Some congruences on the hyper-sums of powers of integers involving Fermat quotient and Bernoulli numbers

Abstract

For a given prime p >= 5, let Z(p) denote the set of rational p-integers (those rational numbers whose denominator is not divisible by p) . In this paper, we establish some congruences modulo a prime power p(5) on the hyper-sums of powers of integers in terms of Fermat quotient, Wolstenholme quotient, Bernoulli and Euler numbers.