Some classical polynomials seen from another side
type: Item
instance of: scholarly article
title: Some classical polynomials seen from another side
zbMATH Open document ID: 0934.42015
DOI: 10.1023/A:1004705612927
author: J. Alaya
author: M. J. Atia
published in: Periodica Mathematica Hungarica
publication date: 2000-02-21T00:00:00Z
review text: One characterization of the (very) classical orthogonal polynomials is that besides the sequence \(\{P_n(x)\}_{n\geq 0}\) also the sequence \(\{\frac{P_{n+1}'(x)}{n+1}\}_{n\geq 0}\) is orthogonal. The aim of this paper is to study the sequences \(\{P_n(x)\}_{n\geq 0}\) of orthogonal polynomials for which the sequence \(\{\frac{((x-c)P_n(x))'}{n+1}\}_{n\geq 0}\) is also orthogonal. The result is that these sequences have to be (very) classical, in fact they have to be Laguerre or Jacobi polynomials, where in the case of Laguerre \(c\) has to be zero and in the case of Jacobi \(c\) has to be \(\pm 1\). This result is achieved by considering the Hermite, Laguerre, Bessel and Jacobi polynomials successively.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1404459
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: classical polynomials
reviewed by: Roelof Koekoek
MaRDI profile type: Publication
full work available at URL: a:1004705612927
OpenAlex ID: W1879076357
description: scientific article; zbMATH DE number 1404459