On the Hurwitz-Lerch zeta-function

Let Φ(z, s, α) = Σ^∞_n=0 z ⁿ/(n+α)^s be the Hurwitz-Lerch zeta-function and φ(ξ, s, α) = Φ(e^2πiξ, s, α) for ξ ∈ ℝ its uniformization. Φ(z, s, α) reduces to the usual Hurwitz zeta-function ζ(s, α) when z = 1, and in particular ζ(s) = ζ(s, 1) is the Riemann zeta-function. The aim of this paper is to establish the analytic continuation of Φ(z, s, α) in three variables z, s, α (Theorems 1 and 1*), and then to derive the power series expansions for Φ(z, s, α) 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 ζ(s, α) (Theorem 5) and the special values of φ(ξ, s, α) at s = 0, -1, -2,... (Theorem 6).