Parities and Hypergeometric Function

Remarkable discrete probabilities associated with a finite set of distinct real numbers appear in mathematical statistics, polynomial approximation of a function over this set, and statistical physics. These weights lead to self-dual orthogonal polynomials, whose form motivates this study. The associated probability distributions based on rank parity are shown to be intimately related to the classical hypergeometric function at the specific value. Some combinatorial identities for these distributions are provided, and their asymptotic behavior is investigated.