q-Racah probability distribution

We introduce a certain discrete probability distribution \(P_{n,m,k,l;q}\) having non-negative integer parameters n, m, k, l and quantum parameter q which arises from a zonal spherical function of the Grassmannian over the finite field \(\mathbb {F}_q\) with a distinguished spherical vector. Using representation theoretic arguments and hypergeometric summation technique, we derive the presentation of the probability mass function by a single q-Racah polynomial, and also the presentation of the cumulative distribution function in terms of a terminating \({}_4 \phi _3\)-hypergeometric series.