A product formula for orthogonal polynomials associated with infinite distance-transitive graphs
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 \(G\).
scientific article; zbMATH DE number 1891291
scientific article