A short proof of a symmetry identity for the $q$-Hahn distribution

Abstract

We give a short and elementary proof of a symmetry identity for the $q$-moments of the $q$-Hahn distribution arising in the study of the $q$-Hahn Boson process and the $q$-Hahn TASEP. This identity discovered by Corwin in "The q-Hahn Boson Process and q-Hahn TASEP", Int. Math. Res. Not., 2014, was a key technical step to prove an intertwining relation between the Markov transition matrices of these two classes of discrete-time Markov chains. This was used in turn to derive exact formulas for a large class of observables of both these processes.