Orthogonal polynomials in two variables and second-order partial differential equations
type: Item
instance of: scholarly article
title: Orthogonal polynomials in two variables and second-order partial differential equations
zbMATH Open document ID: 0889.33007
DOI: 10.1016/S0377-0427(97)00082-4
author: J. Ning
published in: Journal of Computational and Applied Mathematics
publication date: 1998-06-22T00:00:00Z
review text: The authors consider the second order partial differential equations of spectral type of the form \[ L[u]:= Au_{xx}+ 2Bu_{xy}+ Cu_{yy}+ Du_x+ Eu_y= \lambda_nu, \qquad n=0,1,2,\dots\;, \tag{\(*\)} \] where \(A(x,y),\dots, E(x,y)\) are polynomials, independent of \(n\), while \(\lambda_n\) is the eigenvalue parameter, and find a characterization of orthogonal polynomials satisfying it. In the particular case when \(A_y= C_x=0\), they characterize \((*)\) which have a product of two classical orthogonal polynomials in one variable as solutions. Furthermore, they give conditions under which derivatives of any orthogonal polynomial solution to \((*)\) are also orthogonal polynomials satisfying the same type of equations as \((*)\). Notice that some of the results are already known but the proofs given here by the authors, which are based on the use of formal functional calculus on movement functionals, are simpler than those given elsewhere [see for example: \textit{H. L. Krall} and \textit{I. M. Sheffer}, Ann. Math. Pura Appl., IV. Ser. 325-376 (1967; Zbl 0186.38602)].
Mathematics Subject Classification ID: 33C50
Mathematics Subject Classification ID: 35P99
zbMATH DE Number: 1084081
reviewed by: Mithat Idemen
MaRDI profile type: Publication
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full work available at URL: s0377-0427(97)00082-4
OpenAlex ID: W2000160788
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description: scientific article; zbMATH DE number 1084081