Summation identities and special values of hypergeometric series in the $p$-adic setting

TL;DR

This paper establishes hypergeometric summation identities and computes special values for a function related to the $p$-adic gamma function, by counting points on hyperelliptic curves over finite fields.

Contribution

It introduces new hypergeometric summation identities and evaluates special values of a $p$-adic gamma function-based function using algebraic geometry techniques.

Findings

Derived new hypergeometric summation identities in the $p$-adic context

Calculated specific special values of the $p$-adic gamma function-based function

Connected hypergeometric identities with point counting on hyperelliptic curves

Abstract

We prove hypergeometric type summation identities for a function defined in terms of quotients of the $p$-adic gamma function by counting points on certain families of hyperelliptic curves over $\mathbb{F}_{q}$. We also find certain special values of that function.