$q$-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 ${}_4\ddot{I}{•}_3$-hypergeometric series.

The first part of the revision of arXiv:2104.12635 v1. The second version is the manuscript that has been accepted for the publication in Ramanujan Journal. 17pages