Some relationships between the generalized Apostol–Bernoulli polynomials and Hurwitz–Lerch Zeta functions

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Abstract
The main object of this paper is to further investigate the generalized Apostol–Bernoulli polynomials of higher order, which were introduced and studied recently by Luo and Srivastava [2005, Journal of Mathematical Analysis and Applications, 308, 290–302; 2006, Computers and Mathematics with Applications, 51, 631–642]. Here, we first derive an explicit representation of these generalized Apostol–Bernoulli polynomials of higher order in terms of a generalization of the Hurwitz–Lerch Zeta function and then proceed to establish a functional relationship between the generalized Apostol–Bernoulli polynomials of rational arguments and the Hurwitz (or generalized) Zeta function. Our results would provide extensions of those given earlier by (for example) Apostol [1951, Pacific Journal of Mathematics, 1, 161–167] and Srivastava [2000, Mathematical Proceedings of the Cambridge Philosophical Society, 129, 77–84].

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