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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 {\it M. Katsurada} [Proc. Japan Acad. 74, No. 10, 167--170 (1998; Zbl 0937.11035)] and {\it D. Klusch} [J. Math. Anal. Appl. 170, No. 2, 513--523 (1992; Zbl 0763.11036)] for the Lipschitz-Lerch zeta function $$ R(a, x, s)\equiv\sum_{k=0}^\infty {e^{2k\pi ix}\over (a+k)^s}, \quad s, x, a \in \Bbb C, \quad 1-a\notin \Bbb N, \quad \Im x \geq 0, $$ to the Hurwitz-Lerch zeta function $$ \Phi(z, s, a)\equiv\sum_{k=0}^\infty {z^k\over (a+k)^s}, \quad 1-a\notin \Bbb N, \quad \vert z\vert <1, $$ 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 $$ \Phi(z, s, a)={1\over \Gamma(s)}\int_0^\infty {x^{s-1}e ^{-ax}\over 1-ze^{-x}}\,d x, \quad \Re a>0, \quad \Re s>0, \quad z\notin[1, \infty), $$ given in [{\it H. M. Srivastava} and {\it 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 $z\in\Bbb C\setminus[1, \infty)$ if $\Re a>0$, and $z\in\{z\in \Bbb C, \vert z\vert <1\}$ if $\Re a \leq 0$, $a\in \Bbb C\setminus\Bbb R^-$. 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.

11M35Hurwitz and Lerch zeta functions
30D10Representations of entire functions by series and integrals
Full Text: DOI
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