A p-adic analogue of a formula of Ramanujan

TL;DR

This paper proves a p-adic analogue of a Ramanujan formula using hypergeometric series, confirming a conjecture by Van Hamme through a combination of numerical and theoretical methods.

Contribution

It provides the first proof of a Van Hamme conjecture relating binomial sums and p-adic gamma functions using hypergeometric series techniques.

Findings

Proof of a specific Van Hamme p-adic conjecture

Establishment of connections between hypergeometric series and p-adic gamma functions

Validation of numerical conjectures through theoretical proof

Abstract

During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Based on numerical computations, Van Hamme recently conjectured $p$-adic analogues to such formulae. Using a combination of ordinary and Gaussian hypergeometric series, we prove one of these conjectures.