Binomial coefficient-harmonic sum identities associated to supercongruences

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.