A distributional study of discrete classical orthogonal polynomials
type: Item
instance of: scholarly article
title: A distributional study of discrete classical orthogonal polynomials
zbMATH Open document ID: 0853.33009
DOI: 10.1016/0377-0427(93)E0241-D
author: L. Salto
author: A. G. Garcia
author: Francisco Marcellán
published in: Journal of Computational and Applied Mathematics
publication date: 1996-12-16T00:00:00Z
review text: This paper is a systematic study of discrete classical orthogonal polynomial sequences (Charlier, Meixner, Hahn) from a distributional viewpoint. The properties for the continuous case orthogonal polynomial sequences (denoted here by DOPS; Hermite, Laguerre, Jacobi, Bessel) can a.o. be derived from a distributional differential equation \(D (\varphi u) = \psi u\) for the associated linear functional \(u\) (cf. \textit{F. Marcellán}, \textit{A. Branquinho} and \textit{J. Petronilho} [Acta Appl. Math. 34, 283-303 (1994; Zbl 0793.33009)]). The authors replace the differential operator \(D\) and the word `differential' by the difference operator \(\Delta\) and the word `difference' and derive properties analogous to those in the continuous case (Sturm-Liouville differential equation of order 2, \(D(P_n)\) again is a DOPS, Rodrigues formula, structure relation, Pearson differential equation for the associated weight). As stated in the opening sentence of this review the work is done systematically: in a (large) number of small steps (4 definitions, 10 lemmas, 9 propositions, 1 theorem and several remarks) the results are derived. A well written paper, suitable for inclusion in a (graduate) textbook on orthogonal polynomials.
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 770342
zbMATH Keywords: difference equations
zbMATH Keywords: moment functionals
MaRDI profile type: Publication
cites work: ON A CHARACTERIZATION OF MEIXNER'S POLYNOMIALS
cites work: Q4170993
cites work: Über Orthogonalpolynome, die q‐Differenzengleichungen genügen
cites work: Systems of Polynomials Connected with the Charlier Expansions and the Pearson Differential and Difference Equations
cites work: Orthogonal Polynomials Defined by Difference Equations
cites work: Q4297475
cites work: Classical orthogonal polynomials: A functional approach
cites work: Q4853625
cites work: Q3997785
cites work: Q3469372
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description: scientific article; zbMATH DE number 770342