Vol. 295, No. 2, 2018

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Certain character sums and hypergeometric series

Rupam Barman and Neelam Saikia

Vol. 295 (2018), No. 2, 271–289
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