymath}
and then that
\begin{displaymath}
\psi _G(S^{-1}x) \leq g(x), \quad x\in
\real{d}.
\end{displaymath}
As $\psi^*_G=g^{**}$ and $g^{**}$ is the largest closed convex
function less than $g$, we have that
\begin{displaymath}
\psi _G(S^{-1}x) \leq \psi ^*_G(x)\leq g(x), \quad x\in \real{d}.
\end{displaymath}
Putting together the relations above, we obtain that
\begin{displaymath}
\frac 12 S(x,x)=\psi _G(x)\leq \psi ^*_G(Sx)\leq
g(Sx)=\frac{1}{2}S(x,x), \quad x\in G.
\end{displaymath}
For every $x\in \real{d}$, the values of the Fitzpatrick function
$\psi_G$ and of the projection multi-function $P_G$ can now be
written as
\begin{align*}
\psi_G(x) & = \sup_{y\in G} \cbraces{S(x,y) - \psi_G^*(Sy)} =
\sup_{z\in SG} \cbraces{\ip{x}{z} - \psi_G^*(z)}, \\
P_G(x) & = \argmax_{y\in G}\cbraces{S(x,y)-\frac 12S(y,y)} = S^{-1}
\argmax_{z\in SG}\cbraces{\ip{x}{z}-\psi^*_G(z)}.
\end{align*}
The stated relations between $P_G$ and $\partial^S \psi_G$ follow
from Lemma~\ref{lem:5} as soon as we observe that
$P_G(x)= \partial^S \psi_G(x)=\emptyset$ for $x\notin \cl D$.
Finally, Lemma~\ref{lem:4} yields the inclusion of
$\range \nabla \psi_G$ into $SG$. \end{proof}
To facilitate geometric interpretations, we also adapt the concept of a normal vector to the product structure of the $S$-space. Let $H$ be a closed set in $\real{d}$. A vector $w\in \real{d}$ is called $S$-\emph{regular normal to $H$ at $x\in H$} if \begin{displaymath}
\limsup_{\substack{H\ni y\to x \\ y\not= x}}
\frac{S(w,y-x)}{\abs{y-x}} =
\limsup_{\substack{H\ni y\to x \\y\not=
x}} \frac{\ip{Sw}{y-x}}{\abs{y-x}} \leq 0. \end{displaymath} A vector $w\in \real{d}$ is called $S$-\emph{normal to $H$ at $x$} if there exist $x_n\in H$ and an $S$-regular normal vector $w_n$ to $H$ at $x_n$, such that $x_n\rightarrow x$ and $w_n\rightarrow w$. In other words, $w$ is $S$-(regular) normal to $H$ at $x\in H$ if $Sw$ is (regular) normal to $H$ at $x$ in the classical Euclidean sense. It is easy to see that if $x\in P_H(z)$, then the vector $z-x$ is $S$-regular normal to $H$ at $x$.
\begin{Lemma}
\label{lem:7}
Let $S\in \msym{d}{m}$, $j\in \braces{1,\dots,d}$,
$C\in \mathcal{C}^j$, $h$ be given by~\eqref{eq:1} for convex
functions $g_1$ and $g_2$ with linear growth, and $H$ be the
zero-level set of the composition function $h\circ S$:
\begin{displaymath}
H := \descr{x\in \real{d}}{h(Sx) = 0}.
\end{displaymath}
Then $h\in \mathcal{H}^j_C$ if and only if for every $x\in H$, there
exists an $S$-normal vector $w\in C$ to $H$ at $x$. \end{Lemma}
\begin{proof}
Let $z\in H$ and $w\in \real{d}$. Setting
\begin{displaymath}
SH := \descr{Sx}{x\in H} = \descr{x\in \real{d}}{h(x) = 0},
\end{displaymath}
recalling that $S(w,y-z) = \ip{w}{Sy-Sz}$, and using the trivial
inequalities:
\begin{displaymath}
\frac1{\norm{S^{-1}}} \abs{y-z} \leq
\abs{Sy-Sz} \leq \norm{S} \abs{y-z}, \quad y\in \real{d},
\end{displaymath}
where $\norm{A} := \max_{\abs{x}=1} \abs{Ax}$ for a $d\times d$
matrix $A$, we deduce that $w$ is normal to $SH$ at $Sz$ if and only
if $w$ is $S$-normal to $H$ at $z$. Lemma~\ref{lem:1} yields the
result. \end{proof}
A set $A\subset \real{d}$ is called $S$-\emph{isotropic} if \begin{displaymath}
S(x-y,x-y) = 0, \quad x,y \in A. \end{displaymath} We denote by $\mathcal{I}(S)$ the family of all closed $S$-isotropic subsets of $\real{d}$.
Lemma~\ref{lem:6} and Theorem~\ref{th:2} show that the interior and boundary singularities of $\cpartial \psi_G$ are contained in the corresponding singular set of $P_G$. We decompose the latter set as \begin{align*}
\Sigma(P_G) &:= \descr{x\in \dom{P_G}}{P_G(x) \text{~is not a
point}} = \Sigma_0(P_G) \cup
\Sigma_1(P_G), \\
\Sigma_0(P_G) & := \descr{x\in \Sigma(P_G)}{P_G(x) \in
\mathcal{I}(S)}, \\
\Sigma_1(P_G) & := \descr{x\in \Sigma(P_G)}{S(y_1-y_2,y_1-y_2)>0
\text{~for \emph{some}~} y_1,y_2 \in P_G(x)}. \end{align*} We further write $\Sigma_1(P_G)$ as \begin{displaymath}
\Sigma_1(P_G) = \bigcup_{j=1}^d \Sigma_1^j(P_G), \end{displaymath} where $\Sigma_1^j(P_G)$ consists of $x\in \Sigma(P_G)$ such that $S(y_1-y_2,y_1-y_2)>0$ for some $y_1,y_2 \in P_G(x)$ with $y_1^j \not = y_2^j$.
Theorem~\ref{th:3} and Lemma~\ref{lem:7} show that $\Sigma^j_1(P_G)$ can be covered by countable $c-c$ surfaces that have at each point a strictly $S$-positive $S$-normal vector $w$ with $w^j=1$.
\begin{Theorem}
\label{th:3}
Let $S\in \msym{d}{m}$, $G\in \mathcal{M}(S)$,
$j\in \braces{1,\dots,d}$, and assume that
$D:= \interior{\dom{\psi_G}} \not=\emptyset$. For every $n\geq 1$,
there exist a compact set $C_n \subset G-G$ with
\begin{displaymath}
x^j>0 \text{~and~} S(x,x) > 0, \quad x\in C_n,
\end{displaymath}
and a function $h_n \in \mathcal{H}^{j}_{\theta^j(C_n)}$ such that
\begin{displaymath}
\Sigma^j_1(P_G) \subset\bigcup_n \descr{x\in
\real{d}}{h_n(Sx)=0}.
\end{displaymath}
For any $\epsilon>0$, all $C_n$ can be chosen such that
$\diam{\theta^j(C_n)}< \epsilon$. \end{Theorem}
\begin{proof}
Fix $\epsilon >0$. Let $(x_n)$ be a dense sequence in $G$ and
$(r_n)$ be an enumeration of all positive rationals. Denote by
$\alpha:= (m, n,k,l)$ the indexes for which the compact sets
\begin{displaymath}
C^{\alpha}_1 :=\descr{x\in G }{ |x-x_m|\leq r_k}, \quad
C^{\alpha}_2 :=\descr{x\in G}{|x-x_n|\leq r_l},
\end{displaymath}
satisfy the constraints:
\begin{displaymath}
\diam \theta^j(C^{\alpha}_2-C^{\alpha}_1) < \epsilon
\text{~and~} x^j > 0, \; S(x,x) > 0, \; x \in C_2^{\alpha} -
C_1^{\alpha}.
\end{displaymath}
We have that
\begin{displaymath}
\Sigma^j_1(P_G) =
\bigcup_{\alpha}\Sigma_{C^{\alpha}_1, C^{\alpha}_2}(P_G)=
\bigcup_{\alpha} \descr{x\in \real{d}}{P_G(x)\cap
C^\alpha_i\not=\emptyset, \; i=1,2}.
\end{displaymath}
Let $g(x) := \psi_G(S^{-1}x)$, $x\in \real{d}$. From
Lemma~\ref{lem:6} we deduce that
\begin{displaymath}
P_G(x) = \cbraces{S^{-1} \partial \psi_G(x)} \cap G = \partial
g(Sx) \cap G, \quad x\in \real{d}.
\end{displaymath}
It follows that
\begin{displaymath}
x\in \Sigma_{C^{\alpha}_1, C^{\alpha}_2}(P_G) \iff
Sx \in \Sigma_{C^{\alpha}_1, C^{\alpha}_2}(\partial g).
\end{displaymath}
Applying Theorem~\ref{th:1} to each singular set
$\Sigma_{C^{\alpha}_1, C^{\alpha}_2}(\partial g)$, we obtain the
result. \end{proof}
The singular set $\Sigma_0(P_G)$ is included into a larger set \begin{displaymath}
\overline{\Sigma}_0(P_G) = \bigcup_{j=1}^d
\overline{\Sigma}^j_0(P_G), \end{displaymath} where $\overline{\Sigma}^j_0(P_G)$ consists of $x\in \dom{P_G}$ such that $S(y_1-y_2,y_1-y_2)=0$ for \emph{some} $y_1,y_2 \in P_G(x)$ with $y_1^j \not = y_2^j$.
Theorem~\ref{th:4} and Lemma~\ref{lem:7} show that $\overline{\Sigma}_0^j(P_G)$ can be covered, for any $\delta >0$, by countable $c-c$ surfaces that have at each point an $S$-normal vector $w$ such that $w^j=1$ and $0\leq S(w,w)\leq \delta$.
\begin{Theorem}
\label{th:4}
Let $S\in \msym{d}{m}$, $G\in \mathcal{M}(S)$,
$j\in \braces{1,\dots,d}$, and assume that
$D:= \interior{\dom{\psi_G}} \not=\emptyset$. Let $\delta>0$. For
every $n\geq 1$, there exist a compact set $C_{n} \subset G-G$ with
\begin{displaymath}
x^j>0, \; x\in C_n, \text{~and~} 0\leq S(x,x) \leq
\delta, \; x\in \theta^j(C_{n}),
\end{displaymath}
and a function $h_{n} \in \mathcal{H}^{j}_{\theta^j(C_{n})}$ such
that
\begin{displaymath}
\overline{\Sigma}^j_0(P_G) \subset \bigcup_n \descr{x\in
\real{d}}{h_{n}(Sx)=0}.
\end{displaymath}
For any $\epsilon>0$, all $C_{n}$ can be chosen such that
$\diam{\theta^j(C_n)}< \epsilon$. \end{Theorem}
\begin{proof}
The proof is almost identical to that of Theorem~\ref{th:3}. We fix
$\delta, \epsilon >0$ and find a \emph{countable} family
$(C^\alpha_1,C^\alpha_2)$ of pairs of compact sets
$C^\alpha_i \subset G$, $i=1,2$, such that
\begin{gather*}
x^j > 0, \; x \in C_2^{\alpha} - C_1^{\alpha}, \text{~and~}
\diam \theta^j(C^{\alpha}_2-C^{\alpha}_1) < \epsilon,\\
0\leq S(x,x) < \delta, \; x \in \theta ^j(C_2^{\alpha} -
C_1^{\alpha}),
\end{gather*}
and every pair $(y_1,y_2)$ of elements of $G$ with
$S(y_1-y_2,y_1-y_2)=0$ and $y_1^j \not = y_2^j$ is contained in some
$(C^\alpha_1,C^\alpha_2)$. Then
\begin{displaymath}
\overline{\Sigma}^j_0(P_G) \subset
\bigcup_{\alpha}\Sigma_{C^{\alpha}_1, C^{\alpha}_2}(P_G)=
\bigcup_{\alpha} S^{-1} \Sigma_{C^{\alpha}_1, C^{\alpha}_2}(\partial g),
\end{displaymath}
where $g(x) := \psi_G(S^{-1}x)$, $x\in \real{d}$, and
Theorem~\ref{th:1} applied to
$\Sigma_{C^{\alpha}_1, C^{\alpha}_2}(\partial g)$ yields the result. \end{proof}
The geometric description of the zero order singularities becomes especially simple if the index $m=1$. In this case, $\overline{\Sigma}^j_0(P_G)$ can be covered by countable number of hyperplanes whose $S$-normal vectors are $S$-isotropic.
\begin{Theorem}
\label{th:5}
Let $S\in \msym{d}{1}$, $G\in \mathcal{M}(S)$,
$j\in \braces{1,\dots,d}$, and assume that
$D:= \interior{\dom{\psi_G}} \not=\emptyset$. For every $n\geq 1$,
there exist $y_n,z_n \in G$ such that
\begin{equation}
\label{eq:14}
y^j_n-z^j_n >0, \quad S(y_n-z_n,y_n-z_n) = 0,
\end{equation}
and
\begin{displaymath}
\overline{\Sigma}^j_0(P_G) \subset\bigcup_n
\descr{x\in \real{d}}{S(x-z_n,y_n-z_n)= 0}.
\end{displaymath} \end{Theorem}
\begin{proof}
By the law of inertia for quadratic forms, \cite[Theorem~1,
p.~297]{Gant-1:98}, there exists a $d\times d$ matrix $V$ of full
rank such that
\begin{displaymath}
S = V^T \Lambda V,
\end{displaymath}
where $V^T$ is the transpose of $V$ and $\Lambda$ is the diagonal
matrix with diagonal entries $\braces{1,-1,\dots,-1}$. In other
words, $\Lambda$ is the canonical quadratic form for
$\mathcal{S}^d_1$ from Example~\ref{ex:2}. As
$S(x,x) = \Lambda(Vx,Vx)$, we deduce that
$F:= VG \in \mathcal{M}(\Lambda)$ and
\begin{displaymath}
A\in \mathcal{I}(S) \iff VA \in
\mathcal{I}(\Lambda).
\end{displaymath}
As we pointed out in Example~\ref{ex:2},
\begin{displaymath}
F = \graph{f} = \descr{(t,f(t))}{t\in D},
\end{displaymath}
for a $1$-Lipschitz function $\map{f}{D}{\real{d-1}}$ defined a
closed set $D\subset \real{}$.
Let $x=(s,f(s))$ and $y=(t,f(t))$ be distinct elements of $F$, where
$s,t\in D$. We have that $\braces{x,y} \in \mathcal{I}(\Lambda)$ if
and only if $\abs{f(t)-f(s)} = \abs{t-s}$. The $1$-Lipschitz
property of $f$ then implies that
\begin{displaymath}
f(r) = f(s) + \frac{r-s}{t-s} \cbraces{f(t)-f(s)}, \quad r\in D
\cap [s,t].
\end{displaymath}
It follows that the collection of all $\Lambda$-isotropic subsets of
$F$ can be decomposed into an intersection of $F$ with at most
countable union of line segments whose relative interiors do not
intersect.
The same property then also holds for the $S$-isotropic subsets of
$G$. Thus, there exist $y_n,z_n \in G$, $n\geq 1$,
satisfying~\eqref{eq:14} and such that every $S$-isotropic subset of
$G$ having elements with distinct $j$th coordinates is a subset of
some $S$-isotropic line
$L_n := \descr{y_n + t(z_n-y_n)}{t\in \real{}}$. In particular, if
$x\in \overline{\Sigma}_0^j(P_G)$, then $P_G(x)$ intersects some
line $L_n$ at distinct $y$ and $z$. We have that
\begin{displaymath}
2S(x-z,y-z) = S(x-z,x-z) + S(y-z,y-z) - S(x-y,x-y) = 0.
\end{displaymath}
As $y,z\in L_n\in \mathcal{I}(S)$, we obtain that
$S(x-z_n,y_n-z_n) = 0$, as required. \end{proof}
\begin{Example}
\label{ex:3}
Let $d=2$ and $S$ be the standard bilinear form from
Example~\ref{ex:1}:
\begin{displaymath}
S(x,y)=S((x_1,x_2),(y_1,y_2)) = x_1y_2 + x_2y_1.
\end{displaymath}
Let $G\in \mathcal{M}(S)$. As $S(x,x) = 2x_1x_2$, we have that
\begin{displaymath}
\Sigma_1(P_G) = \Sigma^j_1(P_G), \quad j=1,2.
\end{displaymath}
Theorem \ref{th:3} yields convex functions $g_{1,n}$ and $g_{2,n}$
on $\real{}$ and constants $\epsilon_n>0$, $n\geq 1$, such that
($g'=g'(t)$ is the derivative of $g=g(t)$)
\begin{gather*}
\epsilon_n \leq g'_{1,n}(t) - g'_{2,n}(t) \leq \epsilon_n^{-1},
\quad t\in
\dom{g'_{1,n}} \cap \dom{g'_{2,n}}, \\
\Sigma _1 (P_G)\subset \bigcup _n \descr{x\in \real{2}}{x_2=
g_{2,n}(x_1) - g_{1,n}(x_1)}.
\end{gather*}
Theorem \ref{th:5} yields sequences $(x^n_1)$ and $(x^n_2)$ in
$\real{}$ such that
\begin{displaymath}
\overline{\Sigma}^1_0(P_G)\subset
\bigcup _n \descr{x\in \real{2}}{x_2=x^n_2}, \quad
\overline{\Sigma}^2_0(P_G)\subset
\bigcup _n \descr{x\in \real{2}}{x_1=x^n_1}.
\end{displaymath}
These results improve \cite[Theorem~B.12]{KramXu:22}, where $G$ is a
maximal monotone set and $g_n := g_{1,n} - g_{2,n}$ is a strictly
increasing Lipschitz function such that
$\epsilon_n \leq g'_n(t) \leq \epsilon^{-1}_n$, whenever it is
differentiable. \end{Example}
\section*{Acknowledgments}
We thank Giovanni Leoni for pointing out the references~\cite{Zahorski:46} and \cite{FowlPreiss:09}.
\end{document}
|
arXiv
|
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{{\bf Analytic Representation of Finite Quantum Systems}}
\author{S. Zhang and A. Vourdas\\ Department of Computing,\\ University of Bradford, \\ Bradford BD7 1DP, United Kingdom} \maketitle
\begin{abstract} A transform between functions in $\mathbb{R}$ and functions in $\mathbb{Z}_d$ is used to define the analogue of number and coherent states in the context of finite $d$-dimensional quantum systems. The coherent states are used to define an analytic representation in terms of theta functions. All states are represented by entire functions with growth of order $2$, which have exactly $d$ zeros in each cell. The analytic function of a state is constructed from its zeros. Results about the completeness of finite sets of coherent states within a cell, are derived.
\end{abstract}
\section{Introduction}
Quantum systems with finite Hilbert space have been studied originally by Weyl and Schwinger \cite{Weyl} and later by many authors \cite{Auslander,Vourdas1}. A formalism analogous to the harmonic oscillator can be developed where the dual variables that we call `position' and `momentum' take values in $\mathbb{Z}_d$ (the integers modulo $d$). This area of research is interesting in its own right; and has many applications in areas like quantum optics; quantum computing\cite{A000}; two-dimensional electron systems in magnetic fields and the magnetic translation group \cite{A100}; the quantum Hall effect \cite{A200}; quantum maps \cite{A300}; hydrodynamics \cite{A400} mathematical physics; signal processing; etc. For a review see \cite{Vourdas2}.
In this paper we introduce a transform from functions in $\mathbb{R}$ to functions in $\mathbb{Z}_d$. This is related to Zak transform from functions in $\mathbb{R}$ to functions on a circle \cite{Zak1,Zak2,Zak3}; but of course here we have functions in `discretized circle'. This transform enables us to transfer some of the harmonic oscillator formalism into the context of finite systems. For example, we define the analogues of number states and coherent states. Coherent states in the context of finite systems have been previously considered in \cite{L1,coherent}.
Coherent states can be used to define analytic representations. For example, ordinary coherent states of the harmonic oscillator can be used to define the Bargmann analytic representation in the complex plane; $SU(1,1)$ coherent states can be used to define analytic representations in the unit disc (Lobachevsky geometry); $SU(2)$ coherent states can be used to define analytic representations in the extended complex plane (spherical geometry). We use the coherent states in the context of finite systems to define analytic representations. Similar analytic representations have been used in the context of quantum maps in \cite{L1}. We show that the corresponding analytic functions obey doubly-periodic boundary conditions; and therefore it is sufficient to define them on a square cell $S$. Each of these analytic functions has growth of order $2$ and has exactly $d$ zeros in $S$.
We use the analytic formalism to study the completeness of finite sets of coherent states in the cell $S$. This discussion is the analogue in the present context, of the `theory of von Neumann lattice' for the harmonic oscillator \cite{V1,V2,V3,V4} , which is based on the theory of the density of zeros of analytic functions \cite{B}.
In section II, we review briefly the basic theory of finite systems and define some quantities for later use. In section III, we introduce the transform between functions in $\mathbb{R}$ and functions in $\mathbb{Z}_d$. Using this transform we define in section IV number states and coherent states in our context of finite systems, and study their properties. In section V, we use the coherent states to define an analytic representation in terms of Theta functions. We show that the order of the growth of these entire functions is $2$. We also study displacements and the Heisenberg-Weyl group in this language. In section VI we study the zeros of the corresponding analytic functions and use them to study the completeness of finite sets of coherent states within a cell. In section VII we construct the analytic representation of a state from its zeros. We conclude in section VIII with the discussion of our results.
\section{Finite quantum systems} \subsection{Position and momentum states and Fourier transform}
We consider a quantum system with a $d$-dimensional Hilbert space ${\cal H}$.
We use the notation $|s \rangle \rangle$ for the states in ${\cal H}$; and we use the notation $|s \rangle $ for the states in the infinite dimensional Hilbert space $H$ associated with
the harmonic oscillator. Let $|X;m \rangle \rangle$ be an orthonormal basis in ${\cal H}$,
where $m$ belongs to $\mathbb{Z}_d$. We refer to them as `position states'. The $X$ in the notation is not a variable but it simply indicates position states.
The finite Fourier transform is defined as: \begin{eqnarray} \label{DFT1}
F=d^{-1/2} \sum_{m,n} \omega(mn)|X;m\rangle\rangle \langle\langle X;n|;\;\;\;\;\;
\omega(\alpha)=\exp\left(i \frac{2\pi \alpha}{d}\right) \end{eqnarray} \begin{eqnarray} \label{DFT2}
F F^{\dagger}=F^{\dagger} F={\bf 1};\;\;\;\;\;
F^4={\bf 1} \end{eqnarray} Using the Fourier transform we define another orthonormal basis, the `momentum states', as: \begin{equation}
|P;m\rangle\rangle=F |X;m\rangle\rangle=d^{-1/2} \sum_{n=0}^{d-1} \omega(mn)|X;n\rangle\rangle \end{equation} We also define the `position and momentum operators' $x$ and $p$ as \begin{eqnarray}
x=\sum _{n=0}^{d-1}n|X;n\rangle\rangle \langle\langle X;n|;\;\;\;\;\;\;
p=\sum _{n=0}^{d-1}n|P;n\rangle\rangle \langle\langle P;n| \end{eqnarray} It is easily seen that \begin{eqnarray} F x F^{\dagger}=p;\;\;\;\;\;\;\;\;\;F p F^{\dagger}=-x \end{eqnarray}
\subsection{Displacements and the Heisenberg-Weyl group} The displacement operators are defined as: \begin{equation} Z=\exp \left (i \frac{2\pi}{d}x \right );\;\;\;\;\; X=\exp \left (-i \frac{2\pi}{d}p \right ) \end{equation} \begin{equation} X^{d}=Z^{d}={\bf 1};\;\;\;\;\;\; X^\beta Z^\alpha=Z^\alpha X^\beta \omega(-\alpha \beta) \end{equation} where $\alpha$,$\beta$ are integers in $\mathbb{Z}_d$. They perform displacements along the $P$ and $X$ axes in the $X-P$ phase-space. Indeed we can show that: \begin{equation}\label{movez}
Z^\alpha |P;m \rangle\rangle=|P; m+\alpha \rangle\rangle ;\;\;\;\;\;\; Z^\alpha|X; m\rangle\rangle= \omega(\alpha m)|X; m\rangle\rangle \end{equation} \begin{equation}\label{movex}
X^\beta |P; m \rangle\rangle= \omega(-m\beta)|P; m \rangle\rangle ;\;\;\;\;\; X^\beta|X; m\rangle\rangle=|X; m+\beta\rangle\rangle \end{equation} The $X-P$ phase-space is the toroidal lattice $\mathbb{Z}_d \times \mathbb{Z}_d$.
The general displacement operators are defined as: \begin{equation} \label{displc} D(\alpha, \beta)=Z^\alpha X^\beta \omega(-2^{-1}\alpha \beta);\;\;\;\;[D(\alpha, \beta)]^{\dagger}=D(-\alpha, -\beta) \end{equation} It is easy to see \begin{eqnarray} \label{dispx}
D(\alpha, \beta) |X;m \rangle\rangle &=& \omega (2^{-1}\alpha\beta + \alpha m) |X;m+\beta \rangle\rangle \nonumber\\
D(\alpha, \beta) |P;m \rangle\rangle &=& \omega (-2^{-1}\alpha\beta - \beta m) |P;m+\alpha \rangle\rangle \end{eqnarray}
We next consider an arbitrary (normalized) state $|s \rangle\rangle $ \begin{eqnarray}
|s\rangle\rangle = \sum_{m=0}^{d-1} s_m |X;m \rangle\rangle; \;\;\;\; \sum_{m=0}^{d-1} |s_m|^2 =1 \end{eqnarray} and act with the displacement operators to get the states: \begin{eqnarray}
|s;\alpha, \beta \rangle\rangle \equiv D(\alpha, \beta) |s\rangle\rangle
= \sum_{m=0}^{d-1} s_m \omega(2^{-1}\alpha\beta + \alpha m) |X;m \rangle\rangle \end{eqnarray}
Clearly $|s;0,0\rangle\rangle= |s\rangle\rangle$. Using Eq(\ref{movez}) and Eq(\ref{movex}) we easily show that \begin{eqnarray}\label{genres}
d^{-1} \sum_{\alpha,\beta=0}^{d-1} |s;\alpha, \beta \rangle\rangle \langle \langle s;\alpha, \beta | = \textbf{1}_d \end{eqnarray}
This shows that the states $|s;\alpha, \beta \rangle\rangle $ (for a fixed `fiducial' state $|s\rangle\rangle $ and all $\alpha $, $\beta$ in $\mathbb{Z}_d$ ) form an overcomplete basis of $d^2$ vectors in the d-dimensional Hilbert space ${\cal H}$. Eq(\ref{genres}) is the resolution of the identity.
\subsection{General transformations}
In this section we expand an arbitrary operator $\Omega$, in terms of displacement operators. In order to do this we first define its Weyl function as \begin{equation}
\widetilde W_{\Omega}(\alpha,\beta) = \textrm{Tr} [\Omega D(\alpha,\beta)] \end{equation} The properties of the Weyl function and its relation to the Wigner function is discussed in \cite{Vourdas2}. We can prove that \begin{equation} \label{WelyDis}
\Omega = d^{-1} \sum_{\alpha,\beta=0}^{d-1} \widetilde W_{\Omega}(-\alpha,-\beta) D(\alpha,\beta) \end{equation}
\section{A transform between functions in $\mathbb{R}$ and functions in $\mathbb{Z}_d$}
In this section we introduce a map between states in the infinite dimensional harmonic oscillator Hilbert space $H$ and the $d$-dimensional Hilbert space ${\cal H}$. This map is a special case of the Zak transform.
We consider a state $|\psi\rangle$
in $H$ with (normalized) wavefunction in the x-representation $\psi (x)=\langle x|\psi\rangle$. The corresponding state $|\psi\rangle \rangle$ in ${\cal H}$ is defined through the map \begin{eqnarray}
\psi_m = \langle\langle X;m| \psi \rangle\rangle = {\cal N}^{-1/2} \sum_{w=-\infty}^{\infty}
\psi \left[ x=\left( \frac{2\pi}{d} \right)^{1/2} \lambda (m+dw) \right];\;\;\;\;\;\;
\psi_{m+d}=\psi_m
\label{mapx} \end{eqnarray}
where $m\in \mathbb{Z}_d$. ${\cal N}$ is a normalization factor so that $\sum_{m=0}^{d-1} |\psi_m|^2 = 1$ which is given in appendix A. The Fourier transform (on the real line) of $\psi (x)$, is defined as: \begin{eqnarray}
\tilde\psi (p) &=& (2\pi)^{-1/2} \int_{-\infty}^{\infty} \psi (x) \exp(-ipx) dx. \label{Fho} \end{eqnarray} Using the map of Eq(\ref{mapx}) we define \begin{eqnarray}
\tilde\psi_m = {{\cal N}^\prime}^{-1/2} \sum_{w=-\infty}^{\infty}
\tilde\psi \left[ p=\left( \frac{2\pi}{d} \right)^{1/2} \frac{1}{\lambda} (m+dw) \right]. \label{mapp} \end{eqnarray} The tilde in $ \tilde\psi$ indicates that the Fourier transform of $\psi (x)$ has been transformed according to Eq.(\ref{Fho}). The normalization factor ${\cal N}^\prime$ is given in appendix A where it is shown that ${\cal N}^\prime=\lambda^2 {\cal N}$.
We next prove that \begin{eqnarray} \label{eigDFT}
\tilde\psi_m= d^{-1/2} \sum_{n=0}^{d-1} \omega(-mn) \psi_n = \langle\langle P;m| \psi \rangle\rangle \end{eqnarray} This shows that $\tilde \psi_m $ is the finite Fourier transform of $\psi _n$, and therefore the tilde also indicates the above finite Fourier transform. So the tilde in the notation is used for two different Fourier transforms, but they are consistent to each other.
In order to prove Eq.(\ref{eigDFT}) we insert Eq.(\ref{mapx}) into Eq.(\ref{eigDFT}) and use the Poisson formula \begin{eqnarray} \label{comb}
\sum_{w=-\infty}^{\infty} \exp (i2\pi wx) =
\sum_{k=-\infty}^{\infty} \delta (x-k), \end{eqnarray} where the right hand side is the `comb delta function'; and also the relation \begin{eqnarray} \label{modelta}
\frac{1}{d} \sum_{m=0}^{d-1} \omega[m(k-\ell)] = \delta(k,\ell);\;\;\;\;k,\ell\in \mathbb{Z}_d \end{eqnarray} where $\delta(k,k^\prime)$ is a Kronecker delta. These two relations are useful in many proofs in this paper.
The above map is not one-to-one (the Hilbert space $H$ is infinite dimensional while the Hilbert space ${\cal H}$ is $d$-dimensional). Therefore, Eq.(\ref{mapx}) cannot be inverted. In appendix B, we use the full Zak transform and introduce a family of $d$-dimensional Hilbert spaces ${\cal H}(\sigma_1,\sigma_2)$ with twisted boundary conditions. We show that the Hilbert space $H$ is isomorphic to the direct integral of all the ${\cal H}(\sigma_1,\sigma_2)$ (with $0\le\sigma_1<1$, $0\le\sigma_2<1$) and then an inverse to the relation (\ref{mapx}) can be found. However the formalism of this paper is valid only for ${\cal H}\equiv {\cal H}(0,0)$ (which has periodic boundary conditions).
\section{Quantum states}
\subsection{Number eigenstates}
In the harmonic oscillator, number states are eigenstates of the Fourier operator $\exp (ia^\dagger a)$ where $a, a^\dagger $ are the usual annihilation and creation operators. In this section we apply the transformation of Eq. (\ref{mapx}) with $\lambda =1$ and we show that the resulting states are eigenstates of the Fourier operator of Eq.(\ref{DFT1}).
We consider the harmonic oscillator number eigenstates $|N\rangle $ whose wavefunction is \begin{eqnarray}
\chi (x,N)= \langle x|N \rangle &=& \left(\frac{1}{\pi^{1/2} 2^N N!}\right)^{1/2} \exp \left( -\frac{1}{2}x^2 \right) H_N(x), \end{eqnarray} It is known that \begin{eqnarray}\label{tr}
\tilde\chi (x, N) &=& i^N \chi (x,N). \end{eqnarray} Using the transforms of Eq(\ref{mapx}) and Eq(\ref{mapp}) with $\lambda =1$, we find: \begin{eqnarray} \label{number}
\chi_{m} (N)= \langle\langle X;m|N \rangle\rangle
&=& {{\cal N}_n(N)}^{-1/2} \sum_{w=-\infty}^{\infty} \chi \left[ x=(m+dw)\left( \frac{2\pi}{d}\right)^{1/2} ,N\right],
\label{eigDFx} \\
\tilde\chi_{m}(N)= \langle\langle P;m|N \rangle\rangle
&=& {{\cal N}_n(N)}^{-1/2}\sum_{w=-\infty}^{\infty} \tilde\chi \left[ x=(m+dw)\left( \frac{2\pi}{d}\right)^{1/2} ,N \right], \label{eigDFp} \end{eqnarray} where ${\cal N}_n(N)$ is the normalization factor for number eigenstates, given by Eq(\ref{norm}) with $\psi$ replaced by $\chi$. Eq.(\ref{tr}) implies that \begin{eqnarray}
\tilde\chi_{m} (N)&=& i^N \chi_{m} (N)\label{eigrlt}. \end{eqnarray} Using this in conjunction with Eq(\ref{eigDFT}) we prove that \begin{eqnarray} \label{FN}
d^{-1/2} \sum_{n=0}^{d-1} \omega(-mn) \chi _n(N) &=& i^N\chi_m(N);\;\;\rightarrow \;\; F|N\rangle\rangle = i^N |N\rangle\rangle. \end{eqnarray} Therefore the vectors $\chi_m(N)$ are eigenvectors of the Fourier matrix. They have been studied in the context of Signal Processing in \cite{eigen}
Of course, the Fourier matrix is finite and only $d$ of these eigenvectors are linearly independent. Therefore the set of all states $|N\rangle\rangle$ is highly overcomplete. In general the number states $|N\rangle\rangle$ are not orthogonal to each other.
The Fourier matrix has four eigenvalues $i^k$ ($0 \le k \le 3$); and all the states $|N= 4M+k \rangle\rangle$ correspond to the same eigenvalue $i^k$. As an example, we consider the case $d=6$ and using Eq.(\ref{eigDFx}) we calculate the six eigenvectors . Results are presented in table I (we note that $|5\rangle\rangle =-|1\rangle\rangle$).
\begin{center}
\begin{tabular}{cccccc}
\multicolumn{6}{c}{\bf Table I. Eigenvectors of the Fourier Operator F with $d=6$} \\ \hline \hline
\hspace{4mm} $|0\rangle\rangle$ \hspace{7mm} & $|1\rangle\rangle$ \hspace{7mm}
& $|2\rangle\rangle$ \hspace{7mm} & $|3\rangle\rangle$ \hspace{7mm}
& $|4\rangle\rangle$ \hspace{7mm} & $|6\rangle\rangle$ \\ \hline
\hspace{4mm} 0.75971 \hspace{7mm} & 0 \hspace{7mm}
& -0.52546 \hspace{7mm} & 0 \hspace{7mm} & 0.37040 \hspace{7mm} & -0.31449 \\
\hspace{4mm} 0.45004 \hspace{7mm} & 0.65328 \hspace{7mm}
& 0.34071 \hspace{7mm} & -0.27059 \hspace{7mm} & -0.37823 \hspace{7mm} & 0.28578 \\
\hspace{4mm} 0.09373 \hspace{7mm} & 0.27060 \hspace{7mm}
& 0.48131 \hspace{7mm} & 0.65328 \hspace{7mm} & 0.37471 \hspace{7mm} & -0.15803 \\
\hspace{4mm} 0.01365 \hspace{7mm} & 0 \hspace{7mm}
& 0.16851 \hspace{7mm} & 0 \hspace{7mm} & 0.54393 \hspace{7mm} & 0.82934 \\
\hspace{4mm} 0.09373 \hspace{7mm} & -0.27060 \hspace{7mm}
& 0.48131 \hspace{7mm} & -0.65328 \hspace{7mm} & 0.37471 \hspace{7mm} & -0.15803 \\
\hspace{4mm} 0.45004 \hspace{7mm} & -0.65328\hspace{7mm}
& 0.34071 \hspace{7mm} & 0.27059 \hspace{7mm} & -0.37823 \hspace{7mm} & 0.28578 \\ \hline
\end{tabular}\\ \end{center}
\subsection{Coherent states}
We consider the harmonic oscillator coherent states $|A \rangle$ whose wavefunction is \begin{eqnarray}
\psi(x,A)= \langle x|A \rangle &=& \pi^{-1/4} \exp \left( -\frac{1}{2} x^2 + Ax - \frac{1}{2} A_R A \right), \label{csx} \end{eqnarray}
where $A=A_R +iA_I$. Using the transformation of Eq(\ref{mapx}) we introduce coherent states $|A \rangle\rangle $ in the finite Hilbert space as: \begin{eqnarray} \label{fcs}
\psi_m(A)&=& \langle\langle X;m|A \rangle\rangle = {{\cal N}_C(A)}^{-1/2} \pi^{-1/4}
\exp \left[ -\frac{\pi \lambda^2 m^2}{d} + A m \lambda \left( \frac{2\pi}{d} \right)^{1/2} - \frac{1}{2}A_R A \right ] \nonumber \\
&& \times \Theta_3 \left [ i\pi m \lambda^2 -i A \lambda \left( \frac{d \pi}{2} \right)^{1/2} ; id \lambda^2 \right ] \nonumber \\
&=& {{\cal N}_C(A)}^{-1/2} \pi^{-1/4} d^{-1/2} \lambda^{-1}
\exp \left( \frac{i}{2}A_IA \right )
\cdot \Theta_3 \left [ \frac{\pi m}{d}-\frac{A}{\lambda} \left( \frac{\pi}{2d} \right)^{1/2}; \frac{i}{d \lambda^2} \right ]. \end{eqnarray} where $\Theta _3$ are theta functions \cite{theta}, defined as \begin{eqnarray}
\Theta_3 (u;\tau) &=& \sum_{n=-\infty}^{\infty} \exp(i\pi\tau n^2+i2nu). \end{eqnarray} and the relation: \begin{eqnarray}
\Theta_3 (u;\tau) &=& (-i\tau)^{-1/2} \exp\left(\frac{u^2}{\pi i \tau}\right)
\cdot \Theta_3 \left( \frac{u}{\tau};-\frac{1}{\tau} \right ), \end{eqnarray} has been used in Eq.(\ref{fcs}). The normalization factor is: \begin{eqnarray}
{\cal N}_C(A) &=& \pi^{-1/2} \lambda^{-2}
\cdot \left\{ \Theta_3\left[ \frac{A_R}{\lambda} (2\pi d)^{1/2}; \frac{2id}{\lambda^2} \right]
\Theta_3\left[ A_I i \lambda^{-1} \left( \frac{2 \pi}{d}\right)^{1/2};\frac{2i}{d\lambda^2} \right]
\right. \nonumber\\
&& + \left. \Theta_2\left[ \frac{A_R}{\lambda} (2\pi d)^{1/2}; \frac{2id}{\lambda^2} \right]
\Theta_2\left[ A_I i \lambda^{-1} \left( \frac{2 \pi}{d}\right)^{1/2};\frac{2i}{d \lambda^2} \right]
\right\} \end{eqnarray}
The $\psi_m(A)$ obeys the relations: \begin{eqnarray} \label{quasi}
\psi_m \left[ A+(2\pi d)^{1/2} \lambda \right] &=& \psi_m(A) \exp \left[ i A_I \lambda \left( \frac{\pi d}{2}\right)^{1/2} \right];\nonumber \\
\psi_m \left[ A+i \frac{(2\pi d)^{1/2}}{\lambda}\right] &=& \psi_m(A) \exp \left[ -i \frac{A_R}{\lambda} \left( \frac{\pi d}{2}\right)^{1/2} \right]. \end{eqnarray} The zeros of the Theta function $\Theta_3(u;\tau)$ are given by: \begin{eqnarray} \label{thetazero}
u = (2k+1)\frac{\pi}{2}+(2l+1)\frac{\pi\tau}{2}, \end{eqnarray} where $k$,$l$ are integers. Therefore \begin{equation}
\psi_m(A_{kl})= \langle\langle X;m|A _{kl}(m)\rangle\rangle =0;\;\;\;\
A_{kl} (m)= \left( \frac{2\pi}{d} \right)^{1/2} \left[ \left( kd+\frac{d}{2}+m \right) \lambda +\frac{(2l+1)i}{2\lambda} \right]. \end{equation}
It is seen that the states $|A_{kl}(m)\rangle\rangle$ are orthogonal to the position states $|X;m\rangle\rangle$.
The `vacuum state' $|0 \rangle\rangle$ is defined as \begin{equation} \label{fcs7}
\langle\langle X;m|0 \rangle\rangle = {{\cal N}_C(0)}^{-1/2} \pi^{-1/4} d^{-1/2}
\cdot \Theta_3 \left( \frac{\pi m}{d}; \frac{i}{d \lambda^2 } \right), \end{equation} where \begin{eqnarray}
{\cal N}_C(0) &=& \pi^{-1/2} \lambda^{-2}
\cdot \left\{ \Theta_3\left[ 0; \frac{2id}{\lambda^2} \right]
\Theta_3\left[ 0;\frac{2i}{d\lambda^2} \right]
+ \Theta_2\left[ 0; \frac{2id}{\lambda^2} \right]
\Theta_2\left[ 0;\frac{2i}{d \lambda^2} \right]
\right\} \end{eqnarray}
The coherent states $|A \rangle\rangle $ satisfy the following resolution of the identity \begin{eqnarray} \label{res}
\lambda (2\pi d)^{-1/2}\int _S d^2A
{\cal N}_C(A) |A \rangle\rangle \langle\langle A | = \textbf{1}_d ;\;\;
\;\;\;S=\left[ a, a+(2\pi d)^{1/2}\lambda \right)_R \times \left[ b, b+\frac{(2\pi d)^{1/2}}{\lambda} \right)_I \end{eqnarray} We integrate here over the cell $S$. The periodicity of Eq.(\ref{quasi}) implies that the cell can be shifted everywhere in the complex plane and this is indicated with the arbitrary real numbers $a$, $b$. The proof of Eq(\ref{res}) is based on the resolution of the identity for ordinary (harmonic oscillator) coherent states, in conjunction with the map of Eq(\ref{mapx}).
The set of all coherent states in the cell $S$ is highly overcomplete. Indeed using Eq.(\ref{genres}) we easily show another resolution of identity which involves only $d^2$ coherent states in the cell $S$: \begin{eqnarray} \label{res2}
d^{-1}\sum_{\alpha, \beta =0}^{d-1} \left. \left\vert A+\left( \frac{2\pi}{d} \right)^{1/2}(\beta \lambda + \frac{\alpha}{\lambda} i) \right\rangle \right\rangle
\left \langle \left \langle A+\left( \frac{2\pi}{d} \right)^{1/2}(\beta\lambda + \frac{\alpha}{\lambda} i) \right\vert \right . = \textbf{1}_d \end{eqnarray}
We calculate overlap of two coherent states odd, \begin{eqnarray}
\langle\langle A_1|A_2 \rangle\rangle &=& \pi^{-1/2} \lambda^{-2} {\cal N}_C(A_1)^{-1/2} {\cal N}_C(A_2)^{-1/2}
\exp\left( -\frac{i}{2}A_{1I}A_1^\ast + \frac{i}{2}A_{2I}A_2 \right) \nonumber \\
&& \times \left \{ \Theta_3\left[ \frac{A_1^\ast+A_2}{\lambda} \left( \frac{\pi d}{2}\right)^{1/2}; \frac{2id}{\lambda^2} \right]
\Theta_3\left[ \frac{A_1^\ast-A_2}{\lambda} \left( \frac{\pi}{2d}\right)^{1/2}; \frac{2i}{d\lambda^2} \right] \right. \nonumber \\
&& + \left. \Theta_2\left[ \frac{A_1^\ast+A_2}{\lambda} \left( \frac{\pi d}{2}\right)^{1/2}; \frac{2id}{\lambda^2} \right]
\Theta_2\left[ \frac{A_1^\ast-A_2}{\lambda} \left( \frac{\pi}{2d}\right)^{1/2}; \frac{2i}{d\lambda^2} \right] \right\}. \end{eqnarray} and particularly find that when $d$ is an even number \begin{eqnarray}
\langle\langle A_1|A_2 \rangle\rangle &=& \pi^{-1/2} \lambda^{-2} {\cal N}_C(A_1)^{-1/2} {\cal N}_C(A_2)^{-1/2}
\exp\left( -\frac{i}{2}A_{1I}A_1^\ast + \frac{i}{2}A_{2I}A_2 \right) \nonumber \\
&& \times \Theta_3\left[ \frac{A_1^\ast+A_2}{\lambda} \left( \frac{\pi d}{8}\right)^{1/2}; \frac{id}{2\lambda^2} \right]
\Theta_3\left[ \frac{A_1^\ast-A_2}{\lambda} \left( \frac{\pi}{2d}\right)^{1/2}; \frac{2i}{d\lambda^2} \right]. \end{eqnarray} In the case, the first theta function in the above relation is zero when \begin{eqnarray} A_2-A_1^\ast=\left ( l+ \frac{1}{2} \right) (2\pi d)^{1/2} \lambda + i(2k+1)\left( \frac{2\pi}{d} \right)^{1/2} \lambda^{-1} \end{eqnarray} and the second is zero when \begin{eqnarray} A_2+A_1^\ast= \left(\frac{2 \pi}{d} \right)^{1/2} \lambda ( 2k+1 )
+ \left(\frac{\pi d}{2} \right)^{1/2} \lambda^{-1} (2l+1)i \end{eqnarray} The corresponding coherent states in these two cases are orthogonal to each other.
There is a relation between the coherent states in a finite Hilbert space studied in this section and the number states studied earlier: \begin{equation}\label{nu}
|A\rangle\rangle = \exp \left(-\frac{|A|^2}{2} \right) \sum_{N=0}^{\infty} \frac{A^N}{\sqrt{N!}}
\left[ \frac{{\cal N}_n(N)}{{\cal N}_C(A)} \right]^{1/2} |N\rangle\rangle. \end{equation} This is analogous to the relation between coherent states and number states in the infinite dimensional Hilbert space for the harmonic oscillator. We have explained earlier that only $d$ of the number states appearing in the right hand side of Eq. (\ref{nu}) are independent.
Introducing the displacement operator defined in Eq(\ref{displc}), we can prove that \begin{eqnarray}
D(\alpha, \beta)|A\rangle \rangle = \left. \left\vert A+\left( \frac{2\pi}{d} \right)^{1/2}(\beta \lambda + \alpha \lambda^{-1} i) \right\rangle \right\rangle
\cdot \exp \left[ -iA_I \lambda \left( \frac{\pi}{2d}\right)^{1/2} \beta + iA_R \lambda^{-1} \left( \frac{\pi}{2d}\right)^{1/2} \alpha \right], \end{eqnarray} where both $\alpha$ and $\beta$ are integers. We might be tempted to use the above equation as a definition for displacement operators with real values of $\alpha$, $\beta$. It can be shown that in this case $D$ depends on $A$; and only for integer $\alpha$, $\beta$ the $D$ is independent of $A$.
\section{Analytic Representation}
\subsection{Quantum states}
Let $|f\rangle\rangle$ be an arbitrary(normalized) state \begin{eqnarray}\label{st}
|f\rangle\rangle = \sum_{m=0}^{d-1} f_m |X;m\rangle\rangle \;\;\;\;\; \sum_{m=0}^{d-1} |f_m|^2 =1. \end{eqnarray} We shall use the notation \begin{eqnarray}
|f^\ast \rangle\rangle = \sum_{m=0}^{d-1} f_m^\ast |X;m\rangle\rangle \nonumber\\
\langle\langle f|=\sum_{m=0}^{d-1} f_m^\ast \langle \langle X;m| \nonumber \\
\langle\langle f^*| = \sum_{m=0}^{d-1} f_m \langle \langle X;m| \end{eqnarray}
We define the analytic representation of $|f\rangle\rangle$, as: \begin{eqnarray}\label{ana}
f(z) &\equiv & {{\cal N}_C(z)}^{1/2} d^{1/2} \lambda \exp \left(-\frac{i}{2} z_Iz \right) \langle\langle z^\ast |f \rangle\rangle \nonumber\\
&=& \pi^{-1/4} \sum_{m=0}^{d-1} \Theta_3 \left [ \frac{\pi m}{d}-\frac{z}{\lambda}\left( \frac{\pi}{2d}\right)^{1/2}; \frac{i}{d\lambda^{2}} \right ] f_m \label{analy} \end{eqnarray}
where $|z\rangle\rangle$ is a coherent state. It is easy to see \begin{eqnarray} \label{periodicity}
f\left[ z+ (2\pi d)^{1/2} \lambda \right] = f(z); \;\;\;
\;\; f\left[ z+ i (2\pi d)^{1/2} \lambda^{-1} \right] = f(z)\exp \left[ \frac{\pi d}{\lambda^2} - i (2\pi d)^{1/2}z\lambda^{-1} \right]. \end{eqnarray} The $f(z)$ is an entire function.
If $M(R)$ is the maximum modulus of $f(z)$ for $|z|=R$,then \begin{equation} \rho =\lim _{R\to \infty }\sup \frac {\ln \ln M(R)}{\ln R} \end{equation} is the order of the growth of $f(z)$ \cite{B}. It is easily seen that in our case the order of the growth is $\rho =2$.
Due to the periodicity our discussion below is limited to a single cell $S$ (defined in Eq.(\ref{res}) ). The scalar product is given by \begin{eqnarray}
\langle\langle f^\ast| g \rangle\rangle = (2\pi)^{-1/2} d^{-3/2} \lambda^{-1} \int_S d^2z \exp \left( - z_I^2 \right) f(z) g(z^\ast). \end{eqnarray}
As special cases, we derive the analytic representation of the position states: \begin{eqnarray}
|X;m\rangle\rangle \;\; &\rightarrow \;\; & \pi^{-1/4}
\cdot \Theta_3 \left [ \frac{\pi m}{d}-z \lambda^{-1} \left( \frac{\pi}{2d}\right)^{1/2}; \frac{i}{d \lambda^{2}} \right ] \end{eqnarray} momentum states: \begin{eqnarray}
|P;m\rangle\rangle \;\; &\rightarrow \;\;& \lambda \pi^{-1/4} \exp \left(-\frac{1}{2}z^2\right)
\cdot \Theta_3 \left [ \frac{\pi m}{d}- \lambda zi\left( \frac{\pi}{2d}\right)^{1/2}; \frac{i\lambda^2}{d} \right ] \end{eqnarray} and the coherent states: \begin{eqnarray}
|A\rangle\rangle \;\; &\rightarrow& \nonumber\\
f(z,A) &=& \pi^{-1/2} \lambda^{-1} d^{1/2} {\cal N}_C(A)^{-1/2} \exp\left( \frac{i}{2} A_I A\right) \nonumber\\
&& \times \left \{ \Theta_3\left[ \frac{z+A}{\lambda} \left( \frac{\pi d}{2}\right)^{1/2}; \frac{2id}{\lambda^2} \right]
\Theta_3\left[ \frac{z-A}{\lambda} \left( \frac{\pi}{2d}\right)^{1/2}; \frac{2i}{d\lambda^2} \right] \right. \nonumber \\
&& + \left. \Theta_2\left[ \frac{z+A}{\lambda} \left( \frac{\pi d}{2}\right)^{1/2}; \frac{2id}{\lambda^2} \right]
\Theta_2\left[ \frac{z-A}{\lambda} \left(
\frac{\pi}{2d}\right)^{1/2}; \frac{2i}{d\lambda^2} \right]
\right\}. \end{eqnarray} Again, when $d$ is even, it can be simplified as \begin{eqnarray}
f(z,A) &=& \pi^{-1/2} \lambda^{-1} d^{1/2} {\cal N}_C(A)^{-1/2} \exp\left( \frac{i}{2} A_I A\right) \nonumber\\
&& \times \Theta_3\left[ \frac{z+A}{\lambda} \left( \frac{\pi d}{8}\right)^{1/2}; \frac{id}{2\lambda^2} \right]
\Theta_3\left[ \frac{z-A}{\lambda} \left( \frac{\pi}{2d}\right)^{1/2}; \frac{2i}{d\lambda^2} \right]; \end{eqnarray}
\subsection{Displacements and the Heisenberg-Weyl group} In this section we express the displacement operators $X$ and $Z$ in the context of analytic representations. Eqs.(\ref{movex}),(\ref{movez}) are written as \begin{eqnarray}
Xf(z)=f \left[ z-\left( \frac{2\pi}{d} \right)^{1/2} \lambda \right]; \;\;\;\;\;\;
Zf(z)=f\left[ z+i\left( \frac{2\pi}{d} \right)^{1/2} \lambda^{-1} \right ] \exp\left[iz \lambda^{-1}\left( \frac{2\pi}{d} \right)^{1/2}-\frac{\pi}{d\lambda^{2}}\right]. \end{eqnarray} Therefore $X$ and $Z$ are given by: \begin{eqnarray}
X&=&\exp \left[ -\left( \frac{2\pi}{d} \right)^{1/2} \lambda \partial_z \right] \nonumber\\
Z&=&\exp \left[ iz \lambda^{-1}\left( \frac{2\pi}{d} \right)^{1/2}-\frac{\pi}{d\lambda^{2}}\right]
\exp \left[ i\left( \frac{2\pi}{d} \right)^{1/2} \lambda^{-1} \partial_z \right] \end{eqnarray} and the general displacement operator is: \begin{eqnarray}
D(\alpha , \beta ) = \omega (-2^{-1/2}\alpha \beta ) \exp\left[i\alpha z \lambda^{-1}\left( \frac{2\pi}{d} \right)^{1/2}-\frac{\alpha^2 \pi}{d\lambda^{2}}\right]
\exp \left[ (i\alpha\lambda^{-1}-\beta\lambda ) \left( \frac{2\pi}{d} \right)^{1/2}\partial_z \right] \end{eqnarray}
where $\alpha, \beta$ are integers in $\mathbb{Z}_d$. Acting with this operator on the state $|f\rangle\rangle$ of Eq.(\ref{st}) represented by the analytic function $f(z)$ of eq(\ref{ana}), we get \begin{eqnarray}
D(\alpha , \beta ) f(z) &=& \pi^{-1/4} \exp \left[ i\alpha z \lambda^{-1} \left( \frac{2\pi}{d} \right)^{1/2}
- \frac{\alpha^2 \pi}{d\lambda^2} - \frac{2^{1/2} \alpha \beta \pi i}{d} \right] \nonumber \\
&& \times \sum_{m=0}^{d-1} f_m \Theta_3 \left[ \frac{\pi m}{d} - \frac{z}{\lambda} \left( \frac{\pi}{2d} \right)^{1/2}
- \frac{\alpha \pi i}{d\lambda} + \frac{\beta \pi}{d}; \frac{i}{d\lambda^{2}} \right]. \end{eqnarray}
\subsection{General transformations}
We have seen in Eq.(\ref{WelyDis}) that an arbitrary operator $\Omega$ can be expanded in terms of displacement operators and using this we can express $\Omega$ as: \begin{eqnarray}
\Omega &=& d^{-1} \sum_{\alpha, \beta=0}^{d-1} \omega (-2^{-1/2}\alpha \beta ) \widetilde W_{\Omega}(-\alpha, -\beta)
\exp\left[i\alpha z \lambda^{-1} \left( \frac{2\pi}{d} \right)^{1/2}-\frac{\alpha^2 \pi}{d\lambda^2}\right] \nonumber \\
&& \exp \left[ (i\alpha \lambda^{-1}-\beta \lambda ) \left( \frac{2\pi}{d} \right)^{1/2}\partial_z \right] \end{eqnarray}
Alternatively the operator $\Omega = \sum_{m,n} \Omega_{mn}|X;m\rangle\rangle \langle\langle X;n|$ can be represented with the kernel \begin{eqnarray}
\Omega(z,\zeta^*) &\equiv& {{\cal N}_C(\zeta)}^{1/2}{{\cal N}_C(z)}^{1/2} \lambda^2
\exp \left( -\frac{i}{2}z_I z + \frac{i}{2}\zeta_I \zeta^* \right)
\langle\langle z^* |\Omega| \zeta^* \rangle\rangle \\
&=& \pi^{-1/2}d^{-1} \sum_{m,n=0}^{d-1} \Omega_{mn}
\Theta_3\left[ \frac{\pi m}{d}-\frac{z}{\lambda}\left( \frac{\pi}{2d} \right)^{1/2}; \frac{i}{d\lambda^2} \right]
\Theta_3\left[ \frac{\pi n}{d}-\frac{\zeta^*}{\lambda}\left( \frac{\pi}{2d} \right)^{1/2}; \frac{i}{d\lambda^2} \right] \end{eqnarray} and we easily prove that \begin{eqnarray}
\Omega | f\rangle\rangle\rightarrow (2\pi d)^{-1/2} \lambda^{-1} \int_S d^2 \zeta \exp \left( -{\zeta_I}^* \right)
\Omega (z,\zeta^*) f(\zeta) \end{eqnarray} It is easily seen that \begin{eqnarray}
\Omega[z+(2\pi d)^{-1/2}\lambda \alpha,\zeta^*+(2\pi d)^{-1/2}\lambda \beta]&=& \Omega[z,\zeta^*] \nonumber\\
\Omega[z+i(2\pi d)^{-1/2}\lambda^{-1} \alpha,\zeta^*] &=& \Omega[z,\zeta^*]
\exp \left[ \frac{\pi d}{\lambda^2}\alpha^2 -i(2\pi d)^{1/2}z\alpha \lambda^{-1} \right] \nonumber\\
\Omega[z,\zeta^*+i(2\pi d)^{-1/2}\lambda^{-1} \beta] &=& \Omega[z,\zeta^*]
\exp \left[ \frac{\pi d}{\lambda^2}\beta^2 -i(2\pi d)^{1/2}\zeta^*\beta \lambda^{-1} \right] \end{eqnarray} where $\alpha$ and $\beta$ are integers.
\section{Zeros of the analytic representation and their physical meaning}
If $z_0$ is a zero of the analytic representation $f(z)$, then Eq.(\ref{analy})
shows that the coherent state $|z_0\rangle\rangle$ is orthogonal to the state $|f\rangle\rangle$.
Using the periodicity of Eq(\ref{periodicity}) we easily prove that \begin{equation}\label{1}
\frac{1}{2\pi i} \oint_\Gamma \frac{f^\prime(z)}{f(z)} \; \textrm{d}z = d , \end{equation} where $\Gamma$ is the boundary of the cell $S$. The above integral is in general equal to the number of zeros minus the number of poles of the function $f(z)$ inside $\Gamma$. Since our functions have no poles, we conclude that the analytic representation of any state has exactly $d$ zeros in the square $S$ (zeros will be counted with their multiplicities). The area of $S$ is $2\pi d$, and therefore there is an average of one zero per $2\pi$ area of the complex plane, in this analytic representation.
As an example we show in Fig.1 the zeros of the coherent states $|0 \rangle \rangle$ and
$|1+i \rangle \rangle$ for the case $d=4$.
\begin{figure}
\caption{The zeros within a cell of the coherent states $|0 \rangle \rangle$ (circles) and
$|1+i \rangle \rangle$ (triangles) for the case $d=4$.}
\label{Fig1}
\end{figure}
A direct consequence of this result is the fact that any set of $d+1$ coherent states in the cell $S$ is at least complete. Indeed if it is not complete, then there exists some state which is orthogonal to all these coherent states. But such a state would have $d+1$ zeros, which is not possible. A set of $d+1$ states in a $d$-dimensional space which is at least complete is in fact overcomplete; in the sense that there exist a state which we can take out and be left with a complete set of $d$ states. We note that if we take out an arbitrary state we might be left with an undercomplete set of $d$ states.
A set of $d-1$ coherent states is clearly undercomplete, because our Hilbert space is $d$-dimensional.
A set of $d$ distinct coherent states $\{|z_i\rangle\rangle; i=1,...,d \}$ will be complete or undercomplete depending on whether it violates or satisfies the constraint \begin{equation}\label{sumz}
\sum _{i=1}^dz_i= \left( \frac{\pi}{2}\right)^{1/2} d^{3/2} (\lambda+i\lambda^{-1})
+(2\pi d)^{1/2}(M\lambda+iN\lambda^{-1}). \end{equation} where $M,N$ are integers. In order to prove this we use the periodicity of Eq(\ref{periodicity}) to prove that \begin{equation}\label{sumzero}
\frac{1}{2\pi i} \oint_\Gamma \frac{z f^\prime(z)}{f(z)} \; \textrm{d}z = \left( \frac{\pi}{2}\right)^{1/2} d^{3/2} (\lambda+i\lambda^{-1})
+(2\pi d)^{1/2}(M\lambda+iN \lambda^{-1}). \end{equation} The above integral is in general equal to the sum of zeros minus the sum of poles (with the multiplicities taken into account) of the function $f(z)$ inside $\Gamma$. Since our functions have no poles, we conclude that the sum of zeros is equal to the right hand side of Eq(\ref{sumzero}). Eqs(\ref{1}),(\ref{sumzero}) have also been given in \cite{L1}.
If the $d$ coherent states considered violate Eq.(\ref{sumz}), then
clearly they form a complete set because there exists no state which is orthogonal to all of them. If however they do satisfy the constraint (\ref{sumz}), then there exists a state which is orthogonal to all of them. To construct such a state we simply take the first $d-1$ coherent states $\{|z_1\rangle\rangle,...,|z_{d-1}\rangle\rangle\}$
(which form an undercomplete set because the space is $d$ -dimensional) and find a state $|g\rangle\rangle$ which is orthogonal to them. The corresponding analytic function $g(z)$ will have $d$ zeros which will be the
$z_1,...,z_{d-1}$ and an extra one which has to obey the constraint (\ref{sumz}) and therefore has to be $z_d$. Therefore $|g\rangle\rangle$ will be orthogonal to $|z_d\rangle\rangle$ also, and consequently the set of $\{|z_1\rangle\rangle,...,|z_d\rangle\rangle\}$ is undercomplete.
\section{Construction of the analytic representation of a state from its zeros}
We have proved in the last section that for an arbitrary state $|f\rangle\rangle$, the analytic representation $f(z)$ has $d$ zeros in the cell $S$ of Eq.(\ref{res}). In this section we assume that the zeros $z_1,z_2,...z_d$ in the cell $S$, are given (subject to the constraint of Eq(\ref{sumz})) and we will construct the function $f(z)$. We note that some of the zeros might be equal to each other.
We first consider the product \begin{equation}
Q(z) = \prod_{j=1}^d \Theta_3 \left[ (z-z_j+w) \left( \frac{\pi}{2d} \right)^{1/2} \lambda^{-1}; \frac{i}{\lambda^2} \right]
;\;\;\;\;\;w= \left( \frac{\pi d}{2} \right)^{1/2} (\lambda+\lambda^{-1}i) \end{equation} It is easily seen that $Q(z)$ has the given zeros. The ratio $f(z)/Q(z)$ is entire function with no zeros and therefore it is the exponential of an entire function: \begin{equation} f(z)=Q(z)\exp (P(z)) \end{equation} Taking into account the periodicity constraints of Eq.(\ref{periodicity}) we conclude that \begin{eqnarray} \label{periodicity1}
P\left[ z+ (2\pi d)^{1/2} \lambda \right]& = &P(z)+i2\pi K\nonumber\\ P\left[ z+ i (2\pi d)^{1/2} \lambda^{-1} \right] & = &P(z)+\frac {2\pi N}{\lambda ^2}+i2\pi \Lambda. \end{eqnarray} Here $N$ is the integer entering the constraint of Eq.(\ref{sumz}); and $K$,$\Lambda$ are arbitrary integers. We have explained earlier that the growth of $f(z)$ is of order $2$. The order of $Q(z)$ is $2$; therefore the $P(z)$ is a polynomial of maximum possible degree $2$. Eq.(\ref{periodicity1}) shows that in fact $P(z)$ is \begin{eqnarray}
P(z) = -\left( \frac{2\pi}{d} \right)^{1/2} N \lambda^{-1}z i +C. \end{eqnarray} where $C$ is a constant. Therefore \begin{eqnarray}\label{500}
f(z) = C' \cdot \exp \left[ -\left( \frac{2\pi}{d} \right)^{1/2} N \lambda^{-1}z i \right]Q(z). \end{eqnarray} where the constant $C'$ is determined by the normalization condition.
\section{Discussion}
The harmonic oscillator formalism with phase space $\mathbb{R} \times \mathbb{R}$ has been studied extensively in the literature. Equally interesting is quantum mechanics on a circle, with phase space $\mathbb{S} \times \mathbb{Z}$\cite{cir1,cir2}; and finite quantum systems, with phase space $ \mathbb{Z}_d \times \mathbb{Z}_d $. Most of the results for physical systems on a circle or circular lattice (which is the case here), are intimately related to Theta functions; and well known mathematical results for Theta functions can be used to derive interesting physical results for these systems.
In this paper we have introduced the transform of Eq.(\ref{mapx})
between functions in $\mathbb{R}$ and functions in $\mathbb{Z}_d$. The aim is to create a harmonic oscillator-like formalism in the context of finite systems.
We have defined the analogue of number states for finite quantum systems in Eq.(\ref{number}); and of coherent states in Eq.(\ref{fcs}). The properties of these states have been discussed.
Using the coherent states we have defined the analytic representation of Eq.(\ref{ana}) in terms of Theta functions. In this language we have studied displacements and the Heisenberg-Weyl group; and also more general transformations. Symplectic transformations are also important for these systems. Especially in the case where $d$ is the power of a prime number, there are strong results (e.g., \cite{Vourdas2}). Further work is needed in order to express these results in the language of analytic representations used in this paper.
The analytic functions (\ref{ana}) have growth of order $2$ and they have exactly $d$ zeros in each cell $S$. If the zeros are given we can construct the analytic representation of the state using Eq.(\ref{500}). Therefore we can describe the time evolution of a system through the paths of the $d$ zeros of its analytic representation, in the cell $S$.
Based on the theory of zeros of analytic functions we have shown that a set of $d+1$ coherent states in the cell $S$ is overcomplete; and a set of $d-1$ coherent states is undercomplete. A set of $d$ coherent states in the cell $S$, is complete if the constraint of Eq.(\ref{sumz}) is violated; and undercomplete if the constraint of Eq.(\ref{sumz}) is obeyed. These results are analogous to the ` theory of von Neumann lattice' in our context of finite quantum systems.
Our results use the powerful techniques associated to analytic representations in the context of finite systems.
\section{Appendix A} The normalization factor appearing in Eq.(\ref{mapx}) is given by \begin{eqnarray}
{\cal N} = \sum_{m=0}^{d-1} \left \{ \sum_{w=-\infty}^{\infty} \psi^\ast \left[ x=\left( \frac{2\pi}{d} \right)^{1/2}\lambda(m+dw) \right]
\right \} \left \{
\sum_{w^\prime=-\infty}^{\infty} \psi \left[ x=\left( \frac{2\pi}{d} \right)^{1/2}\lambda(m+dw^\prime) \right]
\right \}\label{norm} \end{eqnarray} The normalization factor appearing in Eq.(\ref{mapp}) is given by \begin{eqnarray}
{\cal N}^\prime = \sum_{m=0}^{d-1}
\left \{ \sum_{w=-\infty}^{\infty} \tilde\psi^\ast \left[ p=\left( \frac{2\pi}{d} \right)^{1/2}\lambda^{-1}(m+dw) \right] \right \}
\left \{ \sum_{w^\prime=-\infty}^{\infty} \tilde\psi \left[ p=\left( \frac{2\pi}{d} \right)^{1/2}\lambda^{-1}(m+dw^\prime) \right] \right \}
\label{normp} \end{eqnarray} We insert Eq(\ref{Fho}) into Eq(\ref{normp}), and use Eqs(\ref{comb}), (\ref{modelta}) to prove that ${\cal N}^\prime=\lambda^{2}{\cal N}$.
\section{Appendix B}
In this appendix we use the full Zak transform to introduce a family of $d$-dimensional Hilbert space ${\cal H}(\sigma_1,\sigma_2)$ (with ${\cal H}\equiv {\cal H}(0,0)$). We generalize Eq.(\ref{mapx}) into \begin{eqnarray}\label{genmap}
\psi_m(\sigma_1,\sigma_2) =
[{\cal N}(\sigma_1,\sigma_2)]^{-1/2}
\sum_{w=-\infty}^{\infty} \exp(-2\pi i \sigma_1 w) \psi \left[ \left( \frac{2\pi}{d} \right )^{1/2} \lambda
(m+\sigma_2 + dw) \right], \end{eqnarray} where ${\cal N}(\sigma_1,\sigma_2)$ is a normalization factor. The Hilbert space ${\cal H}(\sigma_1,\sigma_2)$ is spanned by the states corresponding to $\psi_m(\sigma_1,\sigma_2)$. These spaces and the corresponding twisted boundary conditions of the wavefunctions, have been studied in \cite{L2}. The Hilbert space $H$ is isomorphic to the direct integral of all the ${\cal H}(\sigma_1,\sigma_2)$ (with $0\le\sigma_1<1$, $0\le\sigma_2<1$). In this case Eq.(\ref{mapx}) can be inverted as follows: \begin{equation}
\psi \left[ x= \left( \frac{2\pi}{d}\right)^{1/2} \lambda (m+\sigma_2 + dw) \right] = \int_{0}^{1} [{\cal N}(\sigma_1,\sigma_2)]^{1/2}
\psi_m(\sigma_1,\sigma_2) \exp(2\pi i \sigma_1 w) \textrm{d}\sigma_1. \end{equation} The formalism of this paper is valid for the space ${\cal H}\equiv {\cal H}(0,0)$.
\end{document}
|
arXiv
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