Diagonal orthogonal polynomial sequences
type: Item
instance of: scholarly article
title: Diagonal orthogonal polynomial sequences
zbMATH Open document ID: 1025.42013
published in: Methods and Applications of Analysis
publication date: 2002-06-19T00:00:00Z
review text: Let \((p_n)_{n\geq 0}\) be a sequence of polynomials, \(\phi(x)\) a polynomial of degree \(t\) and \(s\geq 0\) an integer such that \[ \phi(x)p_n(x)=\sum_{k=n-s}^{n+t} \lambda_{n,k} p_{k+1}'(x),\quad n\geq 0, \] where \(\lambda_{n,k}\) are some constants such that \(\lambda_{n,n-s}\neq 0\) for \(n\geq s\). Let pose the following problem: To characterize all orthogonal polynomial sequences satisfying the above relation. The case \(\phi(x)=1\), i.e., \(t=0\), leads to the classical orthogonal polynomials (see \textit{F. Marcellán, A. Branquinho}, and \textit{J. Petronilho}, ``Classical orthogonal polynomials: A functional approach'', Acta Appl. Math. 34, 283-303 (1994; Zbl 0793.33009)]). The aim of this paper is the detailed study of the case \(t=1\) (\(\phi(x)=x+1\)). In fact, the authors find that the arising sequences are the classical forms and some perturbations of them (e.g., addition of delta Dirac measures, multiplication by first degree polynomials).
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 1757331
zbMATH Keywords: semiclassical orthogonal polynomials
zbMATH Keywords: coherent pairs
author: P. Maroni
author: Ridha Sfaxi
reviewed by: Renato Álvarez-Nodarse
MaRDI profile type: Publication
DOI: 10.4310/MAA.2000.V7.N4.A9
description: scientific article; zbMATH DE number 1757331