On the Hurwitz—Lerch zeta-function
On the Hurwitz—Lerch zeta-function
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关于 Hurwitz-Lerch zeta 函数
DOI：
10.1007/pl00000117
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发表时间：
2000
影响因子：
0.8
通讯作者：
M. Yoshimoto
中科院分区：
文献类型：
--
作者：
S. Kanemitsu;M. Katsurada;M. Yoshimoto
摘要
$ \ phi（z，s，\ alpha）= \ sum \ limits^\ infty_ {n = 0} {z^n \ over（n + \ alpha）^s} $ be hurwitz-lerch-lerch zeta-function和
$ \ phi（\ xi，s，\ alpha）= \ phi（e^{2 \ pi i \ xi}，s，\ alpha）$ for for
$ \ xi \ in {\ bbb r} $其均匀化。
$ \ phi（z，s，\ alpha）$降低到通常的hurwitz zeta功能
$ \ zeta（s，\ alpha）$当z = 1时，特别是
$ \ zeta（s）= \ zeta（s，1）$是riemann zeta功能。
$ \ phi（z，s，\ alpha）$在三个变量z，s，α（定理1和1*）中
$ \ phi（z，s，\ alpha）$在第一个和第三个变量方面（推论1*和2*）作为我们的主要结果的应用。
$ \ zeta（s，\ alpha）$（定理5）和特殊值
$ \ phi（\ xi，s，\ alpha）$ at
$ s = 0，-1，-2，\ ldots $（定理6）。
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).