Scaled asymptotics for \(q\)-orthogonal polynomials
type: Item
instance of: scholarly article
title: Scaled asymptotics for \(q\)-orthogonal polynomials
zbMATH Open document ID: 1110.33011
author: Mourad E. H. Ismail
author: Ruiming Zhang
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2007-03-02T00:00:00Z
review text: The \(q^{-1}\)-Hermite polynomials, the Stieltjes-Wigert polynomials and the \(q\)-Laguerre polynomials are typical cases of polynomials which satisfy three-term recurrence relations whose coefficients grow exponentially. The authors summarize results concerning scaled asymptotics for the above \(q\)-orthogonal polynomials. For \(0<q<1\), \(t>0\) and \(u\) a complex number, \(u\neq 0\), their scaling for the \(q^{-1}\) -Hermite polynomials is \(x_{n}(t,u)=(q^{-nt}u-q^{nt}/u)/2\) and for Stieltjes-Wigert polynomials and the q-Laguerre polynomials is \(x_{n}(t,u)=q^{-nt}u\). Their bulk asymptotics involve theta functions and exhibit a chaotic behavior when \(t\) is irrational and periodic behavior when \(t\) is rational.
Mathematics Subject Classification ID: 33D45
Mathematics Subject Classification ID: 34E05
zbMATH DE Number: 5130334
zbMATH Keywords: q-othogonal polynomials
zbMATH Keywords: asymptotic expansions
reviewed by: Chrysoula G. Kokologiannaki
MaRDI profile type: Publication
full work available at URL: j.crma.2006.11.018
OpenAlex ID: W1981428239
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DOI: 10.1016/J.CRMA.2006.11.018
description: scientific article; zbMATH DE number 5130334