Gaussian Hypergeometric Series and Combinatorial Congruences

In a recent paper [A-O], the author and K. Ono study the “Gaussian” hypergeometric series 4F3(1)p over the finite field Fp. They describe relationships between values of these series, Fourier coefficients of modular forms, and the arithmetic of a certain algebraic variety. These relationships, together with tools from p-adic analysis and some unexpected combinatorial identities, lead to the proof of one of Beukers “supercongruence” conjectures for the Apéry numbers A(n) := ∑n k=0 ( n k )2(n+k k )2 . Our purpose in this paper is to investigate similar phenomena for the hypergeometric series 3F2(λ)p. We begin by recalling some definitions. If p is an odd prime, then let Fp be the field with p elements. We extend each multiplicative character χ of Fp to Fp by defining χ(0) := 0. If A and B are two such characters, then we define the normalized Jacobi sum ( A B ) by ( A B ) := B(−1) p J(A, B̄) = B(−1) p ∑