×

Some series of the zeta and related functions. (English) Zbl 0919.11056

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.

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M35 Hurwitz and Lerch zeta functions
33B15 Gamma, beta and polygamma functions

Software:

Mathematica
PDF BibTeX XML Cite
Full Text: DOI