Supercongruences for truncated hypergeometric series and p-adic gamma function

TL;DR

This paper establishes new supercongruences linking truncated hypergeometric series with the p-adic Gamma function, generalizing known results and confirming conjectures in the field.

Contribution

It introduces three new general supercongruences connecting hypergeometric series and p-adic Gamma functions, extending previous results and confirming conjectures.

Findings

Proved three new supercongruences involving hypergeometric series and p-adic Gamma functions.

Derived a supercongruence for the _7F6 hypergeometric series similar to existing results.

Confirmed a conjecture by Rodriguez-Villegas using the new supercongruences.

Abstract

We prove three more general supercongruences between truncated hypergeometric series and $p$-adic Gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez-Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for ${_7}F_6$ truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.