Certain character sums and hypergeometric series

TL;DR

This paper establishes new transformations for $p$-adic hypergeometric series by evaluating character sums and relating them to hypergeometric functions, revealing new identities and special values in the $p$-adic setting.

Contribution

It introduces novel $p$-adic hypergeometric transformations analogous to classical identities, expanding the understanding of $p$-adic special functions.

Findings

Derived two key $p$-adic hypergeometric transformations.

Connected character sums to hypergeometric series to prove identities.

Obtained new special values and finite field hypergeometric function results.

Abstract

We prove two transformations for the $p$-adic hypergeometric series which can be described as $p$-adic analogues of a Kummer's linear transformation and a transformation of Clausen. We first evaluate two character sums, and then relate them to the $p$-adic hypergeometric series to deduce the transformations. We also find another transformation for the $p$-adic hypergeometric series from which many special values of the $p$-adic hypergeometric series as well as finite field hypergeometric functions are obtained.