On sieved orthogonal polynomials. IV. Generating functions
type: Item
instance of: scholarly article
title: On sieved orthogonal polynomials. IV. Generating functions
zbMATH Open document ID: 0645.33017
DOI: 10.1016/0021-9045(86)90066-3
author: Mourad E. H. Ismail
published in: Journal of Approximation Theory
publication date: 1986-01-01T00:00:00Z
review text: The most elementary sieved orthogonal polynomials satisfy the three term recurrence relation \[ 2xa_ nP_ n(x)=b_ nP_{n+1}(x)+c_ nP_ n(x)+d_ nP_{n-1}(x) \] with \(a_ n=b_ n=d_ n=1\), \(c_ n=0\) unless k-1 divides n, and in this case the coefficients are either all polynomials in n or else all polynomials in \(q^ n\). Here k is an integer at least 2 and q is real. Generating functions of the form \(\sum^{\infty}_{n=0}P_{kn+\ell}(x)r^ n\), \(\ell =0,1,...,k-1\), are considered and are shown to be expressible in terms of the case \(\ell =0\). For some polynomials this generating function for \(\ell =0\) can be found, and from this the asymptotics of \(P_ n(x)\) can be found. From this the orthogonality can sometimes be found. One interesting example treated in this paper is \[ a_ n=d_ n=1,\quad c_ n=0,\quad b_{mk- 1}=1-q^ m,\quad b_{mk+\ell}=1,\quad \ell =0,1,...,m-2. \] When \(k=1\) these are the continuous q-Hermite polynomials of Rogers. The orthogonality for \(k=2,3,..\). is given for the first time in this paper.
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 4052019
zbMATH Keywords: sieved polynomials
zbMATH Keywords: Generating functions
zbMATH Keywords: q-Hermite polynomials
reviewed by: R. Askey
MaRDI profile type: Publication
full work available at URL: 0021-9045(86)90066-3
OpenAlex ID: W2067750967
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description: scientific article; zbMATH DE number 4052019