Binomial coefficient-harmonic sum identities associated to supercongruences

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 pk (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.