An introduction to the \(H_q\)-semiclassical orthogonal polynomials
type: Item
instance of: scholarly article
title: An introduction to the \(H_q\)-semiclassical orthogonal polynomials
zbMATH Open document ID: 1058.33018
published in: Methods and Applications of Analysis
publication date: 2004-10-25T00:00:00Z
review text: The study of \(q\)-polynomials has known an increasing interest in the last years, specially the Askey-Wilson polynomials and the polynomials related to them. In this paper the author considers the so-called \(q\)-semiclassical polynomials, i.e., an extension of the classical \(q\)-polynomials [see e.g. \textit{J. C. Medem, R. Álvarez-Nodarse}, and \textit{F. Marcellán}, J. Comput. Appl. Math. 135, No. 2, 157--196 (2001; Zbl 0991.33007)] introduced by Hahn in 1949. The approach is based on the distributional \(q\)-Pearson equation \[ H_q(\Phi u)+\Psi u=0, \] where \(\Phi,\Psi\) are polynomials of degree \(t\) and \(p\) such that the leading coefficient of \(\Psi\) is different from \([n+1]=(q^{n+1}-1)/(q-1)\) (i.e., \((\Phi,\Psi)\) is an admissible pair). Using a technique introduced and developed by Maroni the author establish several important properties such as the class of the \(H_q\) functional \(u\). In particular, several characterizations are proved among which a second-order linear difference equation with polynomial coefficients (that depends also on \(n\), the degree of the polynomials) is obtained. Finally, an example of \(q\)-semiclassical polynomials of class 1 is considered in details. Some of the results presented here already appeared in the Doctoral dissertation by J. C. Medem in 1996.
Mathematics Subject Classification ID: 33D99
zbMATH DE Number: 2110024
zbMATH Keywords: \(q\)-semiclassical orthogonal polynomials
author: Lotfi Khériji
reviewed by: Renato Álvarez-Nodarse
MaRDI profile type: Publication
full work available at URL: maa.2003.v10.n3.a5
OpenAlex ID: W1547447742
DOI: 10.4310/MAA.2003.V10.N3.A5
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description: scientific article; zbMATH DE number 2110024