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Asymptotic expansions of the Hurwitz–Lerch zeta function. (English) Zbl 1106.11034

In the paper, a generalization of the asymptotic expansions obtained by M. Katsurada [Proc. Japan Acad. 74, No. 10, 167–170 (1998; Zbl 0937.11035)] and D. Klusch [J. Math. Anal. Appl. 170, No. 2, 513–523 (1992; Zbl 0763.11036)] for the Lipschitz-Lerch zeta function \[ R(a, x, s)\equiv\sum_{k=0}^\infty {e^{2k\pi ix}\over (a+k)^s}, \quad s, x, a \in \mathbb C, \quad 1-a\notin \mathbb N, \quad \operatorname{Im} x \geq 0, \] to the Hurwitz-Lerch zeta function \[ \Phi(z, s, a)\equiv\sum_{k=0}^\infty {z^k\over (a+k)^s}, \quad 1-a\notin \mathbb N, \quad | z| <1, \] is presented. Note that \(\Phi({\text e}^{2\pi i x}, s, a)=R(a, x, s)\). First, using an integral formula for the Hurwitz-Lerch zeta function \[ \Phi(z, s, a)={1\over \Gamma(s)}\int_0^\infty {x^{s-1}e ^{-ax}\over 1-ze^{-x}}\,d x, \quad \operatorname{Re} a>0, \quad \operatorname{Re} s>0, \quad z\notin[1, \infty), \] given in [H. M. Srivastava and J. Choi, Series associated with the zeta and related functions. Dordrecht: Kluwer Academic Publishers (2001; Zbl 1014.33001)], the authors obtain an integral representation which gives the analytical continuation of the function \(\Phi(z, s, a)\) to the region \(z\in\mathbb C\setminus[1, \infty)\) if \(\operatorname{Re} a>0\), and \(z\in\{z\in \mathbb C, | z| <1\}\) if \(\operatorname{Re} a \leq 0\), \(a\in \mathbb C\setminus\mathbb R^-\). From this they deduce three complete asymptotic expansions for either large or small \(a\) and large \(z\) with error bounds. Moreover, the numerical examples for these bounds are presented.

MSC:

11M35 Hurwitz and Lerch zeta functions
30D10 Representations of entire functions of one complex variable by series and integrals
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