Fourth-order difference equation for the associated classical discrete orthogonal polynomials
type: Item
instance of: scholarly article
title: Fourth-order difference equation for the associated classical discrete orthogonal polynomials
zbMATH Open document ID: 0928.33006
DOI: 10.1016/S0377-0427(98)00050-8
author: W. Koepf
author: M. Foupouagnigni
author: A. Ronveaux
published in: Journal of Computational and Applied Mathematics
publication date: 2000-01-18T00:00:00Z
review text: Let \(\{P_n(x)\}_n\) be the sequence of the monic orthogonal polynomials satisfying the recurrence \(P_{n+1}(x)= (x-\beta_n)P_n(x)-\gamma_n P_{n-1}(x)\), \(n\geq 1\), \(\gamma\neq 0\), \(P_0(x)=1\), \(P_1(x)= x-\beta_0\). The associated orthogonal polynomials \(P_n^{(r)}(x)\) of order \(r\) are defined by the shifted recurrence obtained by substituting \(n\) with \(n+r\) in \(\beta\) and \(\gamma\). In this paper the authors give a fourth-order difference equation which holds for all integer \(r\) and for all classical discrete orthogonal polynomials. The coefficients of this equation are given in terms of the polynomials \(\sigma(x)\) and \(\tau(x)\) which appear in the second-order difference equation \(\sigma(x)\nabla \Delta y(x)+ \tau(x)\Delta y(x)+ \lambda_n y(x)=0\), with \(\Delta y(x)= y(x+1)-y(x)\), \(\nabla u(x)= y(x)-y(x-1)\) and \(2\lambda_n=- n[(n-1)\sigma'' +2\tau']\).
reviewed by: Luigi Gatteschi
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1326321
zbMATH Keywords: discrete orthogonal polynomials
zbMATH Keywords: difference equation
MaRDI profile type: Publication
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full work available at URL: s0377-0427(98)00050-8
OpenAlex ID: W2186919933
description: scientific article; zbMATH DE number 1326321