Hypergeometric type identities in the $p$-adic setting and modular forms

TL;DR

This paper establishes hypergeometric identities involving the $p$-adic gamma function and applies them to prove a supercongruence conjecture linking hypergeometric series to modular form coefficients.

Contribution

It introduces new hypergeometric identities in the $p$-adic setting and proves a conjecture connecting hypergeometric series with modular forms.

Findings

Proved hypergeometric identities in the $p$-adic gamma function context.

Validated a supercongruence conjecture of Rodriguez-Villegas.

Connected hypergeometric series to Fourier coefficients of modular forms.

Abstract

We prove hypergeometric type identities for a function defined in terms of quotients of the $p$-adic gamma function. We use these identities to prove a supercongruence conjecture of Rodriguez-Villegas between a truncated $_4F_3$ hypergeometric series and the Fourier coefficients of certain weight four modular form.