Gaussian Hypergeometric Series and Extensions of Supercongruences

Let p be an odd prime. The purpose of this paper is to refine methods of Ahlgren and Ono [2] and Kilbourn [13] in order to prove a general mod p congruence for the Gaussian hypergeometric series n+1Fn(λ) where n is an odd positive integer. As a result, we extend three recent supercongruences. The first is a result of Ono and Ahlgren [2] on a supercongruence for Apéry numbers which was conjectured by Beukers in 1987. The second is one of Mortenson [18] which relates truncated hypergeometric series to the number of Fp points of some family of Calabi-Yau manifolds. Finally, the third is a result of Loh and Rhodes [16] on congruences between coefficients of modular forms corresponding to a particular class of elliptic curves and combinatorial objects. Additionally, we discuss the non-trivial methods of the computer summation package Sigma which were used to find explicit evaluations of two strange combinatorial identities involving generalized Harmonic sums.