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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) k=0 e 2kπix (a+k) s ,s,x,a,1-a,x0,

to the Hurwitz-Lerch zeta function

Φ(z,s,a) k=0 z k (a+k) s ,1-a,|z|<1,

is presented. Note that Φ(e 2πix ,s,a)=R(a,x,s). First, using an integral formula for the Hurwitz-Lerch zeta function

Φ(z,s,a)=1 Γ(s) 0 x s-1 e -ax 1-ze -x dx,a>0,s>0,z[1,),

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 Φ(z,s,a) to the region z[1,) if a>0, and z{z,|z|<1} if a0, a - . 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:
11M35Hurwitz and Lerch zeta functions
30D10Representations of entire functions by series and integrals