  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.
