$q$-Racah probability distribution

Abstract: 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.