  Let p be an odd prime. In 1984, Greene introduced the notion of
hypergeometric functions over finite fields. Special values of these functions
have been of interest as they are related to the number of F_p points on
algebraic varieties and to Fourier coefficients of modular forms. In this
paper, we explicitly determine these functions modulo higher powers of p and
discuss an application to supercongruences. This application uses two
non-trivial generalized Harmonic sum identities discovered using the computer
summation package Sigma. We illustrate the usage of Sigma in the discovery and
proof of these two identities.
