Laguerre-Sobolev orthogonal polynomials
type: Item
instance of: scholarly article
title: Laguerre-Sobolev orthogonal polynomials
zbMATH Open document ID: 0855.33015
DOI: 10.1016/0377-0427(95)00234-0
author: Miguel A. Piñar
author: Francisco Marcellán
author: Teresa E. Pérez
published in: Journal of Computational and Applied Mathematics
publication date: 1996-07-28T00:00:00Z
review text: The authors consider the polynomials \(Q_n^{(\alpha)} (x)\) \((n= 0, 1, \dots)\) which are orthogonal with respect to the inner product \[ (f, g)_S= \int_0^{+\infty} f(x) g(x) x^\alpha e^{-x} dx+ \lambda \int_0^{+\infty} f' (x) g' (x) x^\alpha e^{-x} dx, \tag \(*\) \] and satisfy the relations \[ (Q_n^{(\alpha)}, x^m)_S= 0, \qquad m=0, 1, \dots, n-1. \tag \(**\) \] Here one supposes \(\alpha> -1\), \(\lambda\geq 0\) and \(Q_0^{\alpha} (x)=1\). The case \(\lambda =0\) gives the classical Laguerre polynomials \(L_n ^{(\alpha)} (x)\). Their aim consists of deriving several relations which are satisfied by these polynomials themselves or connect these polynomials to the classical Laguerre polynomials (four-term recurrence relation, difference-differential relation, Christoffel-Darboux-type relation, Rodrigues-type relation, etc). The zeros of the polynomials \(Q_n^{(\alpha)} (x)\) and their localizations with respect to the zeros of the Laguerre polynomials are also studied. The paper involves 22 Lemmas, theorems, propositions and corollaries, and indicates some open problems.
Mathematics Subject Classification ID: 33E30
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 910315
zbMATH Keywords: difference-differential relation
zbMATH Keywords: Christoffel-Darboux-type relation
zbMATH Keywords: Rodrigues-type relation
zbMATH Keywords: Laguerre polynomials
reviewed by: Mithat Idemen
MaRDI profile type: Publication
full work available at URL: 0377-0427(95)00234-0
OpenAlex ID: W2005988701
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description: scientific article; zbMATH DE number 910315