Moments of Gaussian hypergeometric functions over finite fields

Ankan Pal; Bidisha Roy; Mohammad Sadek

paper

Moments of Gaussian hypergeometric functions over finite fields

arXiv:2111.08393

Abstract

We prove explicit formulas for certain first and second moment sums of families of Gaussian hypergeometric functions $_{n + 1} F_{n}$ , $n \geq 1$ , over finite fields with $q$ elements where $q$ is an odd prime. This enables us to find an estimate for the value $_{6} F_{5} (1)$ . In addition, we evaluate certain second moments of traces of the family of Clausen elliptic curves in terms of the value $_{3} F_{2} (- 1)$ . These formulas also allow us to express the product of certain $_{2} F_{1}$ and $_{n + 1} F_{n}$ functions in terms of finite field Appell series which generalizes current formulas for products of $_{2} F_{1}$ functions. We finally give closed form expressions for sums of Gaussian hypergeometric functions defined using different multiplicative characters.