On the Hurwitz-Lerch zeta-function

Shigeru Kanemitsu, Masanori Katsurada, Masami Yoshimoto

Research output: Contribution to journalArticle

29 Citations (Scopus)

Abstract

Let Φ(z, s, α) = Σn=0 z n/(n+α)s be the Hurwitz-Lerch zeta-function and φ(ξ, s, α) = Φ(e2πiξ, 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).

Original languageEnglish
Pages (from-to)1-19
Number of pages19
JournalAequationes Mathematicae
Volume59
Issue number1-2
DOIs
Publication statusPublished - 2000 Jan 1

Keywords

  • Hurwitz zeta-function
  • Lerch zeta-function
  • Power series expansion
  • Special values of zeta-functions

ASJC Scopus subject areas

  • Mathematics(all)
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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