ON SUMS OF BINOMIAL COEFFICIENTS MODULO p2

Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum Sigma(pa-1)(k=0) ((hpa-1)(k)) ((2k)(k))/m(k) mod p(2), where h and m are p-adic integers with m not equivalent to 0 (mod p). For example, we show that if h not equivalent to 0 (mod p) and p(a) > 3, then Sigma(pa-1)(k=0) ((hpa - 1)(k)) ((2k)(k)) (-h/2)(k) equivalent to (1 - 2h/p(a)) (1 + h((4 - 2/h)(p-1) -1)) (mod p(2)), where (divided by) denotes the Jacobi symbol. Here is another remarkable congruence: If p(a) > 3 then Sigma(pa-1)(k=0) ((pa-1)(k)) ((k)(2k)) (-1)(k) equivalent to 3(p-1) (p(a)/3) (mod p(2)).