Classical orthogonal polynomials: A functional approach
type: Item
instance of: scholarly article
title: Classical orthogonal polynomials: A functional approach
zbMATH Open document ID: 0793.33009
DOI: 10.1007/BF00998681
author: J. Petronilho
author: Francisco Marcellán
author: A. Branquinho
published in: Acta Applicandae Mathematicae
publication date: 1994-08-16T00:00:00Z
review text: The Rodriguez formula for the classical orthogonal polynomials (here Hermite, Laguerre, Jacobi and Bessel) leads to a distributional differential equation for the corresponding moment functional \(u\) of the form \(D(\Phi u)=\Psi u\) with polynomials \(\Phi,\Psi\) and \(\deg \Phi \leq 2\), \(\deg \Psi=1\). Moreover, the moment functionals of the classical orthogonal polynomials are characterized by this formula. In the present paper, the authors use this characterization and derived further characterizations for the classical orthogonal polynomials. Some of these characterizations are known, but the new proofs given here are quite natural. Moreover, the approach of this paper leads to some new characterizations. One of them is as follows: Let \((P_ n)_{n \geq 0}\) be a system of monic orthogonal polynomials with moment functional \(u\). Then this system is classical if and only if \(P_ n=Q_ n+r_ n Q_{n-1}+s_ n Q_{n-2}\) \((n \geq 2)\) with \(Q_ n=P_{n+1}' /(n+1)\) for suitable constants \(r_ n,s_ n\).
reviewed by: Michael Voit
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33D70
zbMATH DE Number: 572240
zbMATH Keywords: characterization by distributional
zbMATH Keywords: differential equation
zbMATH Keywords: classical orthogonal polynomials
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00998681
OpenAlex ID: W1984567188
description: scientific article; zbMATH DE number 572240