A probabilistic counterpart of the Askey scheme for continuous polynomials
type: Item
instance of: scholarly article
title: A probabilistic counterpart of the Askey scheme for continuous polynomials
zbMATH Open document ID: 1242.33016
author: Jean-Renaud Pycke
published in: Advances in Pure and Applied Mathematics
publication date: 2012-05-07T00:00:00Z
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 60E05
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 6031455
zbMATH Keywords: Askey scheme
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: convergence of orthogonality measures
zbMATH Keywords: convergence of densities
MaRDI profile type: Publication
full work available at URL: apam.2011.014
OpenAlex ID: W2316384233
cites work: A risk model driven by Lévy processes
DOI: 10.1515/APAM.2011.014
review text: Consider the part of the Askey scheme of orthogonal polynomials which belongs to orthogonality measures with Lebesgue densities. For all polynomials there (Wilson, continuous Hahn, continuous dual Hahn, Jacobi, Meixner-Pollaczek, Laguerre, Hermite), the densities of the associated orthogonality measures are known where these measures are normalized as probability measures. The author checks by hand for each limit transition for the polynomials that the densities converge (after suitable rescaling) pointwise and that hence, by Scheffe's theorem, the associated probability measures converge w.r.t. total variation norm.NEWLINENEWLINEFor the more classical limits these results are known for a long time (even with estimates for the rate of convergence). The reviewer expects that at least a part of the limits should be obtained by a general argument from the convergence of the polynomials.
reviewed by: Michael Voit
description: scientific article; zbMATH DE number 6031455