On the Hurwitz—Lerch zeta-function

Summary.

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).