A product formula for orthogonal polynomials associated with infinite distance-transitive graphs
type: Item
instance of: scholarly article
title: A product formula for orthogonal polynomials associated with infinite distance-transitive graphs
zbMATH Open document ID: 1016.43009
DOI: 10.1016/S0021-9045(02)00039-4
published in: Journal of Approximation Theory
publication date: 2003-04-02T00:00:00Z
review text: This paper considers graphs that are infinite, locally finite, undirected and connected having a distance transitive metric (for all vertices \(v_1,v_2, v_3,v_4\) with \(d(v_1,v_2)= d(v_3,v_4)\) there is an automorphism \(g\) of the graph satisfying \(g(v_1)= g(v_3)\) and \(g(v_2)= g(v_4))\). These graphs have been characterized by \textit{H. D. MacPherson} [Combinatorica 2, 63-69 (1982; Zbl 0492.05036)]. Such a graph \(\Gamma(a,b)\) depends on two integer parameters \(a,b\geq 2\). The author studies certain orthogonal polynomials associated with \(\Gamma (a,b)\), which are in fact orthogonal with respect to a measure supported on the interval \([-1,1]\). For \(G\), a locally compact group acting on \(\Gamma(a,b)\) in a distance-transitive way, the author proves a discrete analogue of results on rank one non-compact symmetric spaces. Furthermore, he obtains explicit product formulas for the orthogonal polynomials, associating them with positive definite functions on \(G\).
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 43A62
zbMATH DE Number: 1891291
zbMATH Keywords: automorphism groups
zbMATH Keywords: distance transitive graphs
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: locally compact group
zbMATH Keywords: positive definite functions
author: Michael Voit
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1891291