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Summing one- and two-dimensional series related to the Euler series. (English) Zbl 1002.11021

The authors compute double series appearing in Feynman diagram calculations. A typical example is

$\sum _{n=1}^{+\infty }\sum _{j=1}^{kn}\frac{1}{{n}^{2}j}=\left(\frac{{k}^{2}}{2}+\frac{3}{2k}\right)\zeta \left(3\right)+\pi \sum _{j=1}^{k-1}j{Cl}_{2}\left(\frac{2\pi j}{k}\right),$

where the Clausen function ${\text{Cl}}_{2}$ is defined by ${\text{Cl}}_{2}\left(\theta \right)={\sum }_{p=1}^{+\infty }\frac{sinp\theta }{{p}^{2}}$.

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