Certain character sums and hypergeometric series

Rupam Barman and Neelam Saikia

Vol. 295 (2018), No. 2, 271–289

DOI: 10.2140/pjm.2018.295.271

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.

Keywords

character sum, hypergeometric series, $p$-adic gamma function

Mathematical Subject Classification 2010

Primary: 11S80, 11T24, 33E50, 33C99

Milestones

Received: 23 June 2017

Revised: 6 February 2018

Accepted: 6 February 2018

Published: 11 April 2018