Sums of products of Apostol-Bernoulli numbers

Abstract

By expressing the sums of products of the Apostol-Bernoulli polynomials in terms of the special values of multiple Hurwitz-Lerch zeta functions at non-positive integers, we obtain the sums of products identity for the Apostol-Bernoulli numbers which is an analogue of the classical sums of products identity for Bernoulli numbers dating back to Euler.