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aequationes mathematicae

, Volume 59, Issue 1–2, pp 1–19 | Cite as

On the Hurwitz—Lerch zeta-function

  • S. Kanemitsu
  • M. Katsurada
  • M. Yoshimoto

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).

Keywords. Hurwitz zeta-function, Lerch zeta-function, power series expansion, special values of zeta-functions. 

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Copyright information

© Birkhäuser Verlag, Basel, 2000

Authors and Affiliations

  • S. Kanemitsu
    • 1
  • M. Katsurada
    • 2
  • M. Yoshimoto
    • 3
  1. 1.Kayanomori 11-6, Iizuka, Fukuoka 820-8555, Japan, e-mail: kanemitu@fuk.kindai.ac.jp JP
  2. 2.Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan, e-mail: ma396031@math.kyushu-u.ac.jp JP
  3. 3.Mathematics Hiyoshi Campus, Keio University, 4-1-1 Hiyoshi, Kouhoku-ku, Yokohama 223-8521, Japan, e-mail: masanori@math.hc.keio.ac.jp JP

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