Generating functions on extended Jacobi polynomials from Lie group view point
type: Item
instance of: scholarly article
title: Generating functions on extended Jacobi polynomials from Lie group view point
zbMATH Open document ID: 0862.33012
DOI: 10.5565/PUBLMAT_40196_01
author: M. C. Mukherjee
published in: Publicacions Matemàtiques
publication date: 1997-06-02T00:00:00Z
full work available at URL: https://eudml.org/doc/41257
review text: Let \(_2F_1(a,b;c,t)\) be the Gauss hypergeometries series. The extended Jacobi polynomials in this paper are defined by \(F_n(\alpha, \beta;x) =_2F_1 (-n,n+ \alpha+ \beta+1, \alpha+1; (x-a)/(b-a))\). The author introduces a Lie algebra of first order differential operators and calculates their exponential actions on the extended Jacobi polynomials. The author thus derives various generating functions of the polynomials. As some special and limiting cases the author recovers the generating functions of Laguerre, Hermite and Bessel polynomials. The differential operators in this paper are essentially the hypergeometric shift operators, which have been studied for Jacobi polynomials of several variables by \textit{G. J. Heckman} and \textit{E. M. Opdam} [An elementary approach to the hypergeometric shift operators of Opdam, Invent. Math. 103, 341-350 (1991; Zbl 0721.33009)].
reviewed by: Genkai Zhang
Mathematics Subject Classification ID: 33C80
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 911922
zbMATH Keywords: generating functions
zbMATH Keywords: Jacobi polynomials
zbMATH Keywords: Hermite polynomials
zbMATH Keywords: Laguerre polynomials
zbMATH Keywords: Bessel polynomials
MaRDI profile type: Publication
Wikidata QID: Q115159261
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description: scientific article; zbMATH DE number 911922