Values of $p$-adic hypergeometric functions, and $p$-adic analogue of Kummer's linear identity

Abstract

Let $p$ be an odd prime and ${\mathbb{F}}_p$ be the finite field with $p$ elements. This paper focuses on the study of values of a generic family of hypergeometric functions in the $p$-adic setting which we denote by ${}_{3n-1}{G}_{3n-1}(p,t),$ where $n\ge 1$ and $t\in {\mathbb{F}}_p$. These values are expressed in terms of numbers of zeros of certain polynomials over ${\mathbb{F}}_p$. These results lead to certain $p$-adic analogues of classical hypergeometric identities. Namely, we obtain $p$-adic analogues of particular cases of a Gauss' theorem and a Kummer's theorem. Moreover, we examine the zeros of these functions. For instance, if $n$ is odd then we obtain zeros of ${}_{3n-1}{G}_{3n-1}(p,t)=0$ under certain condition on $t$. In contrast we show that if $n$ is even then the function ${}_{3n-1}{G}_{3n-1}(p,t)$ has no non-trivial zeros for any prime $p$.