Differential operators having symmetric orthogonal polynomials as eigenfunctions
type: Item
instance of: scholarly article
title: Differential operators having symmetric orthogonal polynomials as eigenfunctions
zbMATH Open document ID: 0938.34074
DOI: 10.1016/S0377-0427(99)00094-1
author: H. Bavnick
author: J. Koekoek
published in: Journal of Computational and Applied Mathematics
publication date: 1999-08-22T00:00:00Z
review text: From authors' abstract: ``Let the polynomials \(\{P_n(x)\}^\infty_{n=0}\), orthogonal with respect to a symmetric positive definite moment functional \(\sigma\), be eigenfunctions of a linear differential operator \({\mathbf L}\). The authors consider the orthogonal polynomials \(\{P^\mu_n(x)\}^\infty_{n= 0}\) and \(\{P^{\mu,\nu}_n(x)\}^\infty_{n= 0}\), which are obtained by adding one resp. two symmetric (Sobolev type) terms to \(\sigma\). In all the cases they derive a representation for the polynomials and show that they are eigenfunctions of one or more linear differential operators (mostly of infinite order) of the form \({\mathbf L}+\mu{\mathbf A}\) resp. \({\mathbf L}+\mu{\mathbf A}+\nu{\mathbf B}+\mu\nu{\mathbf C}\). Further, it is investigated to what extend the eigenvalues can be chosen arbitrarily and finally expressions are given for the other eigenvalues''.
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 34A35
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1326541
zbMATH Keywords: differential operators
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: linear differential operators of infinite order
reviewed by: Ludwig Elsner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1326541