Higher recurrences for Apostol-Bernoulli-Euler numbers

We present explicit formulas for sums of products of Apostol-Bernoulli and Apostol-Euler numbers of the form [GRAPHICS] where N and n are positive integers, B-m(q)stand for the Apostol-Bernoulli numbers, Em( q) for the Apostol- Euler numbers, and (M1, ... , mN) = n!/m1! ... , mN!. Our formulas involve Stirling numbers of the first kind. We also derive results for Apostol- Bernoulli and Apostol- Euler polynomials. As an application, for q = 1 we recover results of Dilcher, and our paper can be regarded as a q- extension of that of Dilcher.