Certain product formulas and values of Gaussian hypergeometric series

TL;DR

This paper derives finite field analogues of classical hypergeometric product formulas, expresses products of Gaussian hypergeometric series as higher-order series, and evaluates special values using properties of Gauss sums and elliptic curves.

Contribution

It introduces new finite field product formulas for Gaussian hypergeometric series and connects these to evaluations of special values, extending classical identities to finite fields.

Findings

Derived finite field analogues of classical hypergeometric product formulas.

Expressed products of two ${_2}F_1$ series as ${_4}F_3$ and ${_3}F_2$ series.

Evaluated special values of Gaussian hypergeometric series using elliptic curve point counts.

Abstract

In this article we find finite field analogues of certain product formulas satisfied by the classical hypergeometric series. We express product of two ${_2}F_1$-Gaussian hypergeometric series as ${_4}F_3$- and ${_3}F_2$-Gaussian hypergeometric series. We use properties of Gauss and Jacobi sums and our earlier works on finite field Appell series to deduce these product formulas satisfied by the Gaussian hypergeometric series. We then use these transformations to evaluate explicitly some special values of ${_4}F_3$- and ${_3}F_2$-Gaussian hypergeometric series. By counting points on CM elliptic curves over finite fields, Ono found certain special values of ${_2}F_1$- and ${_3}F_2$-Gaussian hypergeometric series containing trivial and quadratic characters as parameters. Later, Evans and Greene found special values of certain ${_3}F_2$-Gaussian hypergeometric series containing arbitrary characters as parameters from where some of the values obtained by Ono follow as special cases. We show that some of the results of Evans and Greene follow from our product formulas including a finite field analogue of the classical Clausen's identity.