Hypercomplex systems originating from orthogonal polynomials
type: Item
instance of: scholarly article
title: Hypercomplex systems originating from orthogonal polynomials
zbMATH Open document ID: 0638.43004
DOI: 10.1007/BF01056816
author: A. A. Kalyuzhnyi
author: Yurij M. Berezansky
published in: Ukrainian Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: This is a mostly expository article discussing discrete hypergroups associated to families of orthogonal polynomials of one variable. Suppose \(\{P_ n(x):\) \(n=0,1,2,...\}\) is a suitably normalized family of such polynomials associated to the positive measure \(d\sigma\) (usually supported in a bounded interval), so that \(P_ n\) is of degree n, \(n=0,1,2,... \). The hypergroup is determined by the structural constants \(\gamma\) (m,n,k), where \[ P_ m(x)P_ n(x)=\sum^{m+n}_{k=| m- n|}\gamma (m,n,k)P_ k(x),\quad x\in {\mathbb{R}}. \] The relation between the \(\gamma\) (m,n,k) and the coefficients of the three-term recurrence for \(\{P_ n(x)\}\) is investigated. Some interesting results are possible even if \(\gamma\) (m,n,k) satisfies a boundedness rather than a positivity condition. This includes the situation of \(d\sigma (t)=h(t)(1-t)^{\alpha}(1+t)^{\beta}dt\) on \(-1<t<1\), where \(h(t)>0\) and satisfies certain smoothness and boundedness conditions (note that \(h\equiv 1\) is the case of Jacobi polynomials).
Mathematics Subject Classification ID: 43A20
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 4040528
zbMATH Keywords: discrete hypergroups
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: measure
zbMATH Keywords: Jacobi polynomials
reviewed by: Charles F. Dunkl
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01056816
OpenAlex ID: W2071990415
description: scientific article; zbMATH DE number 4040528