A product formula for orthogonal polynomials associated with infinite distance-transitive graphs Original Research Article

The infinite, locally finite distance-transitive graphs form an extension of homogeneous trees and are described by two discrete parameters. The associated orthogonal polynomials may be regarded as spherical functions of certain Gelfand pairs or as characters of some polynomial hypergroups; they are certain Bernstein polynomials and admit a discrete nonnegative product formula. In this paper we use the graph-theoretic origin of these polynomials to derive the existence of positive dual continuous product and transfer formulas. The dual product formulas will be computed explicitly.