\section{Introduction}

Our manipulations rely heavily on plethystic notation and the terminology used in \cite{GHRY}. In \cite{HRW15}, the reader can find detailed explanations for all of the notations used in this paper.

Recall that Dyck paths in the $n\times n$ lattice square $R_n$ are   paths from $(0,0)$ to $(n,n)$ proceeding by north and east  unit steps, always remaining  weakly above the main diagonal of $R_n$.  These paths are usually represented by their area sequence
$(a_1,a_2,\ldots, a_n)$, where $a_i$ counts the number of complete cells
 between the north step in the $i^{th}$ row and the diagonal. Notice that the $x$-coordinate of the north step in the $i^{th}$ row is simply the difference $u_i=i-1-a_i$.

 A parking function $PF$ supported by the Dyck path $D\in R_n$ is obtained by labeling the north steps of $D$ with $1,2,\ldots ,n$ (usually referred as ``cars''), where the labels increase along the north segments of $D$. Parking functions can be represented as two line arrays
$$
PF=\begin{pmatrix}
c_1 & c_2  &  \cdots  &   c_n\\
a_1 & a_2  &  \cdots  &   a_n\\
\end{pmatrix}
$$
 with cars $c_i$ and area numbers $a_i$ listed from bottom to top.
We also set
$$
area(PF)= \sum_{i=1}^n a_i,
\enskip\ess\enskip
dinv(PF)= \hskip -.08in \sum_{1\le i<j\le n}\hskip -.08in \Big(
\chi(c_i<c_j \enskip\&\enskip a_i=a_j)\, + \,
\chi(c_i>c_j \enskip\&\enskip a_i=a_j+1)
\Big).
$$
Moreover, the word $w(PF)$ is the permutation obtained by reading the cars
in the two line array by decreasing area numbers and from right to left.

 This given, the Haglund factor of a Dyck path $D$ is obtained by setting
$$
H_D(z;t)\enskip = \enskip \prod_{i=2}^{n}\big(1+{z\over t^{a_i}}\big)^{\chi(u_{i-1}= u_i)}.
$$
The $LLT$ polynomial constructed
from the Dyck path $D$ is  obtained by setting
$$
LLT_D(X;q,t)\enskip = \enskip \sum_{D(PF)=D}t^{area(PF)}q^{dinv(PF)}s_{p\big(ides(w(PF)\big)}[X]
$$
where the sum is over parking functions supported by $D$ and the last factor is the Schur function indexed by the composition giving the descent set of the inverse of $w(PF)$.

The special version of the Delta Conjecture of \cite{HRW15} we refer to here is the equality
\begin{equation}\label{eq:theDC}
\Delta'_{e_{k-1}} e_n\enskip = \enskip
\sum_{D\in R_n}
LLT_D(X;q,t)\enskip
H_D(z;t)\, \Big|_{z^{n-k}}
\end{equation}
where $\Delta'_{F}$ is the eigen-operator of the modified Macdonald polynomial
defined by setting for any symmetric function $F$
$$
\Delta'_{F}{\widetilde H}_\mu[X;q,t]\enskip = \enskip
F\big[B_\mu(q,t)-1\big]\, {\widetilde H}_\mu[X;q,t]
\enskip\ess\enskip
(\hbox{for all $\mu$}).
$$
As mentioned previously, it was proved in \cite{GHRY} that
the equality in (\ref{eq:theDC}) is valid when both sides are evaluated at $q=0$. Since the left hand side is easily shown to be symmetric in $q$ and $t$, then it must also remain valid when both sides are evaluated at $t=0$.

The main result in \cite{GHRY} is the equality of the symmetric functions on the right hand sides of
the following two equations
\begin{equation}\label{eq:24fromGHRY}
\sum_{\lambda\vdash n}LHS_{k,\lambda}s_{\lambda'}[x(1-q)]=
\sum_{\mu\vdash n}q^{- n(\mu)}
P_\mu[X;1/q]
\Big[{ \ell(\mu)-1\atop k-1 }\Big]_q
(q;q)_{\ell(\mu)},
\end{equation}
and
\begin{equation}\label{eq:25fromGHRY}
\sum_{\lambda\vdash n}RHS_{k,\lambda}s_{\lambda'}[X(1-q)]\enskip = \enskip
q^{-k(k-1)}
(q;q)_k\sum_{\substack{\mu\vdash n\\ \ell(\mu)=k}}
q^{n(\mu)}
 P_\mu[X;q],
\end{equation}
(these are labeled   (24) and (25) in that paper).
Where
\[
LHS_{k,\lambda}=q^{-\binom{k}{2}}\langle\Delta_{e_{k-1}} ' e_n, s_\lambda\rangle\Big|_{\substack{t=0}}
,
\enskip\ess\enskip
RHS_{k,\lambda}= \sum_{D\in R_n}
\big\langle LLT_D(X;q,t) ,s_\lambda\big\rangle
H_D(z;t) \Big|_{z^{n-k}}\Big|_{t=0},
\]
and $ P_\mu[X;q]$, $ Q_\mu[X;q]$ are the Hall-Littlewood polynomials with Cauchy Kernel
\[
\sum_{\mu\vdash n}P_\mu[X;q]\, Q_\mu[Y;q]
\,=\, h_n\big[XY(1-q)\big]
\]

The present work was started by Jeff Remmel who sadly passed away before its completion. Remmel proposed the possibility of extending the Delta Conjecture when
the symmetric function side
``$\Delta'_{e_k}e_n$'' is replaced by
``$\Delta'_{s_\nu}e_n$'' , with $\nu$ an arbitrary partition. Remmel asked
the first author to obtain computer data to see if there was any similarity to the data that was obtained in the classical case.  One of the most surprising features of the classical case
is the discovery that the polynomial in (\ref{eq:25fromGHRY}) contains only hook Schur functions in its Schur expansion.
It is precisely this experimental discovery that made the proof of the equality of the polynomials in (\ref{eq:24fromGHRY}) and (\ref{eq:25fromGHRY}) substantially less challenging.

This given, we began an exploration of the Schur expansion of the polynomial
\begin{equation}\label{eq:SymSide}
LHS_{\nu,n}[X;q]\enskip = \enskip\sum_{\lambda\vdash n}\big\langle\Delta'_{s_\nu}e_n\hskip .25em , \hskip .25em
s_\lambda\big\rangle\Big|_{t=0} s_{\lambda'}[X(1-q)].
\end{equation}
To our surprise, this polynomial also yielded Schur expansions containing only hook Schur functions.
\vskip .06truein

A crucial feature of \cite{GHRY} was the discovery of a new method for proving the equality of two symmetric functions.
More precisely, the equality of the functions in (\ref{eq:24fromGHRY}) and (\ref{eq:25fromGHRY}) as well as their hook Schur function expansion
was obtained simply by showing that both could be expressed as linear combinations of the following shifted Cauchy kernel, using  the same coefficients $c_i(q)$
\[
\sum_{\mu\vdash n}P_\mu[X;q]\, Q_\mu[\tttt{ 1-q^i \over  1-q };q]
\,=\,h_n\big[X(1-q^i)\big]
\hskip .5truein (\hbox{for $1\le i\le n$})
\]

The data obtained, in the present case, suggested that all these desired features are present only when
$\nu$ is restricted to be a hook partition $(m-k,1^k)$  with $m<n$.
This discovery prompted us to study the symmetric function
\begin{equation}\label{eq:polyToStudy}
{\it LHS}_{k,m,n}[X,q]\enskip = \enskip \omega\Big(\Delta_{s_{m-k,1^k}}'e_n\Big|_{t=0}
\Big) [X(1-q)].
\end{equation}
Following the basic steps carried out in \cite{GHRY}  we prove here  that (\ref{eq:polyToStudy}) is equivalent to the identity
\begin{equation}\label{eq:equivalentPoly}
{\it LHS}_{k,m,n}[X,q]= q^{m+{k+1\choose2}}
\sum_{\mu\vdash  n}   q^{-n(\mu)}(q;q)_{\ell (\mu)}
\Big[{  m -1\atop k}\Big]_q
\Big[{m +\ell(\mu) -(k+2)\atop m}\Big]_q  P_\mu [X;q^{-1}].
\end{equation}

 To mimic the methods used in the classical case, we now need   to produce a ``combinatorial side''.  A simple comparison of the right hand sides of (\ref{eq:24fromGHRY}) and (\ref{eq:25fromGHRY}) shows that, in the case of the Delta Conjecture,
  the symmetric function produced by the ``combinatorial side'' could be obtained by expanding the symmetric function side in terms of the basis
$\{P_\mu[X;q]\}_\mu$.
\vskip .06truein

This led to the decision to declare the symmetric function obtained by expanding
the polynomial in (\ref{eq:SymSide}) in terms of the basis $\{P_\mu[X;q]\}_\mu$ as the ``combinatorial side'' of (\ref{eq:SymSide}). This decision led us to conjecture the following  ``combinatorial side'' of
(\ref{eq:polyToStudy}).
\begin{equation}\label{eq:combSide}
RHS_{k,m,n}[X;q]
=q^m\sum_{j=2+k}^{m+1}
q^{{k+2\choose 2}-(k+2)j+1}
\Big[{j-2\atop k}\Big]_q\Big[{m-1\atop j-2}\Big]_q
(q;q)_j\hskip -.15in\sum_{\mu\vdash n;\ell(\mu)=j}
\hskip -.15in q^{n(\mu)}P_\mu[X;q].
\end{equation}
In this paper, we first prove that
\begin{equation}\label{eq:symSideEqualsCombSide}
LHS_{k,m,n}[X;q]\enskip = \enskip  RHS_{k,m,n}[X;q].
\end{equation}
\vskip .06truein

Jeff Remmel succeeded in formulating many of the conjectures needed to prove (\ref{eq:symSideEqualsCombSide}) by precisely following the methods developed in \cite{GHRY}.
In the first section we will outline the proof of (\ref{eq:symSideEqualsCombSide}) and walk through the steps
used by Jeff Remmel to formulate his conjectures needed to complete this proof.  In the second section, we present the technical details carried out by the remaining authors to prove Remmel's conjectures and ultimately prove (\ref{eq:symSideEqualsCombSide}).
\vskip .06truein

After this project was completed, we learned that Brendon Rhoades and Mark Shimozono had already constructed, for any partition $\nu$, a symmetric function to be viewed as the ``combinatorial side'' and conjectured it to be equal to the
polynomial
\begin{equation}\label{eq:RSSymSide}
{\it LHS}_{\nu,n}[X,q]\enskip = \enskip \omega\Big(\Delta_{s_{\nu}}'e_n\Big|_{t=0}
\Big)[X].
\end{equation}
Even more importantly, Jim Haglund communicated to us that he was able to prove the Rhoades-Shimozono conjectures using solely the results in  \cite{GHRY}.  We show here that an appropriate modification of Haglund's argument proves that the polynomial in (\ref{eq:RSSymSide}) plethystically evaluated at $X(1-q)$ expands only in terms of hook Schur functions for all $\nu$.  This confirms our original experimental findings about the polynomial in (\ref{eq:SymSide}).
\vskip .06truein

These truly surprising circumstances demanded  at least two additional investigations.
The first was to determine whether or not there was any relation between our method of predicting a ``combinatorial side'' and
the Rhoades-Shimozono conjectures.  The second was to find a symmetric function reason explaining Haglund's result.  In the final section of  the paper, we present our comments about  these two problems. Here we will add a few words.
\vskip .06truein

For the first problem the evidence we gathered confirms that in this case
 our combinatorial side predicts the Rhoades-Shimozono combinatorial side.
\vskip .06truein

To be precise, we show that the symmetric function
\[
LHS_{\nu,n}[X,q]\enskip = \enskip \omega\Big(\Delta'_{s_\nu}\, e_n\Big|_{t=0}\Big)[X(1-q)]
\]
expands in terms of the $\{P_\mu[X, q^{-1}]\}_\mu$ basis as
\begin{equation}\label{eq:expSymSidePmu}
LHS_{\nu,n}[X,q]
\enskip = \enskip
q^{|\nu| } \sum_{\mu\vdash n}
s_\nu\big[\tttt{1-q^{{\ell}(\mu)-1}\over  1-q }\big]
q^{-n(\mu)}
(q;q)_{{\ell}(\mu) }
  P_\mu(X,q^{-1}).
\end{equation}
Expanding the polynomial in (\ref{eq:expSymSidePmu}) in terms of the basis $\{P_\mu[X,q]\}_\mu$ yielded our conjectured ``combinatorial side'' to  be the symmetric function
\begin{align*}
RHS_{\nu,n}[X,q]
=&
q^{|\nu| }
\sum_{k={\ell}(\nu)}^{|\nu|}(q;q)_k
\sum_{|\rho|=|\nu|, , \,
{\ell}(\rho)=k}
\,\,
{K_{\nu,\rho}(q)
\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}
q^{n(\rho)}\,\times
\\
&\hskip .5truein\bigsp\hskip .5truein
\times \,
q^{-k(k+1)}
(q;q)_{k+1}\hskip -.08in\uu\sum_{\multi{\mu\vdash n\, ;\, {\ell}(\mu)=k+1}}\hskip -.08in\uu
q^{n(\mu)}
 P_\mu[X;q]
\end{align*}
It turns out that this is precisely the Rhoades-Shimozono ``combinatorial side'' plethystically evaluated at $X(1-q)$.

\section{Jeff Remmel's conjectures in the hook case}

In this section, we will outline the steps followed by Remmel to formulate the conjectures necessary to establish the equality in (\ref{eq:symSideEqualsCombSide}), that is,
\[
LHS_{k,m,n}[X;q]\enskip = \enskip  RHS_{k,m,n}[X;q]
\]
with the polynomial in (\ref{eq:equivalentPoly}) :
\[
{\it LHS}_{k,m,n}[X,q]= q^{m+{k+1\choose2}}
\sum_{\mu\vdash  n}   q^{-n(\mu)}(q;q)_{\ell (\mu)}
\Big[{  m -1\atop k}\Big]_q
\Big[{m +\ell(\mu) -(k+2)\atop m}\Big]_q  P_\mu [X;q^{-1}]
\]
as the ``symmetric function  side'', and the polynomial in (\ref{eq:combSide}):
\[
RHS_{k,m,n}[X;q]
=q^m\sum_{j=2+k}^{m+1}
q^{{k+2\choose 2}-(k+2)j+1}
\Big[{j-2\atop k}\Big]_q\Big[{m-1\atop j-2}\Big]_q
(q;q)_j\hskip -.15in\sum_{\mu\vdash n;\ell(\mu)=j}
\hskip -.15in q^{n(\mu)}P_\mu[X;q]
\]
as  the ``combinatorial side''.

To follow the classical case, Remmel used the identity
\begin{equation}\label{eq:CauchyKernelIdentity}
{h_n[X(1-q^i)]\over  1-q^i}\enskip = \enskip\sum_{\mu\vdash n}
q^{n(\mu)} P_\mu[X;q]
\prod_{j=2}^{{\ell}(\mu)}
(1-q^{i- j+1 }  )
\end{equation}
and then tried to solve for the $c_i^{k,m}(q)$ in the equations
\[
RHS_{k,m,n}[X;q]\enskip = \enskip
\sum_{i=1}^n c_i^{k,m}(q)\,
\sum_ {\mu\vdash n }q^{n(\mu)}  P_\mu[X;q]
\prod_{r=2}^{\ell(\mu)}
(1-q^{i- r+1 }  ),
\]
which may be best rewritten as
\begin{equation}\label{eq:RHSexp}
RHS_{k,m,n}[X;q]=\sum_ {\mu\vdash n }q^{n(\mu)}  P_\mu[X;q]
\sum_{i=1}^n c_i^{k,m}(q)\,
\prod_{r=2}^{\ell(\mu)}
(1-q^{i- r+1 }  ).
\end{equation}
Likewise (\ref{eq:combSide}) may also be rewritten as
\begin{equation}\label{eq:RHSaltExp}
RHS_{k,m,n}[X;q]  =
\sum_  {\mu\vdash n }
q^{n(\mu)} P_\mu[X;q] q^m q^{{k+2\choose 2}-(k+2)\ell(\mu)+1}\Big[{\ell(\mu)-2\atop k}\Big]_q\Big[{m-1\atop \ell(\mu)-2}\Big]_q
(q;q)_j.
\end{equation}
Since $\{P_\mu[X;q]\}_\mu$ is a  basis, the equality of (\ref{eq:RHSexp}) and (\ref{eq:RHSaltExp}) can be true if and only if we have
\begin{equation}\label{eq:RemmelConj1}
\sum_{i=1}^n c_i^{k,m}(q)\,
\prod_{r=2}^{j}
(1-q^{i- r+1 })
=
q^{m+{k+2\choose 2}-(k+2)j+1}\Big[{j-2\atop k}\Big]_q\Big[{m-1\atop j-2}\Big]_q
(q;q)_j.
\end{equation}

A careful examination of computer data led  Jeff Remmel to conjecture that the solution of the equations in (\ref{eq:RemmelConj1}) are the
coefficients
\begin{equation}\label{eq:coeffs}
c_{s}^{k,m}(q)
\enskip = \enskip
(-1)^{m+1-s}q^{{ m+1-s \choose 2}-(k+1)m+{k+1\choose 2}}
\Big[
{ m-1 \atop k
}
\Big]_q
\Big[
{ k+2 \atop  m+1-s
}
\Big]_q (1-q^s)
\end{equation}

It turns  out that the proof of the  Remmel conjecture is an easy consequence of the nature of the equations in (\ref{eq:RemmelConj1}). This gives the validity of (\ref{eq:RemmelConj1}) with the $c_i^{k,m}(q)$ given by (\ref{eq:coeffs}). This also proves  the identity
\[
\sum_{i=1}^n c_{i}^{k,m}(q)
{h_n\big[X(1-q^i)\big]\over 1-q^i
}\enskip = \enskip RHS_{k,m,n}[X;q]
\enskip\ess\enskip
(\hbox{for all $1\le k\le m-1 $ and
$m<n$})
\]
This given, to prove (\ref{eq:symSideEqualsCombSide}) we only
need to show that we also have
\[
\sum_{i=1}^n c_{i}^{k,m}(q)
{h_n\big[X(1-q^i)\big]\over 1-q^i
}\enskip = \enskip LHS_{k,m,n}[X;q]
\enskip\ess\enskip
(\hbox{for all $1\le k\le m-1 $ and
$m<n$}).
\]
However here,  as in the classical case, rather than the expression in (\ref{eq:CauchyKernelIdentity}) Remmel was forced to use the equivalent expression
\[
{h_n[X(1-q^i)]\over  1-q^i}\enskip = \enskip
\sum_{\mu\vdash n}
q^{-n(\mu)} P_\mu[X;1/q]
\prod_{j=2}^{{\ell}(\mu)}
(1-  q^{i+ j-1 }  ).
\]
This given, his next goal was to
 prove the identity
\begin{align*}
&
\sum_{\mu\vdash n}
q^{-n(\mu)} P_\mu[X;1/q]
\sum_{i=1}^n c_{i}^{k,m}(q)
\prod_{j=2}^{{\ell}(\mu)}
(1-  q^{i+ j-1 }  )\enskip = \enskip
\\
\enskip = \enskip
&
\sum_{\mu\vdash  n}
 q^{-n(\mu)}P_\mu [X;q^{-1}]
q^{m-k-1+{k+2\choose2}}
\Big[{  m -1\atop k}\Big]_q
\Big[{m +\ell(\mu) -(k+2)\atop m}\Big]_q(q;q)_{\ell (\mu)}.
\end{align*}
Since $\{P_\mu [X;q^{-1}]\}_\mu $ is a symmetric function basis, equating the coefficients of $P_\mu [X;q^{-1}]$ on both sides reduced us to verifying the following $q$-identity for all $1\le {\ell}\le n$
\begin{equation}\label{eq:RemmelConj2}
\sum_{i=1}^n c_{i}^{k,m}(q)
\prod_{j=2}^{{\ell}}
(1-  q^{i+ j-1 }  )=
q^{m+{k+1\choose2}}
\Big[{  m -1\atop k}\Big]_q
\Big[{m +{\ell} -(k+2)\atop m}\Big]_q(q;q)_{{\ell}}.
\end{equation}
Actually, in order to  prove (\ref{eq:symSideEqualsCombSide}), we need only show that by means of the Remmel's coefficients defined in (\ref{eq:coeffs}), both of his conjectures (\ref{eq:RemmelConj1}) and (\ref{eq:RemmelConj2}) hold.  The following section contains all the details needed to carry this out.

\section{Technical details}

In this section, we provide the technical details that are needed to prove the Remmel conjectures.  We begin with a particular $q$-binomial identity.

\begin{prop}\label{prop:mainqidentity}
  Given nonnegative integers $m,k,\ell$ with $k+2\leq \ell\leq m+1$,
  \[ \sum_{i=0}^{\min(k+2,m+1-\ell)}(-1)^i q^{\binom{i}{2}} \qbin{k+2}{i}{q} \qbin{m+1-i}{\ell}{q}=q^{(k+2)(m+1-\ell)} \qbin{m-k-1}{\ell-2-k}{q}.\]
\end{prop}
\begin{proof}
  We will show that the proposition is a consequence of a well known hypergeometric series identity.  First, we put it in standard form.  Let $$t_j=(-1)^j q^{\binom{j}{2}} \qbin{k+2}{j}{q} \qbin{m+1-j}{\ell}{q}.$$  Then, the ratio of consecutive terms in the summation is $\frac{t_{j+1}}{t_j}$ which after some simplification can be shown to be equal to $\frac{-q^{k-\ell+2}(1-q^{-2-k}q^j)(1-q^{\ell-m-1}q^j)}{(1-q^{j+1})(1-q^{-m-1}q^j)}.$  Thus we can write the summation appearing on the left hand side of the proposition as a hypergeometric series, \begin{equation}\label{eq:hgs} \setlength\arraycolsep{1pt}
\qbin{m+1}{\ell}{q}{}_2 \Phi_1\left(\begin{matrix}q^{-2-k}, &q^{\ell-m-1}\\& q^{-m-1}\end{matrix}\bigg\rvert q;q^{k-\ell+2}\right).\end{equation}
 The $q$-Vandermonde hypergeometric series identity asserts that
 $${}_2 \Phi_1\left(\begin{matrix}A, &q^{-n}\\& C\end{matrix}\bigg\rvert q;\frac{C}{Aq^{-n}}\right)=\frac{(\frac{C}{A};q)_n}{(C,q)_n}.$$  Applying this to (\ref{eq:hgs}) yields \begin{equation}\label{eq:evalhgs} \qbin{m+1}{\ell}{q}\frac{(q^{k-m+1};q)_{m-\ell+1}}{(q^{-m-1};q)_{m-\ell+1}}.\end{equation}
 Using the identity
 \begin{equation}(q^{-n};q)_m=q^{m(m-2n-1)/2}(-1)^m(q^{n-m+1};q)_m, \end{equation}
  equation (\ref{eq:evalhgs}) can be simplified to
  \[ q^{(k+2)(m+1-\ell)} \qbin{m+1}{\ell}{q}\frac{(q^{\ell-k-1};q)_{m-\ell+1}}{(q^{\ell+1};q)_{m-\ell+1}}, \] which can easily be manipulated to become the right hand side of the proposition.
\end{proof}
The identity given in Proposition (\ref{prop:mainqidentity}) gives rise to the following corollary under the substitution $m\rightarrow m-1+\ell$.
\begin{cor}\label{cor:mainqidentity}
\[ \sum_{i=0}^{\min(k+2,m)} (-1)^iq^{\binom{i}{2}}\qbin{k+2}{i}{q}\qbin{m+\ell-i}{\ell}{q}=q^{(k+2)m}\qbin{m+\ell-(k+2)}{\ell-(k+2)}{q}. \]
\end{cor}
What follows next is a proposition which completely verifies Remmel's conjectures.  Namely, that given the coefficients defined in (\ref{eq:coeffs}), both (\ref{eq:RemmelConj1}) and (\ref{eq:RemmelConj2}) hold.
\begin{prop}\label{prop:simplifyingproduct}Given nonnegative integers $k,m,n,\ell$ with $k+2\leq \ell \leq m+1\leq n$,
\begin{enumerate}
\item $\displaystyle \sum_{i=1}^n c_i^{k,m}\prod_{j=2}^{\ell}(1-q^{i-j+1})=q^{m+\binom{k+2}{2}-(k+2)\ell+1}\qbin{\ell-2}{k}{q}\qbin{m-1}{\ell-2}{q}(q;q)_\ell$ \label{prop:simpa}
\item $\displaystyle \sum_{i=1}^n c_i^{k,m}\prod_{j=2}^{\ell}(1-q^{i+j-1})=q^{m+\binom{k+1}{2}}\qbin{m-1}{k}{q}\qbin{m+\ell-(k+2)}{m}{q}(q;q)_\ell$ \label{prop:simpb}
\end{enumerate}
    \end{prop}
\begin{proof}
First, it is worth noting that by our definitions $c_i^{k,m}=0$ when either $i>m+1$ or $i<m-k-1$.
To prove part~\ref{prop:simpa}, notice that when $i<\ell$ the product contains a 0   term.  Thus,
\begin{eqnarray*}
&&\sum_{i=1}^n c_i^{k,m}\prod_{j=2}^{\ell}(1-q^{i-j+1})\\
&=&\sum_{i=\max(m-k-1,\ell)}^{m+1} c_i^{k,m}\prod_{j=2}^{\ell}(1-q^{i-j+1})\\
&=&\sum_{i=\max(m-k-1,\ell)}^{m+1} c_i^{k,m}\frac{(1-q)\cdots(1-q^{i-1})}{(1-q)\cdots(1-q^{i-\ell})}\\
&=&\sum_{i=\max(m-k-1,\ell)}^{m+1} c_i^{k,m}\qbin{i}{\ell}{q}\frac{(q;q)_\ell}{1-q^i}\\
&=&\sum_{i=\max(m-k-1,\ell)}^{m+1} (-1)^{m+1-i}q^{\binom{m+1-i}{2}-(k+1)m+\binom{k+1}{2}}\qbin{m-1}{k}{q}\qbin{k+2}{m+1-i}{q}\qbin{i}{\ell}{q}(q;q)_\ell\\
&=&\sum_{i=0}^{\min(k+2,m+1-\ell)} (-1)^{i}q^{\binom{i}{2}-(k+1)m+\binom{k+1}{2}}\qbin{m-1}{k}{q}\qbin{k+2}{i}{q}\qbin{m+1-i}{\ell}{q}(q;q)_\ell\\
&=&q^{\binom{k+1}{2}-m(k+1)}\qbin{m-1}{k}{q}\sum_{i=0}^{\min(k+2,m+1-\ell)} (-1)^{i}q^{\binom{i}{2}}\qbin{k+2}{i}{q}\qbin{m+1-i}{\ell}{q}(q;q)_\ell.\\
\end{eqnarray*}
Then using Proposition \ref{prop:mainqidentity},
\begin{eqnarray*}
&=&q^{\binom{k+1}{2}-m(k+1)}\qbin{m-1}{k}{q} q^{(k+2)(m+1-\ell)} \qbin{m-k-1}{\ell-2-k}{q}(q;q)_\ell\\
&=&q^{m+\binom{k+2}{2}-(k+2)\ell+1} \qbin{\ell-2}{k}{q}\qbin{m-1}{\ell-2}{q}(q;q)_\ell.\\
\end{eqnarray*}
This completes the proof of part~\ref{prop:simpa}.  To prove part~\ref{prop:simpb},
\begin{eqnarray*}
&&\sum_{i=1}^n c_i^{k,m}\prod_{j=2}^{\ell}(1-q^{i+j-1})\\
&=&\sum_{i=\max(m-k-1,1)}^{m+1} c_i^{k,m}\prod_{j=2}^{\ell}(1-q^{i+j-1})\\
&=&\sum_{i=0}^{\min(k+2,m)} c_{m+1-i}^{k,m}\prod_{j=2}^{\ell}(1-q^{m+j-i})\\
&=&q^{\binom{k+1}{2}-(k+1)m}\qbin{m-1}{k}{q}\sum_{i=0}^{\min(k+2,m)} (-1)^iq^{\binom{i}{2}}\qbin{k+2}{i}{q}(1-q^{m+1-i})\prod_{j=2}^{\ell}(1-q^{m+j-i})\\
&=&q^{\binom{k+1}{2}-(k+1)m}\qbin{m-1}{k}{q}\sum_{i=0}^{\min(k+2,m)} (-1)^iq^{\binom{i}{2}}\qbin{k+2}{i}{q}\qbin{m+\ell-i}{\ell}{q}(q;q)_\ell\\
&=&q^{\binom{k+1}{2}-(k+1)m}\qbin{m-1}{k}{q}q^{(k+2)m}\qbin{m+\ell-(k+2)}{\ell-(k+2)}{q}(q;q)_\ell\\
&=&q^{m+\binom{k+1}{2}}\qbin{m-1}{k}{q}\qbin{m+\ell-(k+2)}{m}{q}(q;q)_\ell
\end{eqnarray*}
The next to last step is justified by Corollary \ref{cor:mainqidentity}.
\end{proof}

\section{Additional investigations}
To begin the investigation of whether our ``combinatorial side'' was related to that of Rhodes-Shimozono, we first   expanded the symmetric function $\omega\Big(\Delta_{s_{\nu}}'e_n\Big|_{t=0}\Big)[X(1-q)]$ in terms of the basis $\{P_\mu(X,q^{-1})\}_\mu$.  This is done in the following theorem.
\begin{thm}\label{thm:genSymSideExp}
\begin{equation}\label{eq:thmGenSymSideExp}
\omega\Big(\Delta_{s_{\nu}}'e_n\Big|_{t=0}
\Big)[X(1-q)]
 \enskip = \enskip q^{|\nu| } \sum_{\mu\vdash n}
s_\nu\big[\tttt{1-q^{{\ell}(\mu)-1}\over  1-q }\big]
q^{-n(\mu)}
(q;q)_{{\ell}(\mu) }
  P_\mu(X,q^{-1})
\end{equation}
\end{thm}
\begin{proof}
We begin with the following expansion of $e_n$ (Lemma 2.1 in \cite{GHRY}),
\[ e_n (X) = \sum_{\mu\vdash n}\frac{(1-q)(1-t)\widetilde{H}_\mu (X;q,t) \Pi_\mu ^{\prime}(q,t)B_\mu (q,t) }{w_\mu (q,t)}.
\]
Recognizing that the left hand side does not contain the indeterminates $q$ and $t$, we can interchange them and obtain
\[ e_n (X) = \sum_{\mu\vdash n}\frac{(1-q)(1-t)\widetilde{H}_\mu (X;t,q) \Pi_\mu ^{\prime}(t,q)B_\mu (t,q) }{w_\mu (t,q)}.
\]
Then using the definition of $\Delta'$, and setting $t=0$, we have
\begin{equation}\label{eq:expOfEn}\Delta_{s_{\nu}}'e_n\Big|_{t=0} = \sum_{\mu\vdash n}\frac{(1-q)s_{\nu}\big[B_\mu(0,q) -1 \big]\widetilde{H}_\mu (X;0,q) \Pi_\mu ^{\prime}(0,q)B_\mu (0,q) }{w_\mu (0,q)}.
\end{equation}
In \cite{GHRY}, it was noted that
\begin{align*}
B_\mu (0,q) &= 1+q+\dotsb +q^{\ell (\mu)-1}=\frac{1-q^{\ell(\mu)}}{1-q},\\
 \Pi_\mu ' (0,q) &= (q;q)_{\ell (\mu)-1},\\
w_\mu (0,q) &= \prod_{c\in \mu}q^{l(c)} \cdot \prod_{\substack{c\in \mu\\ a(c)=0}}(1-q^{l(c)+1})\cdot \prod_{\substack{c\in \mu \\ a(c)>0}}(-q^{l(c)+1})\\
&= (-1)^{n-\ell (\mu)}q^{2n(\mu)+n-\sum_i \binom{m_i(\mu) +1}{2}}\prod_{i}(q;q)_{m_i(\mu)},
\end{align*}
where $(q;q)_m = (1-q)\cdots (1-q^m)$.
Substituting these into (\ref{eq:expOfEn}) and simplifying gives
\[
\Delta_{s_{\nu}}'e_n\Big|_{t=0} = \sum_{\mu\vdash n}(-1)^{n-\ell (\mu)}s_{\nu}\big[q+q^2+\dotsb +q^{\ell (\mu)-1} \big]q^{-2n(\mu)-n+\sum_i \binom{m_i(\mu) +1}{2}}\Big[{ {\ell}(\mu) \atop m(\mu) }\Big]_q \widetilde{H}_\mu (X;0,q).
\]
Replacing $X$ by $X(1-q)$ and factoring a $q$ out of the plethystic evaluation, the right hand side becomes
\[
q^{|\nu|}\sum_{\mu\vdash n}(-1)^{n-\ell (\mu)}s_{\nu}\Big[\frac{1-q^{\ell (\mu)-1}}{1-q} \Big]q^{-2n(\mu)-n+\sum_i \binom{m_i(\mu) +1}{2}}\Big[{ {\ell}(\mu) \atop m(\mu) }\Big]_q \widetilde{H}_\mu (X(1-q);0,q),
\]
and then expanding $\widetilde{H}_\mu (X(1-q);0,q)$ yields
\[
q^{|\nu|}\sum_{\mu\vdash n}(-1)^{n-\ell (\mu)}s_{\nu}\Big[\frac{1-q^{\ell (\mu)-1}}{1-q} \Big]q^{-2n(\mu)-n+\sum_i \binom{m_i(\mu) +1}{2}}\Big[{ {\ell}(\mu) \atop m(\mu) }\Big]_q \sum_{\lambda\vdash n}s_\lambda\left[X(1-q)\right]\widetilde{K}_{\lambda , \mu}(q).
\]
But, since $s_\lambda\left[X(1-q)\right]=(-q)^ns_{\lambda'}\left[X(1-1/q)\right]$ and $\widetilde{K}_{\lambda , \mu}(q) = q^{n(\mu)}K_{\lambda,\mu}(q^{-1})$, we can now apply $\omega$ and eventually arrive at
\[
q^{|\nu|}\sum_{\mu\vdash n}(-1)^{\ell (\mu)}s_{\nu}\Big[\frac{1-q^{\ell (\mu)-1}}{1-q} \Big]q^{-n(\mu)+\sum_i \binom{m_i(\mu) +1}{2}}\Big[{ {\ell}(\mu) \atop m(\mu) }\Big]_q \sum_{\lambda\vdash n}s_{\lambda}\left[X(1-q^{-1})\right]K_{\lambda , \mu}(q^{-1}).
\]
We will next need  two  facts stated in \cite{GHRY}:
\begin{equation}\label{eq:fact1fromGHRY}
 Q_\mu(X,q) =  \sum_{\lambda \vdash n}s_\lambda\left[X(1-q)\right]K_{\lambda,\mu}(q)
\end{equation}
and
\begin{equation}\label{eq:fact2fromGHRY}
 P_\mu(X,q^{-1}) =  \frac{(-1)^{\ell(\mu)} q^{\sum_{i}{m_i(\mu)+1\choose 2}}}{\prod (q;q)_{m_i(\mu)}}Q_\mu(X,q^{-1}),
\end{equation}
where $\mu$ is a partition of $n$.
Applying (\ref{eq:fact1fromGHRY}) at $q^{-1}$ gives
\[
q^{|\nu|}\sum_{\mu\vdash n}(-1)^{\ell (\mu)}s_{\nu}\Big[\frac{1-q^{\ell (\mu)-1}}{1-q} \Big]q^{-n(\mu)+\sum_i \binom{m_i(\mu) +1}{2}}
{(q;q)_{{\ell}(\mu)}
\over
\prod_{i=1}^{{\ell}(\mu)}(q;q)_{m_i(\mu)}
}\,\, Q_\mu(X,q^{-1}),
\]
and then applying (\ref{eq:fact2fromGHRY}) we prove the theorem, namely,
$$
\omega\Big(\Delta_{s_{\nu}}'e_n\Big|_{t=0}
\Big)[X(1-q)]
 \enskip = \enskip q^{|\nu| } \sum_{\mu\vdash n}
s_\nu\big[\tttt{1-q^{{\ell}(\mu)-1}\over  1-q }\big]
q^{-n(\mu)}
(q;q)_{{\ell}(\mu) }
  P_\mu(X,q^{-1}).
$$

\end{proof}

\begin{cor}
The identity (\ref{eq:equivalentPoly}), namely,
\[
{\it LHS}_{k,m,n}[X,q]=
q^{m +{k+1\choose2}}
\sum_ {\mu\vdash  n}
\Big[{  m -1\atop k}\Big]_q
\Big[{m +\ell(\mu) -(k+2)\atop m}\Big]_q q^{-n(\mu)}(q;q)_{\ell (\mu)} P_\mu [X;q^{-1}],
\]
is none other but a specialization of Theorem \ref{thm:genSymSideExp}, at $\nu=(m-k,1^k)$.
\end{cor}

\vskip  -.12in
\begin{flushright}
\begin{tikzpicture}[scale=0.46, line width=1pt]
  \draw (0,0) grid (1,-5);
  \draw (1,-4) grid (4,-5);
  \node at (.5,-.5) {-4};
  \node at (.5,-1.5) {-3};
  \node at (.5,-2.5) {-2};
  \node at (.5,-3.5) {-1};
  \node at (.5,-4.5) {0};
  \node at (1.5,-4.5) {1};
  \node at (2.5,-4.5) {2};
  \node at (3.5,-4.5) {3};
  \node[right] at (2,-.5) {$m$-$k$-$1$};
  \draw[->](2,-.5) -- (1,-.5);
  \node[left] at (2.75,-3) {$k$};
  \draw[->](2.5,-3.25) -- (3.5,-4);
  \node at (2,-5.5) {$c(x)$};

  \draw (6,0) grid (7,-5);
  \draw (6,-4) grid (10,-5);
  \node at (6.5,-.5) {1};
  \node at (6.5,-1.5) {2};
  \node at (6.5,-2.5) {3};
  \node at (6.5,-3.5) {4};
  \node at (6.5,-4.5) {8};
  \node at (7.5,-4.5) {3};
  \node at (8.5,-4.5) {2};
  \node at (9.5,-4.5) {1};
  \node[right] at (7.75,-2.25) {$m$-$k$-$1$};
  \draw[->](8,-2.5) -- (6.75,-3.5);
  \node[right] at (7.85,-3.5) {$m$};
  \draw[->](8,-3.5) -- (6.75,-4.5);
  \node at (8,-5.5) {$h(x)$};
\end{tikzpicture}
\end{flushright}

\vskip  -1.6in

\vskip .3in
\begin{proof}

Recall that the definition of the left hand side of (\ref{eq:equivalentPoly}) is
\vskip .1in

\enskip\ess\enskip$
{\it LHS}_{k,m,n}[X,q]= \omega\Big(\Delta_{s_{m-k,1^k}}'e_n\Big|_{t=0} \Big)[X(1-q)]
$

\vskip .1in
\noindent
Now the Macdonald formula for the plethystic evaluation of $s_\lambda$

\vskip .1in
\noindent
at $ 1+q+\cdots+q^{n-1}$  is

\vskip -.12in
$$
s_\lambda[1+q+\cdots+q^{n-1}]
= q^{n(\lambda)}\Big[{n \atop  \lambda'} \Big]_q
$$
\vskip -.16in
\noindent
where
$$
\Big[{n \atop  \lambda } \Big]_q
\enskip = \enskip \prod_{x\in \lambda}{  1-q^{n-c(x)} \over 1-q^{h(x)}}
$$
With $c(x)$ and $h(x)$ the {\it content} and the {\it hook} of cell $x\in \lambda$.
\vskip .06truein

Now for $\lambda=(m-k,1^k)$ we have  $n(\lambda)={k+1\choose 2}$ and
 $\lambda'=(k+1,1^{m-k-1})$. We thus obtain

\begin{align*}
s_{m-k,1^k}[1+q+\cdots+q^{\ell-2}]&=
q^ {k+1\choose 2}
{
(1-q^{\ell-1+0 })\cdots (1-q^{\ell-1 +m-k-1})(1-q^{\ell-1-1})\cdots (1-q^{\ell-1-k})
\over
(q;q)_k(1-q^m)(q;q)_{m-k-1}
}\\
&=
q^ {k+1\choose 2}
{ (q^{{\ell}-k-1},q)_m
\over
(q;q)_k(1-q^m)(q;q)_{m-k-1}
}
\end{align*}
See the illustration above where  the statistics $c(x)$ and $h(x)$ are computed for the hook partition $(m-k,1^k)$
{\vskip .125truein}
\noindent
Notice next that we have
\begin{align*}
\Big[{  m -1\atop k}\Big]_q
\Big[{m +\ell(\mu) -(k+2)\atop m}\Big]_q &\enskip = \enskip
{1
\over
(q;q)_{k}(q;q)_{m-1-k}
}
\enskip
{(q;q)_{m +\ell(\mu) -(k+2)}
\over
(1-q^m)(q;q)_{\ell(\mu) -(k+2)}}
\\
&\enskip = \enskip
{(q^{{\ell}-k-1},q)_m
\over
(q;q)_{k}(1-q^m)(q;q)_{m-1-k}
}
\end{align*}

\noindent
To prove that for $\nu=(m-k,1^k)$
(\ref{eq:thmGenSymSideExp}) reduces to (\ref{eq:equivalentPoly}), we need only verify the equality
\[
q^{m }
s_{m-k,1^k}\big[\tttt{1-q^{{\ell} -1}\over  1-q }\big]\enskip = \enskip q^{m +{k+1\choose2}}
\Big[{  m -1\atop k}\Big]_q
\Big[{m +\ell  -(k+2)\atop m}\Big]_q
\]
However, the above calculations show exactly that.
\end{proof}

Theorem \ref{thm:genSymSideExp} provides an expansion of the symmetric function side in terms of the\\ $\{P_\mu(X,q^{-1})\}$ basis.  We now seek an appropriate ``combinatorial side'' by expanding the same symmetric function in terms of the $\{P_\mu(X,q)\}$ basis.  In order to do this, we will use a special evaluation given in the following theorem.

\begin{thm}\label{thm:specEval}
\begin{equation}\label{eq:thmSpecEval}
s_\nu\Big[\tttt{1-q^{j-1}\over  1-q }\Big]
\enskip = \enskip
\sum_{k={\ell}(\nu)}^{|\nu|}
\sum_{\multi{|\rho|=|\nu|\cr
{\ell}(\rho)=k}}
\,\,
{K_{\nu,\rho}(q)
\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}
q^{n(\rho)}
{(q;q)_{j-1}
\over (q;q)_{j-1-k}}
\end{equation}
\end{thm}
\begin{proof}
Recall that from \cite{Mac}, we get the identity
\[
s_\nu[X]\enskip = \enskip \sum_{\rho\vdash |\nu|}K_{\nu,\rho}(q)P_\rho[X,q],
\]
which can be also written as
\[
s_\nu[X]\enskip = \enskip \sum_{\rho\vdash |\nu|}K_{\nu,\rho}(q){Q_\rho[X,q]\over \prod_{i=1}^m
(q;q)_{m_i(\rho)}}
\]
and $X{ \rightarrow } X/(1-q)$ gives
\[
s_\nu\big[\tttt{X\over 1-q}\big]\enskip = \enskip \sum_{\rho\vdash|\nu|}H_\rho[X ;q] { K_{\nu,\rho}(q)\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}.
\]
Now the replacement $X{ \rightarrow } 1-q^{j-1}$
yields
\[
s_\nu\big[\tttt{ 1-q^{j-1}\over 1-q}\big]\enskip = \enskip \sum_{\rho\vdash|\nu|}H_\rho[ 1-q^{j-1} ;q] { K_{\nu,\rho}(q)\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}
\]
This can be rewritten in the form
\begin{equation}\label{eq:thmSpecEvalRew}
s_\nu\big[\tttt{1-q^{j-1}\over  1-q }\big]
\enskip = \enskip
\sum_{k=1}^{|\nu|}
\sum_{\multi{|\rho|=|\nu|\cr
{\ell}(\rho)=k}}
\,\,H_\rho[1-q^{j-1};q ]
{K_{\nu,\rho}(q)
\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}.
\end{equation}
Now, the Macdonald reciprocity in the Hall-Littlehood case yields
\[
H_\rho[1-u;q ]\enskip = \enskip q^{n(\rho)}
\prod_{s=1}^{{\ell}(\rho)}
(1-u/q^{ s-1 }  ).
\]
In particular, the replacement $u{ \rightarrow } q^{j-1}$ gives (for  ${\ell}(\rho)=k$)
\[
H_\rho[1-q^{j-1};q ]\enskip = \enskip q^{n(\rho)}
\prod_{s=1}^{k}
(1- q^{j -s }  )\enskip = \enskip q^{n(\rho)}(1-q^{j-k})\cdots (1-q^{j-1})
\]
Thus (\ref{eq:thmSpecEvalRew}) becomes
\[
s_\nu\big[\tttt{1-q^{j-1}\over  1-q }\big]
\enskip = \enskip
\sum_{k=1}^{|\nu|}
\sum_{\multi{|\rho|=|\nu|\cr
{\ell}(\rho)=k}}
\,\,
{K_{\nu,\rho}(q)
\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}
q^{n(\rho)}
{(q;q)_{j-1}
\over (q;q)_{j-1-k}}
\]
Since the coefficient $K_{\nu,\rho}(q)$ fails to vanish only when  $\nu\ge\rho$ in dominance,
the hypothesis ${\ell}(\rho)=k$ forces ${\ell}(\nu)\le k$. This given we can write
\[
s_\nu\big[\tttt{1-q^{j-1}\over  1-q }\big]
\enskip = \enskip
\sum_{k={\ell}(\nu)}^{|\nu|}
\sum_{\multi{|\rho|=|\nu|\cr
{\ell}(\rho)=k}}
\,\,
{K_{\nu,\rho}(q)
\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}
q^{n(\rho)}
{(q;q)_{j-1}
\over (q;q)_{j-1-k}}
\]
\end{proof}

\vskip -.35in
Now Theorem \ref{thm:genSymSideExp} gives that our symmetric function side has the expansion
\[
LHS_{\nu,n}[X,q]
\enskip = \enskip
q^{|\nu| } \sum_{\mu\vdash n}
s_\nu\big[\tttt{1-q^{{\ell}(\mu)-1}\over  1-q }\big]
q^{-n(\mu)}
(q;q)_{{\ell}(\mu) }
  P_\mu(X,q^{-1}).
\]
Using Theorem \ref{thm:specEval}, this can be rewritten as
\[
q^{|\nu| } \sum_{\mu\vdash n}
\sum_{k={\ell}(\nu)}^{|\nu|}
\sum_{\multi{|\rho|=|\nu|\cr
{\ell}(\rho)=k}}
\,\,
{K_{\nu,\rho}(q)
\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}
q^{n(\rho)}
{(q;q)_{{\ell}(\mu)-1}
\over (q;q)_{{\ell}(\mu)-1-k}}
q^{-n(\mu)}
(q;q)_{{\ell}(\mu) }
  P_\mu(X,q^{-1}),
\]
or better,
\[
q^{|\nu| }
\sum_{k={\ell}(\nu)}^{|\nu|}(q;q)_k
\sum_{\multi{|\rho|=|\nu\cr
{\ell}(\rho)=k}}
\,\,
{K_{\nu,\rho}(q)
\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}
q^{n(\rho)}\,
\sum_{\mu\vdash n}\Big[{{\ell}(\mu)-1\atop k}\Big]_q
q^{-n(\mu)}
(q;q)_{{\ell}(\mu) }
  P_\mu(X,q^{-1}).
\]

Recall that in \cite{GHRY}, for the classical case of the Delta conjecture at $t=0$, we proved the identity
\[
\sum_{\mu\vdash n}q^{- n(\mu)}
P_\mu[X;q^{-1}]
\Big[{ {\ell}(\mu)-1\atop k }\Big]_q
(q;q)_{{\ell}(\mu)}
\enskip = \enskip q^{-k(k+1)}
(q;q)_{k+1}\hskip -.08in\uu\sum_{\multi{\mu\vdash n\, ;\, {\ell}(\mu)=k+1}}\hskip -.08in\uu
q^{n(\mu)}
 P_\mu[X;q]
\]
This permits us to obtain
the expansion of the symmetric function side in terms of the basis
$\big\{P_\mu[X;q]\big\}_\mu$ and use our recipe to obtain what we would label as the ``combinatorial side''.  Namely,
\begin{align*}
RHS_{\nu,n}[X,q]
\enskip = \enskip&
q^{|\nu| }
\sum_{k={\ell}(\nu)}^{|\nu|}(q;q)_k
\sum_{\multi{|\rho|=|\nu\cr
{\ell}(\rho)=k}}
\,\,
{K_{\nu,\rho}(q)
\over
\prod_{i=1}^m
(q;q)_{m_i(\rho)}}
q^{n(\rho)}\,\times
\\
&\hskip .5truein\bigsp
\times \,
q^{-k(k+1)}
(q;q)_{k+1}\hskip -.08in\uu\sum_{\multi{\mu\vdash n\, ;\, {\ell}(\mu)=k+1}}\hskip -.08in\uu
q^{n(\mu)}
 P_\mu[X;q]
\end{align*}

Additionally, the last sum appearing in ${\it RHS}_{\nu,n}[X,q]$ was proved to have a hook Schur function expansion in \cite{GHRY}.
We have thus proved the following generalization of (\ref{eq:symSideEqualsCombSide}).
\begin{thm}It is not only true that
\begin{equation}\label{eq:thmGeneralSymSideEqualsCombSide}
{\it LHS}_{\nu,n}[X,q]\enskip = \enskip {\it RHS}_{\nu,n}[X,q],
\end{equation}
but also that the Schur expansion of both sides contains only hook
Schur functions.
\end{thm}

\begin{remark}
The right hand side of this identity is none one other than the Rhodes-Shimozono ``combinatorial side'' transformed to our set up, (see the righthand side of  Theorem 1.2 in \cite{HagRhoShi}).
\end{remark}

\begin{remark}
In \cite{GHRY} (see Lemma 4.2) it is shown that
$$
h_n[X(1-u)] \enskip = \enskip (1-u)\sum_{s=0}^{n-1}(-u)^s
s_{(n-s,1^s)}[X]
$$
It follows from this identity that any symmetric polynomial whose Schur functions expansion contains only hook Schur functions may be expanded as  linear combination of the shifted Cauchy  kernel $h_n(X(1-q^i)]$. What forced Remmel to restrict himself to $\Delta_{s_\nu} e_n$ in the hook case of $\nu$ is that in the hook case the needed coefficients are products of $q$-analogues of integers. This facilitated conjecturing their exact nature. With the wisdom of hindsight we can now explain Haglund result as due to the fact that Schur function expansions of  the appropriately modified polynomials
$\Delta_{s_\nu}e_n$ contain only hook Schur functions in full generality. However, this
circumstance is only an artifact of the specialization at $t=0$. In fact, without this specialization, computer data reveals the dimension of the space spanned by the polynomials $\Delta_{s_\nu}e_n$ to be much larger than $n$.
The data suggests that,  more likely,  this dimension is the number of partitions of $n$.
\end{remark}

\section{Acknowledgements}

The authors want to  express their gratitude  to Jim Haglund for making his result  and his argument available to us, (unpublished manuscript). We are also grateful to Dennis Stanton for providing us with the tools we needed to be able to prove (in section 3) Remmel's $q$-binomial conjectures.


\section{Time-evolution in Quantum Mechanics with

A certain climax of our present letter comes when we address the
problem of the time-evolution in Quantum Mechanics with $\Theta
\neq I$ and with a manifest time-dependence in all our operators.
Once we prepare an initial state as a normalized vector
$|\varphi(t) \!\succ \in  {\cal H}_{phys}^{(stand)}$ at $t=0$, we can
rely only on our understanding of the evolution caused by the
auxiliary self-adjoint Hamiltonians $h(t)$.  In particular, we may
immediately solve any time-dependent Schr\"{o}dinger equation
 \begin{equation}
 {\rm i}\,\partial_t |\varphi(t)\!\!\succ \ \ =\ h(t)\,
 |\varphi(t)\!\!\succ\,,\ \ \ \ \ \
 |\varphi(t)\!\!\succ \ \ = \ u(t)\,|\varphi(0)\!\!\succ
 \label{timeq}
 \end{equation}
and we are sure that the related evolution operator given by
equation
 \begin{equation}
 {\rm i}\partial_t u(t)=h(t)\,u(t)\,
 \label{seh}
 \end{equation}
is {\em certainly} unitary in  ${\cal H}_{phys}^{(stand)}$,
 \[
 \prec\! \varphi(t) \,|\,
 \varphi(t)\!\succ=
 \prec\! \varphi(0) \,|\,
 \varphi(0)\!\succ\,.
 \]
In the next step of our considerations we recollect the pull-backs
$|\Phi(t)\rangle=\Omega^{-1}(t)\, |\varphi(t)\!\!\succ$ and $\langle\!\langle
\Phi(t)\,|=\prec\!\!\varphi(t)\,|\,\Omega(t)$ carrying, by
assumption, their own time dependence. It is represented,
formally, by the ``right-action" evolution rule
 \begin{equation}
 |\Phi(t)\rangle=U_R(t)\, |\Phi(0)\rangle\,,\ \ \ \ \ \
 U_R(t)=\Omega^{-1}(t)\,u(t)\,\Omega(0)
 \end{equation}
accompanied by its ``left-action" parallel
 \begin{equation}
 |\Phi(t)\rangle\!\rangle=U_L^\dagger(t)\, |\Phi(0)\rangle\!\rangle\,,\ \ \ \ \ \
 U_L^\dagger(t)=\Omega^\dagger(t)\,u(t)\,
 \left [\Omega^{-1}(0)\right ]^\dagger\,.
 \end{equation}
The respective non-Hermitian analogues of the Hermitian evolution
rule~(\ref{seh}) are now obtained by the elementary
differentiation and insertions yielding the two separate
differential operator equations
 \begin{equation}
 {\rm i}\partial_t U_R(t)=
 -\Omega^{-1}(t)
 \left [{\rm i}\partial_t\Omega(t)
 \right ]\, U_R(t)+H(t)\, U_R(t)\,
 \end{equation}
and
 \begin{equation}
 {\rm i}\partial_t U_L^\dagger(t)=
 H^\dagger(t)\, U_L^\dagger(t)
 +
 \left [{\rm i}\partial_t\Omega^\dagger(t)
 \right ]\,
 \left [
 \Omega^{-1}(t)
 \right ]^\dagger\, U_L^\dagger(t)\,.
 \end{equation}
We are prepared to verify what happens with the norm $ \langle\!\langle
\Phi(t)\,|\,\Phi(t)\rangle $ of states which evolve with time in the
physical space ${\cal H}^{(\Theta)}$. The elementary
differentiation gives
 \begin{eqnarray}
 {\rm i}\partial_t\langle\!\langle \Phi(t)\,|\,\Phi(t)\rangle
 ={\rm i}\partial_t
 \langle\!\langle \Phi(0)\,|\,U_L(t)\,U_R(t)  \,|\,\Phi(0)\rangle=
 \\
 =\langle\!\langle \Phi(0)\,|\,
 \left [ {\rm i}\partial_t
 U_L(t)
 \right ]
 \,U_R(t)  \,|\,\Phi(0)\rangle
 +\langle\!\langle \Phi(0)\,|\,U_L(t)\,
 \left [ {\rm i}\partial_t
 U_R(t)
 \right ]
   \,|\,\Phi(0)\rangle=
   \\
 = \langle\!\langle \Phi(0)\,|\, U_L(t)\,
 \left [ -H(t)+\Omega^{-1}(t)
 \left [{\rm i}\partial_t\Omega(t)
 \right ]
 \right ]
 \,U_R(t)  \,|\,\Phi(0)\rangle+\\
 +\langle\!\langle \Phi(0)\,|\,U_L(t)\,
 \left [ H(t)
  -\Omega^{-1}(t)
 \left [{\rm i}\partial_t\Omega(t)
 \right ]
 \right ]\,U_R(t)
   \,|\,\Phi(0)\rangle=0\,.
   \end{eqnarray}
We see that the norm remains constant also in ${\cal
H}^{(\Theta)}$. In the other words, the time-evolution of the
system specified by the quasi-Hermitian and time-dependent
Hamiltonian $H(t)$ is unitary.

Once we abbreviate $\dot{\Omega}(t)\equiv \partial_t\Omega(t)$,
our latter observation can be rephrased as an explicit
specification of {\em the same} time-evolution generator
 \begin{equation}
 H_{(gen)}(t)=H(t) -{\rm i}\Omega^{-1}(t)
 \dot{\Omega}(t)
 \end{equation}
entering {\em both} the present quasi-Hermitian updates
 \begin{eqnarray}
 {\rm i}\partial_t|\Phi(t)\rangle
 =H_{(gen)}(t)\,|\Phi(t)\rangle\,,
 \label{SEA}\\
 {\rm i}\partial_t|\Phi(t)\rangle\!\rangle
 =H_{(gen)}^\dagger(t)\,|\Phi(t)\rangle\!\rangle\,
 \label{SEbe}
 \end{eqnarray}
of the current textbook time-dependent Schr\"{o}dinger equation
for wave functions. Such a confirmation of the preservation of the
overall unitarity of the evolution (at the cost of a not too
difficult and explicit  redefinition $H \to H_{(gen)}$ of the
generator of time evolution which remains the same for both the
left and right action) may be read as really good news for the
theory.

We arrived at the  answer to the question given in the title. In a
concise discussion we should emphasize that such an answer is
quite surprising because the standard connection between the
Hamiltonian and the time evolution of the system is usually
interpreted as a certain ``first principle" of the quantum
dynamics. In such a context our present constructive argument
against the current postulate $H_{(gen)}=H$ should be perceived as
a formal foundation of an apparently counterintuitive innovative
idea that once we admit some dynamically motivated time dependence
of the Hamiltonian itself, we might also contemplate a parallel
assumption of some equally arbitrary, phenomenologically motivated
time dependence of some other observable quantities.

The key reason for the possible mathematical consistency as well
as for a formal acceptability of such a fairly unusual scenario
should be seen in the deep formal ambiguity of the metric
operator. In principle, this observation (made also in refs.
\cite{Geyer} and \cite{ali}) gives us a huge space and freedom for
the {\em time-dependent} variability of the metric
$\Theta=\Theta(t)$. Let us emphasize that in the generic case with
$\dot{H}(t)\neq 0$ the $t-$dependence of $H\neq H^\dagger$ becomes
immediately transferred to its left and right eigenvectors and,
subsequently (i.e., via formula (\ref{identify})), to the metric
$\Theta=\Theta(t)$ itself. Moreover, in parallel to the freedom of
our choice of the $t-$dependence of the Hamiltonian $H(t)$, there
exists, at least in principle, no mathematical obstruction to  an
unrestricted time-dependence freedom introduced directly in all
the complex coefficients $\mu_n=\mu_n(t)$ (cf., once more,
eq.~(\ref{identify})).

We have to admit that we did not really expect that the ``old"
idea of a mapping $\Omega$ between spaces \cite{Geyer} (and, of
course, between Hamiltonian $H$ and its ``suitable" Hermitian
partner $h$) will survive so easily its transfer to the
time-dependent case with $\Omega=\Omega(t)$. At the same time,
having arrived at our final pair of the generalized quantum
time-evolution differential equations (\ref{SEA}) and (\ref{SEbe})
we find them quite natural and consistent. Summarizing our present
results, the Hamiltonian (i.e., energy-operator $H=H(t)$) was
assumed quasi-Hermitian (i.e., $H^\dagger(t) =
\Theta(t)\,H(t)\,\Theta^{-1}(t)$ at some nontrivial metric
$\Theta(t)=\Theta^\dagger(t)>0$). In parallel, some other
quasi-Hermitian operators $A_j$ of observables were assumed to
carry an independent time-dependence (i.e., we were allowed to
demand that $A_j^\dagger(t) = \Theta(t)\,A_j(t)\,\Theta^{-1}(t)$).
Under these assumptions we demonstrated that the evolution of the
system in question still remains unitary.

The latter observation throws new light upon some applications of
the models with $\Theta \neq I$ (cf., e.g., the ones reviewed in
\cite{Carl}) where the values of all the coefficients
$\mu_n=\mu_n(t)$ are fixed using a suitable phenomenological
postulate. In the other applications reviewed in \cite{Geyer} the
variability of $\mu_n$s is also being removed, step by step, via
an appropriate selection of some other operators decided to play
the role of observables. We may conclude that there is nothing
counterintuitive in the manifest ``decoupling" of $H_{(gen)}(t)$
from $H(t)$. It is fairly easily accepted immediately after one
imagines that the information about the quantum evolution {\em
can} be carried {\em not only} by the manifestly time-dependent
Hamiltonian $H(t)$ {\em but also} by some other manifestly
time-dependent observables $A_j(t)$, $j=1,2, \ldots$. Naturally,
the latter information must also influence the overall evolution
of the system.

In each of the eligible scenarios and approaches, the constraints
imposed upon the variability of the coefficients in $\Omega(t)$
may be considered controllable and at our disposal. After an
acceptance of such a concept of the time-dependence in quantum
dynamics, our present main {\em technical} contribution may be
seen in an almost elementary check of the mutual compatibility and
independence of the two ``input-information" hypotheses involving
$H=H(t)$ {\em and} $\Theta=\Theta(t)$.

\section*{Acknowledgement}

Work supported by GA\v{C}R, grant Nr. 202/07/1307, Institutional
Research Plan AV0Z10480505 and by the M\v{S}MT ``Doppler
Institute" project Nr. LC06002.


\section{Introduction}
The fifth generation (5G) of the cellular systems, which will be introduced in the early 2020s, outperforms the previous generations in terms of data rates and capacity, in addition to supporting new communication protocols to deal with demanding requirements such as latency, reliability and efficiency in MTC~\cite{tullberg2016metis}. In MTC or machine-to-machine communications (M2M), data is automatically exchanged between two MTC nodes or an MTC node and a server where the human cooperation is minimized~\cite{shariatmadari2015machine}.
5G system outperforms current networks in terms of spectrum efficiency via decreasing the latency and guaranteeing reliability up to $99.999 \%$. Authors in~\cite{andrews2014will} discuss about the 5G technology in more details in terms of requirements and challenges such as data rate, latency, energy and cost issues.
\subsection{Machine-Type Communication}
In recent years, MTC has  received much attention due to the vast applications in wireless communications via supporting transmission/reception of short data packets in comparison with the current communication systems which carry long data packets, and also being a cost-effective and energy efficient technology~\cite{lee2016packet}.
MTC technology facilitates wide range of applications and is characterized as massive MTC (mMTC) and ultra-reliable MTC (uMTC)~\cite{bockelmann2016massive}. In mMTC, where the blocklength is short, a huge number of devices in a certain domain are covered with low-rate and low-power connectivity, and also considerable reliability in order to cover critical situations, e.g. in sensor networks, smart meters, actuators, etc,~\cite{7041045},~\cite{singh2016selective}. In addition, timing constraints from few seconds to even extremely low end-to-end deadlines in particular applications, are critical concerns in MTC~\cite{biral2016impact}. In uMTC, the connection is supported by transferring the short data packets with ultra-high reliability and low latency, in the scope of less than a millisecond which can be a requirement in several applications such as cloud connectivity, road safety, industrial control and safe interconnection between vehicles~\cite{bockelmann2016massive},~\cite{7529226}. Therefore, owing to the vast applicability of MTC with short packets in cellular network infrastructure, covering a novel  wireless mode, namely ultra-high reliable communication (URC) is a critical concern in 5G~\cite{schotten2014availability}.
Supporting the ultra-high reliability and low latency are crucial requirements in the upcoming networks in order to improve the security, functionality and the ability of the interaction between different types of communication units such as human-to-human, human-to-machine or machine-to-machine which creates novel business models and applications~\cite{tullberg2016metis}.

In MTC with short packet transmissions, since the length of metadata and information bits are similar, an unsuccessful encoding of the metadata increases the performance loss in the wireless communication systems, particularly under the FB regime~\cite{7529226}. There are several works which have studied different aspects of FB coding. For instance, authors in~\cite{5452208}, provide a tight approximation of achievable coding rate for a specified outage probability under the FB transmissions since majority of the theoretical results assume infinite blocklength (IFB) codes.
In~\cite{iscan2016comparison}, authors provide an overview of some coding schemes for FB which may be used in 5G. They indicate that the performance of wireless communications with short data packets considerably enhances via novel coding schemes with better minimum distance between the codewords; albeit, the decoders become computationally more sophisticated. Furthermore, authors in~\cite{park2012new}, study the impact of FB on the attainable coding rate. They propose a new power allocation strategy, called modified water-filling, over the AWGN channels.
They succeed to maximize the lower bound of the achievable coding rate in comparison to the conventional water-filling technique in FB regime
\subsection{Cooperative Relaying with Finite Blocklength}
In recent years, relaying has become an interesting and challenging research topic. The transmission from the source to the destination is improved with help of intermediate auxiliary nodes. Moreover, relaying is able to combat the wireless fading generated due to the multipath propagation through taking advantage of spatial diversity~\cite{zimmermann2005performance}. The most usual relaying protocol is decode-and-forward (DF), where the relay decodes, encodes and retransmits the message~\cite{zimmermann2005performance}.

Authors in~\cite{hu2015performance} investigate the performance of a two-phase relaying model under the FB and IFB regimes. They consider perfect CSI at the destination where the channel gains of the relaying link and the direct link are combined. They propose different schemes based on the different error scenarios where relaying has a performance advantage over the direct transmission (DT) under the FB regime, particularly when the blocklength is small. Authors in~\cite{swamy2015cooperative} propose cooperative relaying as a way to meet the high reliability and latency requirements.
In~\cite{7569613}, authors examine the performance of a multi-relay DF protocol with FB under the perfect CSI and partial CSI.
They indicate that with perfect CSI, the throughput of FB coding is higher than the throughput of IFB coding.
Furthermore, in our previous work~\cite{parisa2017}, we consider cooperative relaying scenarios with perfect CSI for Rayleigh fading channels. We illustrate how the reliability improves via relaying and how we can meet the URC requirements. We examine the probability of successful transmission as a function of the number of information bits and coded blocklength. Moreover, we provide an approximation to the outage probability in closed form. We show that relaying consumes less transmit power compared to DT in order to enable URC with FB coding.
\subsection{Our Contribution}
In this paper, we further investigate the reliability of relaying under the FB regime. We study protocols, namely selection combining (SC) and maximum ratio combining (MRC), where relaying outperforms DT. Furthermore, we illustrate the superiority of MRC over SC in terms of power consumption, latency and reliability under the FB regime and, different from~\cite{hu2015performance}, we present closed form expressions for the outage probability.
The following are considered the contributions of this paper.
\begin{itemize}
	\item We provide the general expression of the outage probability for each relaying protocol considered in this work and incremental decode-and-forward relaying under general Nakagami-$m$ fading. Therefore, we generalize our previous work in~\cite{parisa2017}.
	\item We extend the work in~\cite{hu2016blocklength} by comparing the MRC scenario with SC relaying. Also, a generalized closed-form expression for the outage probability is attained, thus extending \cite{6888474}.
	\item We show the trade-off between information payload and blocklength under the UR region as a function of the quality of the link, which is reflected by the $m$ parameter of the Nakagami-$m$ distribution. We examine the minimum delay in relaying schemes required to perform in the UR region.
\end{itemize}

{\textbf{Notation}:} Throughout this paper, $f_{W}(\cdot)$ and $F_{W}(\cdot)$ are the probability density function (PDF) and cumulative distribution function (CDF) of a random variable  (RV) W, respectively. $\operatorname{Q}^{-1}(\cdot)$ represents the inverse of the $\operatorname{Q}$-function which is defined as  $\operatorname{Q(w)} = \int_{w}^{\infty} \tfrac{1}{\sqrt{2\pi}}\operatorname{e}^{-t^2/2}dt$~\cite[\S F.2]{miller2012probability}. The outage probability is denoted by  $\epsilon$ and $\operatorname{E[\cdot]}$ is the expectation. $\Gamma(\cdot)$, $\Gamma(\cdot,\cdot)$ and $\operatorname {\cal P}(\cdot,\cdot)$ are the Gamma function~\cite[\S 6.1.1]{abramowitz1964handbook}, incomplete Gamma function~\cite[\S 6.5.2]{abramowitz1964handbook} and Gamma regularized function~\cite[\S 6.5.1]{abramowitz1964handbook}, respectively. $_1F_{1}(a;b;z)$ denotes the regularized hypergeometric function~\cite[\S 15.1.1]{abramowitz1964handbook}.
\section{System Model} \label{sc:system_model}
Consider a source $S$, a destination $D$ and a relay $R$ as illustrated in Fig.~\ref{fig:System Model}. We consider a normalized distance between $S$ and $D$ as $d=1$m, and that $R$ can move in a straight line between $S$ and $D$, while the distance between $D$ and $R$ is denoted by $\beta$. The links denoted as $X$, $Y$ and $Z$ represent the $S$-$R$, $R$-$D$, and $S$-$D$ links respectively, and each transmission takes $n_i$ channel uses where $i \in \{S, R\}$. This means that $n_S$ channel uses for $S$ and $n_R$ channel uses for $R$, respectively. In this scenario, first $S$ sends data to $D$ and $R$ which is known as broadcasting phase. Thereafter, in the relaying phase, $R$ forwards to $D$ only if it decoded the message correctly~\cite{hu2015performance},~\cite{7569613}.
The received signals are written as $y_{1}\!=\!
h_{1}x + w_{1}$ and $y_{2}\!=\! h_{2}x + w_{2}$ at $D$ and $R$ respectively, and if $R$ participates, the received signal at $D$ is $y_{3}\!=\! h_{3}x + w_{3}$, where $x$ is the transmitted signal with power $P$ and $w_i$ is the AWGN noise vector with power $N_0\!=\!1$ where $i\in \{X,Y,Z\}$. Nakagami-$m$ quasi static fading channels in the $S$-$D$, $S$-$R$ and $R$-$D$ links are denoted as $h_1$, $h_2$, and $h_3$, respectively.
\begin{figure}[b!]
	\centering
	\includegraphics[width=\columnwidth]{SystemModel2.pdf}
	\caption{System model for the relaying scenario with a source ($S$), destination ($D$) and a relay ($R$). The links between $S$ to $D$, $S$ to $R$ and $R$ to $D$ are referred as the direct link $h_{1}$, the backhaul link $h_{2}$ and the relaying link $h_{3}$ each of which with $n_{i \in \{S, R\}}$ channel uses respectively.}
	\label{fig:System Model}
\end{figure}
In a DF-based cooperative transmission, the instantaneous SNR for each link depends on the total power constraint $P=P_S+P_R=\eta P+ (1-\eta)P$ which is given by $\Omega_Z \!=\!\eta P|h_{1}|^2/N_{0}$, $\Omega_X \!=\!\eta P|h_{2}|^2/N_{0}$ and $\Omega_Y \!=\! (1-\eta) P|h_{3}|^2/N_{0}$, where $0 <\eta\leqslant1$ and the average SNR in each link is $\gamma_Z =\eta P / N_{0}$, $\gamma_X =\eta P / N_{0}$ and $\gamma_Y = (1-\eta) P / N_{0}$. We consider $\eta$ as the power allocation factor in order to provide a fair comparison between DT and cooperative schemes. Moreover, note that the PDF of Nakagami-$m$ fading is a function of two parameters: the fading parameter $m$ and the scale parameter $\Omega> 0$~\cite{simon2005digital}. The squared envelop is then Gamma distributed as $X \thicksim \mathsf{Gamma}(m_{i} , \Omega_{i} /m_{i} ) $ where $i\in \{X,Y,Z\}$.

\subsection{Coding Rate and Error Probability at Finite Blocklength } \label{sc:Finite Block length}
In this section,  we revisit the concept of coding rate under the FB regime. In a communication system, first $k$ information bits are mapped to a sequence known as the codeword with a blocklength of $n$ symbols. Thereupon, the generated codeword is sent through the wireless channel. Subsequently, the decoder maps the channel outputs into an estimate of the information bits. Therefore, we can define the coding rate ${\cal R}$ as the ratio of the information bits $k$ to the number of channel symbols $n$ as ${\cal R} \!=\! k/n$~\cite{7529226}. The maximum coding rate ${\cal R}^*(n,\epsilon)$ in bits per channel use (bpcu) is~\cite{7529226}
\begin{equation} \label{eq:maximum rate}
{\cal R}^*(n,\epsilon) = C(\rho) - \sqrt{\frac {V(\rho)}{n}}\operatorname{Q}^{-1}(\epsilon)\log_{2}\operatorname{e},
\end{equation}
where $C$ and $V$ are the positive channel capacity and the channel dispersion, respectively, defined as $C(\rho) = \log_{2}(1+\rho)$ and $V(\rho) = \rho (2+\rho) \big/(1+\rho)^2$, where $\rho$ is the average SNR as $P/N_0$.
For the AWGN channel, $h_{i}=1$ and $\frac{1}{n}\sum_{i}^{n}|x_{i}|^2\leq\rho$ holds~\cite{7529226}.
\begin{figure*}[!t]
	\begin{equation}\label{eq:outage_naka}
	\epsilon\!=\! \dfrac{\mu\bigg[\theta\left(\Gamma\left(m,\dfrac{m\varrho}{\Omega}\right)\!-\!\Gamma\left(m,\dfrac{m\vartheta}{\Omega}\right)\right)\!+\!\Omega \bigg(\Gamma\left(1\!+\!m,\dfrac{m\vartheta}{\Omega}\right)
		\!-\!\Gamma\bigg(1\!+\!m,\dfrac{m\varrho}{\Omega}\bigg)\bigg)\bigg]}{\sqrt{2\pi}}\!+\!\dfrac{\operatorname{\cal P}\left(m,\dfrac{m\vartheta}{\Omega}\right)\!+\!\operatorname{\cal P}\left(m,\dfrac{m\varrho}{\Omega}\right)}{2}.
	\end{equation}
	\hrule
\end{figure*}

The outage probability is then the expectation over the instantaneous SNR distribution as follows\footnote{This approximation is accurate for $n$ > 100, as proved for AWGN channels \cite[Figs. 12 and 13]{polyanskiy2010channel}, as well as for fading channels~\cite{yang2014quasi}.}~\cite{hu2016blocklength}
\begin{equation} \label{outage_fading}
\epsilon\approx \operatorname{E}\Bigg[\operatorname{Q}\Bigg(\sqrt{n}\frac{C(\rho|h|^2)-{\cal R}^{*}(n,\epsilon) }{\sqrt{V(\rho|h|^2)}}\Bigg)\Bigg].
\end{equation}
\subsection{Closed-form Expression of the Outage Probability} \label{Q-function}
The outage probability in (\ref{outage_fading}) does not have a closed-form expression, but it can be tightly approximated as we shall see next.
Let us first define $g(x)\!=\!\sqrt{n}\tfrac{C(\rho)-\cal R}{\sqrt{V(\rho)}}$. Then, we resort to a linearization of the Q-function~\cite{7106474},~\cite{6888474}.
\begin{equation}\label{eq:W(t)}
K(x) \approx \operatorname{Q}(g(x)) =
\left\{
\begin{array}{@{}ll@{}}
\ 1 & x\leqslant\varrho\\
\ \dfrac{1}{2}-\dfrac{\mu}{\sqrt{2\pi}}(x-\theta) & \varrho<x< \vartheta  \;\;\;\;,\\
\ 0 & x\geq \vartheta
\end{array}\right.
\end{equation}
where, $\theta = \tfrac{2^{\cal R}-1}{P}$, $\vartheta=\theta+\sqrt{\tfrac{\pi}{2}\mu^{-2}}$, $\varrho=\theta-\sqrt{\tfrac{\pi}{2}\mu^{-2}}$ and $\mu\!=\!\sqrt{\frac{n}{2\pi}}(e^{2\cal R}-1)^{-\tfrac{1}{2}}$.
Therefore, the outage probability in (\ref{outage_fading}) is reformulated as
\begin{equation}\label{outage_fading_linear}
\epsilon= \operatorname{E}[\operatorname{Q}\left(g(x)\right)] = \int_{0}^{\infty}K(x)f_{X}(x)dx,
\end{equation}
where $f_{X}(x)$ represents the PDF of the SNR of link $X$.
\begin{proposition}\label{propose1}
The approximated outage probability of a communication link following Nakagami-$m$ fading is given in closed-form in (\ref{eq:outage_naka}) on top of the next page, where $\varrho$, $\vartheta$, $\mu$ and $\theta$ are defined in (\ref{eq:W(t)}).
\end{proposition}
\begin{proof}
	See Appendix A.
\end{proof}
\begin{corollary}
	Notice that for $m=1$, (\ref{eq:outage_naka}) reduces to the approximated outage probability for the Rayleigh fading as ~\cite[\S 5]{parisa2017}.
\end{corollary}
\noindent Thus, our proposed approximation in (\ref{eq:outage_naka}) includes~\cite[\S 5]{parisa2017}, which was first reported in~\cite{6888474}, as a special case.
\section{Performance Analysis of Relaying}\label{sec:performance}
In this section, we examine the outage probability in all relaying schemes considered in this work, as well as for the case of DT which is used for comparison purposes.
\subsection{Cooperative Transmission} \label{sec:Cooperative schemes}
\subsubsection{Selection Combining (SC)}
In this protocol $D$ tries to decode the messages sent by both $S$ and $R$, separately.
The overall outage probability becomes~\cite{5956530},~\cite{alves2012throughput}
\begin{equation}
\epsilon_{SC}=\epsilon_{Z}\left(\epsilon_{X}+\left(1-\epsilon_{X}\right)\epsilon_{Y}\right),
\end{equation}
where $\epsilon_{Z}$ , $\epsilon_{X}$ and $\epsilon_{Y}$ are the outage probabilities of the $S$-$D$, $S$-$R$ and $R$-$D$ links, respectively. The outages $\epsilon_{Z}$, $\epsilon_{X}$ and $\epsilon_{Y}$ are equal to (\ref{eq:outage_naka}) with $m$ and $\Omega$ replaced by $m_{Z}$ and $\Omega_{Z}$, $m_{X}$ and $\Omega_{X}$, and $m_{Y}$ and $\Omega_{Y}$ for each link, respectively.
\subsubsection{Maximum Ratio Combining (MRC)}\label{sec:MRC}
In this protocol, the transmissions from the source and from the relay are coherently combined at the receiver. Hence, the instantaneous SNR after the source and relay transmissions is $\Omega_{W} = \Omega_{Z} + \Omega_{Y}$~\cite{5956530},~\cite{alves2012throughput}. The  outage probability is~\cite{alves2012throughput}
\begin{align}\label{eq:MRC_outage}
\epsilon_{MRC} =\epsilon_{Z}\left(\epsilon_{X}\!+\!\left(1\!-\!\epsilon_{X}\right)\frac{\epsilon_{SRD}}{\epsilon_{Z}}\right),
\end{align}
where $\epsilon_{SRD}$ is the outage probability after MRC of the transmissions from the source to the destination. Moreover, the ratio $\tfrac{\epsilon_{SRD}}{\epsilon_{Z}}$ comes from the conditioning of $\epsilon_{SRD}$ on the fact that the transmission from the source to the destination failed.
In order to calculate (\ref{eq:MRC_outage}), first we need to define the PDF of $W$, a Gamma random variable equal to $Z + Y$ , the sum of the instantaneous SNRs of the $S$-$D$ and $R$-$D$ links, as follows.
PDF of random variable $W$ is
\begin{equation}\label{pdf_mrc}
\begin{split}
f_{W}(w)&=\operatorname{exp}\left(-\dfrac{w}{\Omega_{y}}\right)w^{-1+m_{z}+m_{y}}\Omega_{z}^{-m_{z}}\Omega_{y}^{-m_{y}}\\ &\times _1F_{1}\bigg(m_{z},m_{z}+m_{y},w\left(\dfrac{1}{\Omega_{y}}-\dfrac{1}{\Omega_{z}}\right)\bigg),
\end{split}
\end{equation}
{\noindent where $\Omega_{Z}$, $\Omega_{Y}$  and $m_{Z}$, $m_{Y}$ are the instantaneous SNRs and the fading severity of the $S$-$D$ and $R$-$D$ links, respectively.}
\begin{proposition}
The outage probability after combining the source and relay transmissions, $\epsilon_{SRD}$, is
\begin{equation}\label{mrc_SRD}
\begin{split}	\epsilon_{SRD}&=\int_{0}^{\varrho}f_W{\left(w|m,\Omega\right)}dw+\int_{\varrho}^{\vartheta}\left(\tfrac{1}{2}\!-\!\tfrac{\mu}{\sqrt{2\pi}}(w\!-\!\theta)\right)\\
&\times f_W{\left(w|m,\Omega\right)}dw.
\end{split}
\end{equation}
where, $\vartheta$ and $\varrho$ are specified in (\ref{eq:W(t)})
\begin{proof}
By plugging (\ref{pdf_mrc}) into (\ref{outage_fading_linear}) and multiplying with $K(x)$, we attain (\ref{mrc_SRD}). Solving (\ref{mrc_SRD}) is difficult and it does not have a closed form expression. Hence, we only work with its numerical integration whenever $m>1$.
\end{proof}
\end{proposition}
Since (\ref{mrc_SRD}) does not have a closed form expression for the outage probability under general Nakagami-$m$, we analyze the particular case when $m=1$, which corresponds to Rayleigh fading. Therefore, the PDF of $W$ becomes~\cite[\S 10]{parisa2017}.

\begin{lemma}\label{proposition2}
	For $m=1$, the outage probability, $\epsilon_{SRD}$, is equal to~\cite[\S 6]{parisa2017}.
\end{lemma}
\begin{proof}
	The proof can be found in~\cite{parisa2017}.
\end{proof}
\subsection{Direct Transmission (DT)}
In DT, the source sends the data packets directly to the destination. Since the channel coefficients are Nakagami-$m$ distributed, the instantaneous  SNR of DT scheme is $\Gamma_{Z} \thicksim \mathsf{Gamma}(m_{Z}, \Omega_{Z} /m_{Z})$. Then, the outage probability of DT is calculated according to (\ref{eq:outage_naka}), where $m = m_{Z}$, $\Omega_{Z} = P_S/ N_0 $, and is denoted as $\epsilon_{DT}$.
\section{Numerical Results}\label{sc:result}
In this paper we study the performance of incremental relaying protocols under the FB regime for the Nakagami-$m$ fading. First, we compare the cooperative schemes with DT as a function blocklength and transmit power. We show that the cooperative relaying protocols outperform the DT in terms of transmit power and the blocklength under the UR region.
The accuracy of our analytical model is proved via the numerical results. Unless stated otherwise, we assume $n=500$ where $n_{S}=n_{R}=n$, $k=250$, average SNR as $10$ dB and that $R$ is exactly midway between $S$ and $D$, with $\beta=\tfrac{1}{2}$.
\begin{figure}[!t]
	\centering
	\includegraphics[width=\columnwidth]{1-c.pdf}
	\vspace{-2mm}
	\caption{Outage probabilities for DT, SC, MRC with $k=250$ and average SNR as 10 dB.}\label{fig:outageVSblocklength}
\end{figure}
\subsection{On the impact of reliability improvement}
Fig.\ref{fig:outageVSblocklength} compares two distinct relaying protocols with DT in terms of the overall outage probability and fading severity. It can be clearly seen that MRC and SC outperform DT and perform closely in the entire range. In addition, higher values of $m$ results in lower outage
probability due to the improvement of LOS. Therefore, the availability of LOS should be taken into account when designing and deploying networks with stringent reliability requirements under the FB regime.
\begin{figure}[!htp]
	\centering
	\includegraphics[width=\columnwidth]{2-c.pdf}
	\vspace{-2mm}
	\caption{Outage probabilities for DT, SC, MRC with $k=250$ and $n=500$. }\label{fig:outageVSpower}
\end{figure}
Fig.\ref{fig:outageVSpower} shows that the cooperative protocols outperform DT in terms of transmit power consumption. The diversity gain achieved by the relaying schemes is more evident at high SNR and MRC performs better than SC. Hence, URC becomes feasible in case of less severe fading (larger $m$) and/or by
utilizing cooperative schemes. Note also that the analytical and numerical results match very well.
\begin{figure*}[!t]
	\centering
	\includegraphics[width=2\columnwidth,height=1.9in]{3-c.pdf}
	\vspace{0mm}
	\caption{Probability of successful transmission in SC and MRC protocols as a function of the blocklength $n$ and payload $k$ with average SNR as 10 dB.}\label{fig:contour}
\end{figure*}
\begin{figure}[!t]
	\centering
	\includegraphics[width=\columnwidth]{4-c.pdf}
	\vspace{-2mm}
	\caption{Impact of power allocation factor $\eta$ on the outage probability of SC and MRC with different blocklengths $n$. We consider $m=2$, $k=250$ and average SNR as $10$ dB. The points marked in red are the minimum outage probabilities of SC and MRC.}\label{fig:powerallocation}
\end{figure}
\subsection{URC requirements}
In Fig.\ref{fig:contour}, we examine the impact of coding rate on the performance of SC and MRC protocols with different values of $m$ under the UR region. We illustrate that MRC protocol outperforms SC under th FB regime. For instance, with $m=1$ and $n=300$, MRC provides $99.99 \%$ reliability with $k=45$, while SC supports equal reliability with $k=40$. Hence, SC is more affected by the coding rate growth compared to the MRC protocol. Moreover, we can clearly see that improving the LOS (larger $m$), considerably improves the reliability in wireless communications; thus, the size of the payload is not a critical concern anymore.

In Fig.\ref{fig:powerallocation}, we show the effect of power allocation factor on the outage probability. The optimal value of $\eta$, where the outage probability is minimized with different blocklengths, is approximately $0.62$ and $0.9$ for SC and MRC, respectively. From the figure, we see that in good LOS conditions more power is allocated to $S$, though $R$ is needed to provide the diversity order with respect to DT and thus achieve the required reliability target.

In the following figure, we compare the delay in relaying schemes with equal power allocation $\eta =0.5$ and optimal power allocation $\eta=0.62$ and $\eta=0.9$ for SC and MRC, respectively. The choice of the minimum delay $\delta$ is in such a way that minimizes the outage probability constrained to a specific interval of interest which gives the optimal values of $n_S$ and $n_R$, and is a nonlinear optimization problem. The optimization problem numerically is solved via the Matlab function $fmincon$\footnote{Interior point algorithm is used to solve the nonlinear optimization problem~\cite{waltz2006interior}.}. Delay $\delta$ is equal to the symbol time $T_s$ multiplied with blocklength $n$ plus the number of retransmissions. For example, in Fig.\ref{fig:totallatency}, where $\eta=0.5$, future releases of LTE foresee a minimum symbol period of $T_s = 8.33\mu$s~\cite{yilmazultra}  and regardless of the retransmissions, $99.99 \%$ reliability is feasible via MRC protocol with $\delta = 3.5 $ms while latency reduces to $\delta = 1.66$ms with $99.9 \%$ reliability. On the other hand, SC protocol provides $99.99 \%$ reliability with $\delta = 3.9$ms when $n_S = n_R$. Note that DT would not be able to cope with such stringent latency requirements. Therefore, we should consider a good trade-off between the reliability and latency in cooperative communication protocols.
Moreover, we indicate that applying the optimal values of $\eta$ shown in Fig.\ref{fig:powerallocation}, slightly reduce the latency in relaying schemes compared to the equal power allocation to $S$ and $R$ under the same assumptions; however, behavior of the protocols alters under the optimal $\eta$. Moreover, in SC with $n_S \neq n_R$ and $\eta = 0.5$, latency reduces in comparison to the case when $n_S = n_R$. For instance, when $n_s=227$ and $n_R=219$,  $99.99 \%$ reliability is feasible with $\delta=3.7$ms compared to the case when $n_S = n_R$. Hence, different parameters such as LOS, transmit power, coding rate of each link, delay and reliability should be all considered in designing the networks to support the URC requirements.
\section{Conclusions and Final Remarks} \label{sc:conc}
Herein we assess the relay communication under the finite blocklength regime. The performance of two relaying schemes, SC and MRC, are compared to the DT in terms of the outage probability under the Nakagami-$m$ fading distribution. We indicate that the relaying improves the communication efficiency which is more obvious at high SNR regime through providing higher order of diversity and consuming less transmit power. Moreover, we show the optimal number of channel uses for each link for all relaying schemes considered in this work in order to meet the latency constraint under the UR region. We show that lower coding rates, improve the reliability up to $99.999 \%$.
\begin{figure}[!t]
	\centering
	\includegraphics[width=\columnwidth, height =2in]{7-c.pdf}
	\vspace{-4mm}
	\caption{Total delay in terms of channel uses with equal power allocation ($\eta=0.5$) and $\eta=0.62$ and $\eta=0.9$ for SC and MRC, respectively with $m=2$, $k=250$ and average SNR as $10$ dB.}\label{fig:totallatency}
\end{figure}
Furthermore, we indicate that increasing the LOS, increases the reliability while the transmit power is kept low. In addition, we show that if $S$ and $R$  transmit with different coding rates in SC scenario, the latency reduces in comparison to the case of having equal coding rates for both direct and relay links.
We also prove the correctness of our analytical model via the numerical results.
All in all, we conclude that the relaying is a desirable technique in order to improve the system quality, particularly in M2M communications where low latency is a concern with short packet transmissions. In addition, relaying requires less transmit power to work under the UR region and increasing the LOS improves the reliability. More
over, MRC outperforms SC and DT in terms of power consumption and latency requirements. Finally, in our future work, we will work on the cooperative relaying under the imperfect CSI with FB coding.
\begin{figure*}[!t]
	\begin{equation}\label{eq:Outage_overall}
	\begin{split}
	\epsilon&\!=\!\dfrac{F_X{\left(\vartheta,m,\Omega\right)}\!+\!F_X{\left(\varrho,m,\Omega\right)}}{2}\!+\!\dfrac{\mu\theta\bigg(F_X{\left(\vartheta,m,\Omega\right)}\!-\!F_X{\left(\varrho,m,\Omega\right)}\bigg)}{\sqrt{2\pi}}\!-\!\dfrac{\mu}{\sqrt{2\pi}}\bigg(\bigg(\bigg(\dfrac{1}{\Omega}\bigg)^{-m} \Omega^{1\!-\!m}\bigg(\Gamma\left(1\!+\!m,\dfrac{m\varrho}{\Omega}\right)\\
	&\!-\!\Gamma\left(1\!+\!m, \dfrac{m\vartheta}{\Omega}\right)\bigg)\bigg)\bigg/ \Gamma\left(1\!+\!m\right)\bigg)\!=\!\dfrac{1}{2}\bigg(\operatorname{\cal P}\left(m,\dfrac{m\vartheta}{\Omega}\right)\!+\!\operatorname{\cal P}\left(m,\dfrac{m\varrho}{\Omega}\right)\bigg)\!+\!\dfrac{\mu m \theta}{\sqrt{2\pi}\Gamma\left(1\!+\!m\right)}\bigg(\Gamma\left(m,\dfrac{m\varrho}{\Omega}\right)\\
	&\!-\!\Gamma\left(m,\dfrac{m\vartheta}{\Omega}\right)\bigg)\!+\!\dfrac{\mu\Omega}{\sqrt{2\pi}\Gamma(1\!+\!m)}\bigg(\Gamma\left(1\!+\!m,\dfrac{m\vartheta}{\Omega}\right)\!-\!\Gamma\left(1\!+\!m,\dfrac{m\varrho}{\Omega}\right)\bigg)\!=\!\dfrac{\mu}{\sqrt{2\pi}}\bigg[\theta\bigg(\Gamma\left(m,\dfrac{m\varrho}{\Omega}\right)\!-\!\Gamma\left(m,\dfrac{m\vartheta}{\Omega}\right)\bigg)\\
	&\!+\!\Omega \bigg(\Gamma\left(1\!+\!m,\dfrac{m\vartheta}{\Omega}\right)\!-\!\Gamma\left(1\!+\!m,\dfrac{m\varrho}{\Omega}\right)\bigg)\bigg]\!+\!\dfrac{1}{2}\bigg(\operatorname{\cal P}\left(m,\dfrac{m\vartheta}{\Omega}\right)\!+\!\operatorname{\cal P}\left(m,\dfrac{m\varrho}{\Omega}\right)\bigg).\\
	\end{split}
	\end{equation}
	\hrule
\end{figure*}
\section*{Appendix A}
\section*{Proof of the Proposition~\ref{propose1}}
\begin{proof}
	Let the PDF of the instantaneous SNR of the link denoted by the random variable $X \thicksim \mathsf{Gamma}(m,\Omega/m)$ be
	%
	\begin{align}\label{eq:pdf_gamma}
	f_X{(x|m,\tfrac{\Omega}{m})} =(\frac{\Omega}{m})^{-m}\frac{\exp(-\frac{mx}{\Omega})x^{m-1}}{\Gamma(m)} &\quad x>0
	\end{align}
	%
	then, plugging (\ref{eq:W(t)}) and (\ref{eq:pdf_gamma}) into (\ref{outage_fading_linear}), we attain the outage probability equal to
	\begin{equation}
	\epsilon\!=\! \int_{0}^{\varrho}f_X{\left(x|m,\Omega\right)}dx\!+\!\int_{\varrho}^{\vartheta}\left(\tfrac{1}{2}\!-\!\tfrac{\mu}{\sqrt{2\pi}}(x\!-\!\theta)\right)f_X{\left(x|m,\Omega\right)}dx.
	\end{equation}
	After some algebraic manipulations we attain (\ref{eq:Outage_overall}) on top of the this page, where  $F_X(x;m,\Omega)\!=\!\operatorname{\cal P}(m,\tfrac{mx}{\Omega})$ and $\tfrac{m\mu\theta\Gamma(m,\tfrac{mx}{\Omega})}{\sqrt{2\pi}\Gamma\left(1\!+\!m\right)}$ can be simplified as $\tfrac{\mu\theta\Gamma\left(m,\tfrac{mx}{\Omega}\right)}{\sqrt{2\pi}\Gamma(m)}$.
\end{proof}

\section*{Acknowledgments}
This work is partially supported by Aka Project SAFE (Grant no. 303532), and by Finnish Funding Agency for Technology and Innovation (Tekes), Bittium Wireless, Keysight Technologies Finland, Kyynel, MediaTek Wireless, Nokia Solutions and Networks, and CNPq (Brazil).