A Gaussian hypergeometric series evaluation and Apéry number congruences

Abstract

If p is prime, then let f p denote the Legendre symbol modulo p and let e p be the trivial character modulo p. As usual, let n1

An extension of the Apéry number supercongruence

“…By Theorem 7 in [1], F (n) = 0 for all positive integers n, proving the second assertion of Proposition 3.1. Finally, by Proposition 2.1, we see that…”

“…If γ = 4[2] − 8 [1], then λ 0 = 2 −8 , the modular form is f (z), and we prove Rodriguez-Villegas' observation in the following theorem. Theorem 1.…”

“…This was proved for primes p such that p ∤ a(p) by Ishikawa [8] and unconditionally by Ahlgren and Ono [1]. The a(p) for odd primes p are also known to be related to the modular Calabi-Yau threefold (1.4) x + 1 x + y + 1 y + z + 1 z + w + 1 w = 0, in the following way.…”

“…By using character sums to calculate the quantity N (p), we can write a(p) in terms of character sums using (1.5), as in [1,Theorem 6]. The Gross-Koblitz formula then transforms the character sums into expressions involving the p-adic gamma function.…”

Congruences for generalized Apéry numbers and Gaussian hypergeometric series

“…One main purpose of this paper is to find similar congruences for the numbers A(f 1 where f 1 , f 2 , m, l ∈ N. Note that this family of sequences includes the Apéry numbers. In his PhD thesis [12], Coster studied the numbers A(f 1 , f 1 , l, m, λ) and proved that…”