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On the Hurwitz—Lerch zeta-function

Research paper by S. Kanemitsu, M. Katsurada, M. Yoshimoto

Indexed on: 01 Feb '00Published on: 01 Feb '00Published in: Aequationes mathematicae



Abstract

Let \( \Phi(z,s,\alpha) = \sum\limits^\infty_{n = 0} {z^n \over (n + \alpha)^s} \) be the Hurwitz-Lerch zeta-function and \( \phi(\xi,s,\alpha)=\Phi(e^{2\pi i\xi},s,\alpha) \) for \( \xi\in{\Bbb R} \) its uniformization. \( \Phi(z,s,\alpha) \) reduces to the usual Hurwitz zeta-function \( \zeta(s,\alpha) \) when z= 1, and in particular \( \zeta(s)=\zeta(s,1) \) is the Riemann zeta-function. The aim of this paper is to establish the analytic continuation of \( \Phi(z,s,\alpha) \) in three variables z, s, α (Theorems 1 and 1*), and then to derive the power series expansions for \( \Phi(z,s,\alpha) \) in terms of the first and third variables (Corollaries 1* and 2*). As applications of our main results, we evaluate in closed form a certain power series associated with \( \zeta(s,\alpha) \) (Theorem 5) and the special values of \( \phi(\xi,s,\alpha) \) at \( s = 0, -1, -2,\ldots \) (Theorem 6).