Supercongruences for truncated hypergeometric series and p-adic gamma function

Rupam Barman, Neelam Saikia

Published 2015-07-27Version 1

We prove several supercongruences for truncated hypergeometric series and $p$-adic Gamma function. Recently, A. Deines, J. Fuselier, L. Long, H. Swisher and F. Tu posed some open problems on supercongruences in \cite{DFLST} based on numeric observations. We prove two of the supercongruences posed in \cite{DFLST}. Our results also generalize certain supercongruences proved in \cite{DFLST}. One of our supercongruence is similar to a supercongruence proved by L. Long and R. Ramakrishna in \cite{long}. The supercongruence conjectured by Rodriguez-Villegas and proved by E. Mortenson \cite{mortenson2} using the theory of Gaussian hypergeometric series follows from one of our more general supercongruence.