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Power series and asymptotic series associated with the Lerch zeta-function. (English) Zbl 0937.11035
Let $$s$$ be a complex variable, $$\alpha$$ and $$\lambda$$ real parameters with $$\alpha>0$$. We define the Lerch zeta-function $\varphi(\lambda, \alpha,s)= \sum_{n=0}^\infty e^{2\pi in\lambda} (n+\alpha)^{-s} \quad (\operatorname {Re}s>1).$ For $$\lambda\in \mathbb{R}\setminus \mathbb{Z}$$, this function is continued to an entire function over the $$s$$-plane. If $$\lambda\in \mathbb{Z}$$, then this function reduces to the Hurwitz zeta-function $$\zeta(s,\alpha)$$. In this paper, based on the Mellin-Barnes type of integral formulae the author studies the power series and asymptotic series for the Lerch zeta-function $$\varphi(\lambda, \alpha,s)$$ in the second parameter, and obtains several asymptotic formulae.

MSC:
 11M35 Hurwitz and Lerch zeta functions
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References:
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