Some new results on products of Apostol–Bernoulli and Apostol–Euler polynomials

Abstract

We perform a further investigation for the Apostol–Bernoulli and Apostol–Euler polynomials and numbers. By making use of an elementary idea used by Euler in the discovery of his famous Pentagonal Number Theorem, we establish some new formulae for products of an arbitrary number of Apostol–Bernoulli and Apostol–Euler polynomials and numbers. These results are the corresponding generalizations of some known formulae including the higher-order convolution ones discovered by Agoh and Dilcher (2014) [5] on the classical Bernoulli and Euler polynomials.