A combinatorial identity and applications

Abstract

An identity for the finite sum 1^N rac{z^n}{q^n-r} is given. Related sums (or series) were studied by Scherk, Clausen, Ramanujan, Shanks, Andrews and others. We use such identity to give new formulas for sum_1^infty rac{z^n}{q^n-r}, the Riemann zeta function and Euler-Mascheroni constant. An irrationality result is also proved.