Umbral ``classical'' polynomials
type: Item
instance of: scholarly article
title: Umbral ``classical'' polynomials
zbMATH Open document ID: 1305.33024
author: Alexei Zhedanov
published in: Journal of Mathematical Analysis and Applications
publication date: 2014-08-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1403.4014
reviewed by: Andrei A. Kapaev
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33C47
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 05A40
zbMATH DE Number: 6334362
zbMATH Keywords: umbral classical polynomials
zbMATH Keywords: Hahn theorem
zbMATH Keywords: Weierstraß\ sigma function
MaRDI profile type: Publication
OpenAlex ID: W2962679231
arXiv ID: 1403.4014
cites work: Q3483793
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cites work: The umbral calculus and orthogonal polynomials
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cites work: On pairs of related orthogonal polynomial sets
cites work: Generalized Hahn's theorem
cites work: Q4438874
cites work: THE KRALL POLYNOMIALS: A NEW CLASS OF ORTHOGONAL POLYNOMIALS
cites work: On the classification of differential equations having orthogonal polynomial solutions
cites work: Q4853625
cites work: Q3997785
cites work: The theory of the umbral calculus. I
cites work: Essays on the theory of elliptic hypergeometric functions
cites work: Elliptic hypergeometric Laurent biorthogonal polynomials with a dense point spectrum on the unit circle
cites work: A ‘missing’ family of classical orthogonal polynomials
cites work: Rational spectral transformations and orthogonal polynomials
DOI: 10.1016/J.JMAA.2014.06.002
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review text: In the mid of 30s of the last century, W.~Hahn classified the orthogonal polynomials whose derivatives or \(q\)-derivatives are also orthogonal. The author of the present paper generalizes Hahn's problem considering the monic polynomials \(P_n(x)\) and their generalized derivatives \(Q_n(x)=\mu_{n+1}^{-1}{\mathcal D}P_{n+1}(x)\), where \({\mathcal D}x^n=\mu_nx^{n-1}\). He finds necessary and sufficient conditions that \(\{P_n(x)\}_{n=0}^{\infty}\) and \(\{Q_n(x)\}_{n=0}^{\infty}\) are both orthogonal with respect to some linear functionals \(\sigma\) and \(\tau\).NEWLINENEWLINEThe key ingredient of the author's approach is the so-called reciprocal operator \({\mathcal R}\) such that \({\mathcal R}Q_n(x)=\nu_{n+1}P_{n+1}(x)\). The necessary and sufficient condition of the author's interest has the form of an infinite system of algebraic equations involving the moments \(g_n=\langle\sigma,x^n\rangle\), \(\tilde g_n=\langle\tau,x^n\rangle\), the defining coefficients \(\mu_n\) and \(\nu_n\) and the matrix elements of the operator \({\mathcal R}\) in the monomial basis. The corresponding orthogonal polynomials appear as the solutions of the eigenvalue problems \(LP_n(x)=\lambda_nP_n(x)\), \(\tilde LQ_n(x)=\lambda_{n+1}Q_n(x)\) for the operators \(L={\mathcal R}{\mathcal D}\) and \(\tilde L={\mathcal D}{\mathcal R}\).NEWLINENEWLINEThe proposed approach is illustrated by two important particular cases. The first case corresponds to the finite-diagonal matrix of \({\mathcal R}\) when both \({\mathcal D}\) and \({\mathcal R}\) become linear combinations of \(q\)-difference and differential operators, moreover in the simplest two-diagonal case, the author reproduces Hahn's result. The second illustrative situation corresponds to the complete degeneration of the operator \(L\) (\(\lambda_1=\lambda_2=\dots=1\)). Now, assuming that \(\mu_n=\mu(n)\), \(g_n=g(n)\) are analytic in \(n\), the author finds these coefficients and orthogonal polynomials \(P_n(x)\), \(Q_n(x)\) in terms…
description: scientific article; zbMATH DE number 6334362