Binomial coefficient-harmonic sum identities associated to supercongruences

TL;DR

This paper derives binomial coefficient and harmonic sum identities using partial fractions, which are crucial for proving supercongruences and connecting hypergeometric series to modular forms in number theory.

Contribution

The paper introduces new identities linking binomial coefficients and harmonic sums, facilitating the proof of advanced supercongruences and extending hypergeometric functions over finite fields to the p-adic setting.

Findings

Established binomial coefficient--harmonic sum identities

Proved supercongruences involving hypergeometric series

Confirmed a conjecture relating hypergeometric series to modular forms

Abstract

We establish two binomial coefficient--generalized harmonic sum identities using the partial fraction decomposition method. These identities are a key ingredient in the proofs of numerous supercongruences. In particular, in other works of the author, they are used to establish modulo $p^k$ ($k>1$) congruences between truncated generalized hypergeometric series, and a function which extends Greene's hypergeometric function over finite fields to the $p$-adic setting. A specialization of one of these congruences is used to prove an outstanding conjecture of Rodriguez-Villegas which relates a truncated generalized hypergeometric series to the $p$-th Fourier coefficient of a particular modular form.