A Gaussian hypergeometric series evaluation and Apéry number congruences

• DOI: 10.1515/crll.2000.004
• Publication Date: 2006-04-18T11:01:09Z
• Authors: S Ahlgren; K. Ono
• Keywords: Congruence relation; Bilateral hypergeometric series; Hypergeometric distribution

If p is prime, then let φp denote the Legendre symbol modulo and be trivial character p. As usual, n+1Fn(x)p := n+1Fn „ φp, . , p, | x « Gaussian hypergeometric series over Fp. For n > 2 non-trivial values of have been difficult to obtain. Here we take first step by obtaining a simple formula for 4F3(1)p. corollary obtain result describing distribution traces Frobenius certain families elliptic curves. We also find that 4F3(1)p satisfies surprising congruences 32 11. establish mod p2 “supercongruence” between Apery numbers coefficients eta-product; this relationship was conjectured Beukers in 1987. Finally, many new generalized numbers.