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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 $$\sum_{n=1}^{+\infty} \sum_{j=1}^{kn} \frac{1}{n^2j}= \biggl( \frac{k^2}{2}+ \frac{3}{2k} \biggr) \zeta(3)+\pi \sum_{j=1}^{k-1}j \operatorname {Cl}_2 \biggl( \frac{2\pi j}{k}\biggr),$$ where the Clausen function $\text{Cl}_2$ is defined by $\text{Cl}_2(\theta)= \sum_{p=1}^{+\infty} \frac{\sin p\theta}{p^2}$.

11B83Special sequences of integers and polynomials
33E20Functions defined by series and integrals
81T18Feynman diagrams
40A30Convergence and divergence of series and sequences of functions
Full Text: DOI
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