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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 ζ(s), the (Hurwitz’s) generalized zeta-function ζ(s,a), the Polygamma function ψ (p) (z)(p=0,1,2,) and the polylogarithmic function Li s (z). The key ingredients in our approach include certain known integral representations for ζ(s) and ζ(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.
11M06ζ(s) and L(s,χ)
11M35Hurwitz and Lerch zeta functions
33B15Gamma, beta and polygamma functions