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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

$R\left(a,x,s\right)\equiv \sum _{k=0}^{\infty }\frac{{e}^{2k\pi ix}}{{\left(a+k\right)}^{s}},\phantom{\rule{1.em}{0ex}}s,x,a\in ℂ,\phantom{\rule{1.em}{0ex}}1-a\notin ℕ,\phantom{\rule{1.em}{0ex}}\Im x\ge 0,$

to the Hurwitz-Lerch zeta function

${\Phi }\left(z,s,a\right)\equiv \sum _{k=0}^{\infty }\frac{{z}^{k}}{{\left(a+k\right)}^{s}},\phantom{\rule{1.em}{0ex}}1-a\notin ℕ,\phantom{\rule{1.em}{0ex}}|z|<1,$

is presented. Note that ${\Phi }\left({\text{e}}^{2\pi ix},s,a\right)=R\left(a,x,s\right)$. First, using an integral formula for the Hurwitz-Lerch zeta function

${\Phi }\left(z,s,a\right)=\frac{1}{{\Gamma }\left(s\right)}{\int }_{0}^{\infty }\frac{{x}^{s-1}{e}^{-ax}}{1-z{e}^{-x}}\phantom{\rule{0.166667em}{0ex}}dx,\phantom{\rule{1.em}{0ex}}\Re a>0,\phantom{\rule{1.em}{0ex}}\Re s>0,\phantom{\rule{1.em}{0ex}}z\notin \left[1,\infty \right),$

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 ${\Phi }\left(z,s,a\right)$ to the region $z\in ℂ\setminus \left[1,\infty \right)$ if $\Re a>0$, and $z\in \left\{z\in ℂ,|z|<1\right\}$ if $\Re a\le 0$, $a\in ℂ\setminus {ℝ}^{-}$. 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 by series and integrals