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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 $\text{Cl}_2(\theta)= \sum_{p=1}^{+\infty} \frac{\sin p\theta}{p^2}$.

MSC:
11B83Special sequences of integers and polynomials
33E20Functions defined by series and integrals
81T18Feynman diagrams
40A30Convergence and divergence of series and sequences of functions
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References:
[1] Abramowitz, M.; Stegun, I. A.: Handbook of mathematical functions. National bureau of standards applied mathematics series 55 (1972) · Zbl 0543.33001
[2] Ahlfors, L. V.: Complex analysis. (1979) · Zbl 0395.30001
[3] Berndt, B. C.: Ramanujan’s notebooks part 1. (1985) · Zbl 0555.10001
[4] Bhabha, H. J.: Scattering of positrons by electrons with exchange in Dirac’s theory of the positron. Proc. roy. Soc. London ser. A 154, 195 (1936) · Zbl 0013.33106
[5] Bjørkevoll, K. S.; Fäldt, G.; Osland, P.: Two-loop ladder-diagram contributions to Bhabha scattering. Nuclear phys. B 386, 303 (1992)
[6] Borwein, D.; Borwein, J. M.: On an intriguing integral and some series related to ${\zeta}$(4). Proc. amer. Math. soc. 123, 1191 (1995) · Zbl 0840.11036
[7] Borwein, D.; Borwein, J. M.; Girgensohn, R.: Explicit evaluation of Euler sums. Proc. Edinburgh math. Soc. 38, 277 (1995) · Zbl 0819.40003
[8] Borwein, J. M.; Girgensohn, R.: Evaluation of triple Euler sums. Electron. J. Combin. 3, R23 (1996) · Zbl 0884.40005
[9] Cheng, H.; Wu, T. T.: Expanding protons: scattering at high energies. (1987)
[10] De Doelder, P. J.: On some series containing ${\psi}(x) - {\psi}(y)$ and $({\psi}(x) - {\psi}(y))$2 for certain values of x and y. J. comput. Appl. math. 37, 125 (1991) · Zbl 0782.33001
[11] Devoto, A.; Duke, D. W.: Table of integrals and formulae for Feynman diagram calculations. Riv. nuovo cimento, No. 3, 1 (1984)
[12] Feynman, R. P.: Space-time approach to quantum electrodynamics. Phys. rev., No. 2, 769 (1949) · Zbl 0038.13302
[13] Gastmans, R.; Troost, W.: On the evaluation of polylogarithmic integrals, Simon stevin. Quart. J. Pure appl. Math. 55, 205 (1981) · Zbl 0477.33011
[14] Hansen, E. R.: A table of series and products. (1975) · Zbl 0438.00001
[15] Lewin, L.: Polylogarithms and associated functions. (1981) · Zbl 0465.33001
[16] Myint-U, T.; Debnath, L.: Partial differential equations for scientists and engineers. 363 (1987) · Zbl 0644.35001
[17] Nielsen, N.: Der eulersche dilogarithmus und seine verallgemeinerungen, nova acta. Abh. der kaiserl. Leop.-carol. Deutschen akademie der naturforscher 90, 123 (1909) · Zbl 40.0478.01
[18] Oberhettinger, F.: Tables of Mellin transforms. (1974) · Zbl 0289.44003
[19] O.M. Ogreid, Ph.D. Thesis, unpublished.
[20] Prudnikov, A. P.; Brychkov, Yu.A.; Marichev, O. I.: 2nd ed. Integrals and series. Integrals and series 3 (1990)
[21] Shen, L. -C.: Remarks on some integrals and series involving the Stirling numbers and ${\zeta}$(n). Trans. amer. Math. soc. 347, 1391 (1995) · Zbl 0828.11044
[22] Sitaramachandrarao, R.; Sivaramasarma, A.: Some identities involving the Riemann zeta function. Indian J. Pure appl. Math. 10, 602 (1979)