Differential equations having orthogonal polynomial solutions
type: Item
instance of: scholarly article
title: Differential equations having orthogonal polynomial solutions
zbMATH Open document ID: 0883.33004
DOI: 10.1016/S0377-0427(96)00096-9
author: O. Diekmann
published in: Journal of Computational and Applied Mathematics
publication date: 1998-02-23T00:00:00Z
review text: The authors study differential operators of the form \[ L_N[y]:=\sum_{i=1}^n\ell_i(x)y^{(i)}(x), \] with polynomial coefficients \(\ell_i(x)\). Necessary and sufficient conditions that the differential operator has as eigenfunctions an orthogonal polynomial sequence (i.e. \(\{P_n(x)\}\) with \(\text{deg } P_n=n\) and \(L_N[P_n(x)]=\lambda_n P_n(x)\), \(n\geq 0\)) were already given by \textit{S. Bochner} for \(N=2\) [Über Sturm-Liouvillesche Polynomsysteme, Math. Z. 29, 730-736 (1929; JFM 55.0260.01] and for general \(N\) by \textit{H. L. Krall} [Certain differential equations for Chebychev polynomials, Duke Math. J. 4, 705-718 (1938; Zbl 0020.02002)]. H. L. Krall also gave a complete classification in the case of \(N=4\). For \(N>4\) the complete classification of the appropriate differential operators still remains open, except for several `isolated' cases (see the references in the paper). The authors now obtain new (and simpler) necessary conditions for the existence of an orthogonal polynomial sequence and moreover give necessary and sufficient conditions on a distribution \(w(x)\) to be an orthogonalizing weight. Finally a connection between the existence of the polynomials and the symmetrizability of the underlying differential operator is looked into.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 34L05
zbMATH DE Number: 1034163
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: weight functions
zbMATH Keywords: symmetrizable differential operator
reviewed by: Marcel G. de Bruin
MaRDI profile type: Publication
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full work available at URL: s0377-0427(96)00096-9
OpenAlex ID: W2032473291
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description: scientific article; zbMATH DE number 1034163