Poisson Kernels as Expansions in $\lowercase{q}$‐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.