Extending Gaussian hypergeometric series to the $p$-adic setting

TL;DR

This paper introduces a p-adic extension of Gaussian hypergeometric series, enabling broader prime-related results and providing tools to prove supercongruence conjectures.

Contribution

The authors define a new p-adic hypergeometric function that extends classical series and establish congruences to prove supercongruence conjectures.

Findings

Established a p-adic hypergeometric function extension.

Derived congruences linking p-adic and classical hypergeometric series.

Provided a framework for proving supercongruences.

Abstract

We define a function which extends Gaussian hypergeometric series to the $p$-adic setting. This new function allows results involving Gaussian hypergeometric series to be extended to a wider class of primes. We demonstrate this by providing various congruences between the function and truncated classical hypergeometric series. These congruences provide a framework for proving the supercongruence conjectures of Rodriguez-Villegas.