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

TL;DR

This paper investigates the values of a family of $p$-adic hypergeometric functions, deriving their connections to polynomial zeros over finite fields and establishing $p$-adic analogues of classical hypergeometric identities, including Kummer's theorem.

Contribution

It introduces a new family of $p$-adic hypergeometric functions, relates their values to polynomial zeros, and establishes $p$-adic analogues of classical hypergeometric identities such as Kummer's theorem.

Findings

Values of ${_{3n-1}G_{3n-1}}(p, t)$ are expressed via zeros of polynomials over $F_p$.

For odd $n$, zeros of ${_{3n-1}G_{3n-1}}(p, t)$ are characterized under certain conditions.

For even $n$, ${_{3n-1}G_{3n-1}}(p, t)$ has no non-trivial zeros for any prime $p$.

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 $n\geq1$ 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 $p$.