On congruences with binomial coefficients and harmonic numbers

Abstract

In this paper, we obtain super congruencesSigma([(p-1)/r])(k=1) (1)(k+1)(( alpha p - 1)(k))H-k(2) (mod p(2)) and Sigma([(p-1)/r])(k=1) (1-())k k((alpha p - 1)(k))H-k(2) (mod p(2)),where r is an element of {1, 2, 3} and alpha is a p-adic integer. Also, we give new congruences involving binomial coefficients and harmonic numbers.