Zeros of hypergeometric functions in the $p$-adic setting

TL;DR

This paper studies the zeros of $p$-adic hypergeometric functions introduced by McCarthy, analyzing their values over finite fields, and identifying conditions for zeros, including infinitely many primes where certain zeros occur.

Contribution

It extends the understanding of $p$-adic hypergeometric functions by characterizing their zeros and values over finite fields, generalizing classical hypergeometric theorems.

Findings

Zeros of ${_nG_n}(t)$ depend on whether $t$ is an $n$-th power residue modulo $p$.

Zeros of ${_n ilde{G}_n}(t)$ depend on solutions to a specific polynomial congruence.

Infinitely many primes satisfy ${_{2k}G_{2k}}(-1)=0$, while ${_{2k} ilde{G}_{2k}}(t)$ has no zeros for $t eq0$.

Abstract

Let $p$ be an odd prime and $\mathbb{F}_p$ be the finite field with $p$ elements. McCarthy \cite{mccarthy-pacific} initiated a study of hypergeometric functions in the $p$-adic setting. This function can be understood as $p$-adic analogue of Gauss' hypergeometric function, and also some kind of extension of Greene's hypergeometric function over $\mathbb{F}_p$. In this paper we investigate values of two generic families of McCarthy's hypergeometric functions denoted by ${_nG_n}(t)$, and ${_n\widetilde{G}_n}(t)$ for $n\geq3$, and $t\in\mathbb{F}_p$. The values of the function ${_nG_n}(t)$ certainly depend on whether $t$ is $n$-th power residue modulo $p$ or not. Similarly, the values of the function ${_n\widetilde{G}_n}(t)$ rely on the incongruent modulo $p$ solutions of $y^n-y^{n-1}+\frac{(n-1)^{n-1}t}{n^n}\equiv0\pmod{p}$. These results generalize special cases of $p$-adic analogues of Whipple's theorem and Dixon's theorem of classical hypergeometric series. We examine zeros of the functions ${_nG_n}(t)$, and ${_n\widetilde{G}_n}(t)$ over $\mathbb{F}_p$. Moreover, we look into the values of $t$ for which ${_nG_n}(t)=0$ for infinitely many primes. For example, we show that there are infinitely many primes for which ${_{2k}G_{2k}}(-1)=0$. In contrast, for $t\neq0$ there is no prime for which ${_{2k}\widetilde{G}_{2k}}(t)=0$.