Harmonic sums, Mellin transforms and integrals.

*(English)*Zbl 0939.65032The 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.

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.

Reviewer: R.S.Dahiya (Ames)

##### MSC:

65D32 | Numerical quadrature and cubature formulas |

65B10 | Numerical summation of series |

65R10 | Numerical methods for integral transforms |

44A15 | Special integral transforms (Legendre, Hilbert, etc.) |

65Z05 | Applications to the sciences |

81T18 | Feynman diagrams |

81-08 | Computational methods for problems pertaining to quantum theory |

65D20 | Computation of special functions and constants, construction of tables |

##### Keywords:

inelastic scattering; Mellin moments; inverse Mellin transformation; Feynman diagram calculations; algorithms; nested symbolic sums; harmonic series; binomial coefficients
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\textit{J. A. M. Vermaseren}, Int. J. Mod. Phys. A 14, No. 13, 2037--2076 (1999; Zbl 0939.65032)

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