# 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).