Certain transformations and values of p-adic hypergeometric functions

TL;DR

This paper establishes new $p$-adic hypergeometric function transformations, including analogues of classical Euler and Clausen transformations, by evaluating character sums and relating them to these functions.

Contribution

It introduces novel $p$-adic hypergeometric transformations and derives special values, expanding the understanding of $p$-adic hypergeometric functions.

Findings

Proved $p$-adic analogues of Euler's and Clausen's transformations.

Evaluated specific character sums related to $p$-adic hypergeometric functions.

Derived special values of certain $p$-adic hypergeometric functions.

Abstract

We prove two transformations for the $p$-adic hypergeometric functions which can be described as $p$-adic analogues of a Euler's transformation and a transformation of Clausen. We first evaluate certain character sums, and then relate them to the $p$-adic hypergeometric functions to deduce the transformations. We use a character sum identity proved by Ahlgren, Ono, and Penniston to deduce the $p$-adic Clausen's transformation. We also deduce special values of certain $p$-adic hypergeometric functions.