Supercongruences involving $p$-adic Gamma functions

TL;DR

This paper proves new supercongruences involving truncated hypergeometric series and $p$-adic Gamma functions, extending known results and proposing conjectures for higher powers of primes.

Contribution

It establishes novel supercongruences for hypergeometric series involving $p$-adic Gamma functions and extends existing supercongruences by Rodriguez-Villegas.

Findings

Proved supercongruences for ${}_2F_1$ and ${}_3F_2$ series

Extended Rodriguez-Villegas supercongruences

Proposed conjectural supercongruences modulo $p^3$

Abstract

We establish some supercongruences for the truncated ${}_2F_1$ and ${}_3F_2$ hypergeometric series involving the $p$-adic Gamma functions. Some of these results extend the four Rodriguez-Villegas supercongruences on the truncated ${}_3F_2$ hypergeometric series. The corresponding conjectural supercongruences modulo $p^3$ are also proposed for further research.