On sums of Apery polynomials and related congruences

The Apery polynomials are given by A(n)(x) = Sigma(n)(k=0)((n)(k))(2)((n+k)(k))(2)x(k) (n=0,1,2,...). (Those A(n) = A(n)(1) are Apery numbers.) Let p be an odd prime. We show that Sigma(p-1)(k=0)(-1)(k)A(k)(x) Sigma(p-1)(k=0)((2k)(k))(3)/16(k) x(k) (mod p(2)), and that Sigma(p-1)(k=0)A(k)(x) (x/p) Sigma(p-1)(k=0)((4k)(k,k,k,k))/(256x)(k) (mod p) for any p-adic integer x not equivalent to 0 (mod p). This enables us to determine explicitly Sigma(p-1)(k=0)(+/- 1)(k)A(k) mod p, and Sigma(p-1)(k=0)(-1)(k)A(k) mod p(2) in the case p 2 (mod 3). Another consequence states that Sigma(p-1)(k=0)(-1)(k)A(k)(-2) {4x(2) - 2p (mod p(2)) if p = x(2) + 4y(2) (x, y is an element of Z), 0(mod p(2)) if p 3 (mod 4). We also prove that for any prime p > 3 we have Sigma(p-1)(k=0)(2k + 1)A(k) p + 7/6p(4)B(p-3) (mod p5) where B-0, B-1, B-2,... are Bernoulli numbers. (C) 2012 Elsevier Inc. All rights reserved.