Exceptional Gegenbauer polynomials via isospectral deformation

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 10 Jun 2022Embargo end date: 01 Jan 2021 Spain English Publisher:WileyJournal:Studies in Applied Mathematics, volume 149, pages 324-363 (issn: 0022-2526, eissn: 1467-9590,

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Authors: María Ángeles García‐Ferrero; David Gómez‐Ullate; Robert Milson; James Munday;

doi: 10.1111/sapm.12510 , 10.48550/arxiv.2110.04059

arXiv: 2110.04059

handle: 2445/193200 , 10498/27007

Exceptional Gegenbauer polynomials via isospectral deformation

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Abstract

AbstractIn this paper, we show how to construct exceptional orthogonal polynomials (XOP) using isospectral deformations of classical orthogonal polynomials. The construction is based on confluent Darboux transformations, where repeated factorizations at the same eigenvalue are allowed. These factorizations allow us to construct Sturm–Liouville problems with polynomial eigenfunctions that have an arbitrary number of real‐valued parameters. We illustrate this new construction by exhibiting the class of deformed Gegenbauer polynomials, which are XOP families that are isospectral deformations of classical Gegenbauer polynomials.

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Spain

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Keywords

Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc., Approximation theory, isospectral deformations, Sturm–Liouville problems, Hypergeometric functions, Polynomials, exceptional polynomials, Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Teoria de l'aproximació, Sturm-Liouville theory, Gegenbauer polynomials, confluent Darboux transformations, Mathematics - Classical Analysis and ODEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Polinomis, Funcions hipergeomètriques, Sturm-Liouville problems, 33C47, 34L10, 34A05

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