Higher recurrences for Apostol-Bernoulli-Euler numbers

Abstract

We present explicit formulas for sums of products of Apostol-Bernoulli and Apostol-Euler numbers of the form $$\sum\limits_{_{m_1 , \cdots ,m_N \geqslant n}^{m_1 + \cdots + m_N = n} } {\left( {_{m_1 , \cdots m_N }^n } \right)B_{m_1 } (q) \cdots B_{m_N } (q),} \sum\limits_{_{m_1 , \cdots ,m_N \geqslant n}^{m_1 + \cdots + m_N = n} } {\left( {_{m_1 , \cdots m_N }^n } \right)E_{m_1 } (q) \cdots E_{m_N } (q),}$$ where N and n are positive integers, B m (q) n stand for the Apostol-Bernoulli numbers, E m (q) for the Apostol-Euler numbers, and $$\left( {\begin{array}{*{20}c} n \\ {m_1 , \cdots ,m_N } \\ \end{array} } \right) = \frac{{n!}} {{m_1 ! \cdots m_N !}}.$$ 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.