Certain character sums and hypergeometric series
Rupam Barman, Neelam Saikia

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 -adic hypergeometric series which can be described as -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 -adic hypergeometric series to deduce the transformations. We also find another transformation for the -adic hypergeometric series from which many special values of the -adic hypergeometric series as well as finite field hypergeometric functions are obtained.
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