ISSN:
1420-8903
Keywords:
Keywords. Hurwitz zeta-function, Lerch zeta-function, power series expansion, special values of zeta-functions.
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/PL00000117
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