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

摘要

In this paper, we first investigate several further interesting properties of the multiple Hurwitz–Lerch Zeta function Φn(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 Φn(z, s, a), the Apostol–Bernoulli polynomials Bk(n)(x;λ) of order n, and the Apostol–Euler polynomials Ek(n)(x;λ) of order n. Relevant connections of the results presented here with those obtained in earlier works are also indicated precisely.