Poisson Kernels as Expansions in q‐Racah Polynomials

This paper concerns stochastic processes on chains of arbitrary length whose Poisson kernel can be expressed in terms of the q‐Racah polynomials, the most general q‐deformed orthogonal polynomials in the discrete series of the Askey scheme. We give a new interpretation of this kernel as the probability transition density for a subordinated Markov process with only nearest neighbor hops. As an application, we give an elementary proof and extend a positivity result for a class of Poisson kernels which Gasper and Rahman established with direct methods.