Orthogonal polynomials satisfying higher-order difference equations
type: Item
instance of: scholarly article
title: Orthogonal polynomials satisfying higher-order difference equations
zbMATH Open document ID: 1264.42006
author: A. J. Durán
published in: Constructive Approximation
publication date: 2013-01-30T00:00:00Z
review text: The author constructs measures with orthogonal polynomials satisfying higher-order (\(>2\)) difference equations with coefficients independent of the degree of the polynomials. These measures are constructed by multiplying the discrete classical weights for the Charlier, Meixner, Krawtchouk and Hahn polynomials by certain well-chosen variants of annihilator polynomials of a finite set of numbers. This leads to some explicit examples of fourth-order difference equations satisfied by generalizations of the discrete classical orthogonal polynomials named after Charlier, Meixner, Krawtchouk and Hahn. Moreover, in the Charlier and Meixner cases, an explicit example is given of a sixth-order difference equation. The method leads to a number of conjectures. Only for the first conjecture (concerning the Meixner and the Hahn weight), a proof is presented for a special case of a generalization of the Meixner weight. More general proofs of this and other conjectures are announced to be given in (at least) two forthcoming papers.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33E30
zbMATH DE Number: 6132831
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: difference equations
zbMATH Keywords: difference operators
zbMATH Keywords: discrete classical polynomials
zbMATH Keywords: Charlier polynomials
zbMATH Keywords: Meixner polynomials
zbMATH Keywords: Krawtchouk polynomials
zbMATH Keywords: Hahn polynomials
reviewed by: Roelof Koekoek
MaRDI profile type: Publication
full work available at URL: s00365-012-9162-2
OpenAlex ID: W2059019091
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DOI: 10.1007/S00365-012-9162-2
description: scientific article; zbMATH DE number 6132831