Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rodrigues' descendants of a polynomial and Boutroux curves   Motivated by the classical Rodrigues' formula, we study below the root
asymptotic of the polynomial sequence $$R_{[\alpha n],n,P}(z)=\frac{d^{[\alpha
n]}P^n(z)}{dz^{[\alpha n]}}, n= 0,1,\dots$$ where ${P(z)}$ is a fixed
univariate polynomial, $\alpha $ is a fixed positive number smaller than deg
$P$, and $[\alpha n]$ stands for the integer part of $\alpha n$.
  Our description of this asymptotic is expressed in terms of an explicit
harmonic function uniquely determined by the plane rational curve emerging from
the application of the saddle point method to the integral representation of
the latter polynomials using Cauchy's formula for higher derivatives. As a
consequence of our method, we conclude that this curve is birationally
equivalent to the zero locus of the bivariate algebraic equation satisfied by
the Cauchy transform of the asymptotic root-counting measure for the latter
polynomial sequence. We show that this harmonic function is also associated
with an abelian differential having only purely imaginary periods and the
latter plane curve belongs to the class of Boutroux curves initially introduced
by Bertola.
  As an additional relevant piece of information, we derive a linear ordinary
differential equation satisfied by $\{R_{[\alpha n],n,P}(z)\}$ as well as
higher derivatives of powers of more general functions.

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The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: bayes-an
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-th
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The categories of this paper are: solv-int
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The categories of this paper are: chem-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: stat
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The categories of this paper are: hep-ph
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The categories of this paper are: funct-an
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The categories of this paper are: quant-ph
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The categories of this paper are: cs
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The categories of this paper are: gr-qc
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The categories of this paper are: patt-sol