Summation identities and transformations for hypergeometric series

TL;DR

This paper derives new summation identities and transformations for $p$-adic and finite field hypergeometric series by evaluating Gauss sums related to counting points on specific algebraic families over finite fields.

Contribution

It expresses point counts on algebraic families in terms of $p$-adic hypergeometric series for all relevant primes and establishes new identities and transformations for these series.

Findings

Derived summation identities for $p$-adic hypergeometric series.

Established transformations and special values for the series.

Provided a summation identity for Greene's finite field hypergeometric series.

Abstract

We find summation identities and transformations for the McCarthy's $p$-adic hypergeometric series by evaluating certain Gauss sums which appear while counting points on the family $$Z_{\lambda}: x_1^d+x_2^d=d\lambda x_1x_2^{d-1}$$ over a finite field $\mathbb{F}_p$. A. Salerno expresses the number of points over a finite field $\mathbb{F}_p$ on the family $Z_{\lambda}$ in terms of quotients of $p$-adic gamma function under the condition that $d|p-1$. In this paper, we first express the number of points over a finite field $\mathbb{F}_p$ on the family $Z_{\lambda}$ in terms of McCarthy's $p$-adic hypergeometric series for any odd prime $p$ not dividing $d(d-1)$, and then deduce two summation identities for the $p$-adic hypergeometric series. We also find certain transformations and special values of the $p$-adic hypergeometric series. We finally find a summation identity for the Greene's finite field hypergeometric series.