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Zbl 1002.11021
Ogreid, Odd Magne; Osland, Per
Summing one- and two-dimensional series related to the Euler series. (English)
[J] J. Comput. Appl. Math. 98, No.2, 245-271 (1998). ISSN 0377-0427

The authors compute double series appearing in Feynman diagram calculations. A typical example is $$\sum_{n=1}^{+\infty} \sum_{j=1}^{kn} \frac{1}{n^2j}= \biggl( \frac{k^2}{2}+ \frac{3}{2k} \biggr) \zeta(3)+\pi \sum_{j=1}^{k-1}j \operatorname {Cl}_2 \biggl( \frac{2\pi j}{k}\biggr),$$ where the Clausen function $\text{Cl}_2$ is defined by $\text{Cl}_2(\theta)= \sum_{p=1}^{+\infty} \frac{\sin p\theta}{p^2}$.
[D.Duverney (Lille)]

MSC 2000:: *11B83 Special sequences of integers and polynomials
33E20 Functions defined by series and integrals
81T18 Feynman diagrams
40A30 Convergence of series and sequences of functions

Keywords: Euler series; hypergeometric series; Riemann zeta-function; psi-function; polylogarithms; Feynman diagram; Clausen function

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