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Zbl 0919.11056
Adamchik, Victor S.; Srivastava, H.M.
Some series of the zeta and related functions. (English)
[J] Analysis, München 18, No.2, 131-144 (1998). ISSN 0174-4747

We propose and develop yet another approach to the problem of summation of series involving the Riemann zeta-function $\zeta(s)$, the (Hurwitz's) generalized zeta-function $\zeta(s,a)$, the Polygamma function $\psi^{(p)} (z)(p=0,1,2,\dots)$ and the polylogarithmic function $Li_s(z)$. The key ingredients in our approach include certain known integral representations for $\zeta(s)$ and $\zeta(s,a)$. The method in this paper is illustrated by numerous examples of closed-form evaluations of series of the aforementioned types; the method developed in Section 2, in particular, has been implemented in Mathematica (Version 3.0). Many of the resulting summation formulas are believed to be new.
[H.M.Srivastava (Victoria)]

MSC 2000:: *11M06 Riemannian zeta-function and Dirichlet L-function
11M35 Other zeta functions
33B15 Gamma-functions, etc.

Keywords: Hurwitz's generalized Zeta function; Riemann Zeta function; Polygamma function; polylogarithmic function; integral representations; summation formulas

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