Moments of Gaussian hypergeometric functions over finite fields

TL;DR

This paper derives explicit formulas for moments of Gaussian hypergeometric functions over finite fields, enabling new estimates and connections to elliptic curves, Appell series, and character sums.

Contribution

It introduces explicit formulas for moments of Gaussian hypergeometric functions over finite fields, linking them to elliptic curves and Appell series, and providing new evaluation methods.

Findings

Established formulas for first and second moments of hypergeometric functions

Derived an estimate for $_6F_5(1)$ value

Connected hypergeometric functions to elliptic curves and Appell series

Abstract

We prove explicit formulas for certain first and second moment sums of families of Gaussian hypergeometric functions $_{n+1}F_n$, $n\ge1$, over finite fields with $q$ elements where $q$ is an odd prime. This enables us to find an estimate for the value $_6F_5(1)$. In addition, we evaluate certain second moments of traces of the family of Clausen elliptic curves in terms of the value $_3F_2(-1)$. These formulas also allow us to express the product of certain $_2F_1$ and $_{n+1}F_n$ functions in terms of finite field Appell series which generalizes current formulas for products of $_2F_1$ functions. We finally give closed form expressions for sums of Gaussian hypergeometric functions defined using different multiplicative characters.