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Harmonic sums, Mellin transforms and integrals. (English) Zbl 0939.65032

The computation of Feynman diagrams has confronted physicists with classes of integrals that are usually hard to be evaluated, both analytically and numerically. Also the newer techniques applied in the more popular computer algebra packages do not offer much relief. Therefore it is good to occasionally study some alternative methods to come to a result. In the case of the computation of structure functions in deep inelastic scattering one is often interested in their Mellin moments. It treats Mellin transforms and inverse Mellin transformation for functions that are encountered in Feynman diagram calculations. This paper also describes algorithms to deal with nested symbolic sums over combinations of harmonic series, binomial coefficients and denominators. Also many sums that had to be evaluated seem to involve new results. Most of the algorithms have been programmed in the language of FORM. The resulting set of procedures is called SUMMER.

This paper contains a number of appendices. Additionally there is an appendix with lists of symbolic sums that are not directly treated by the general algorithm.

##### MSC:
 65D32 Quadrature and cubature formulas (numerical methods) 65B10 Summation of series (numerical analysis) 65R10 Integral transforms (numerical methods) 44A15 Special transforms (Legendre, Hilbert, etc.) 65Z05 Applications of numerical analysis to physics 81T18 Feynman diagrams 81-08 Computational methods (quantum theory) 65D20 Computation of special functions, construction of tables