Some q-extensions of the Apostol–Bernoulli and the Apostol–Euler polynomials of order n, and the multiple Hurwitz zeta function

Abstract

In this paper, we first investigate several further interesting properties of the multiple Hurwitz–Lerch Zeta function Φ n ( z, s, a) which was introduced recently by Choi et al. [J. Choi, D.S. Jang, H.M. Srivastava, A generalization of the Hurwitz–Lerch Zeta function, Integral Transform. Spec. Funct., 19 (2008)]. We then introduce and investigate some q-extensions of the multiple Hurwitz–Lerch Zeta function Φ n ( z, s, a), the Apostol–Bernoulli polynomials B k ( n ) ( x ; λ ) of order n, and the Apostol–Euler polynomials E k ( n ) ( x ; λ ) of order n. Relevant connections of the results presented here with those obtained in earlier works are also indicated precisely.