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Harmonic Sums, Polylogarithms,Special Numbers, and Their Generalizations

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Book cover Computer Algebra in Quantum Field Theory

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

In these introductory lectures we discuss classes of presently known nested sums, associated iterated integrals, and special constants which hierarchically appear in the evaluation of massless and massive Feynman diagrams at higher loops. These quantities are elements of stuffle and shuffle algebras implying algebraic relations being widely independent of the special quantities considered. They are supplemented by structural relations. The generalizations are given in terms of generalized harmonic sums, (generalized) cyclotomic sums, and sums containing in addition binomial and inverse-binomial weights. To all these quantities iterated integrals and special numbers are associated. We also discuss the analytic continuation of nested sums of different kind to complex values of the external summation bound N.

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Notes

  1. 1.

    For a detailed account on the literature on MZVs see [63, 64] and the surveys Ref. [65].

  2. 2.

    The numbers associated with this alphabet are sometimes also called Euler-Zagier values and those of the sub-alphabet \(\{\omega _{0},\omega _{1}\}\) multiple zeta values.

  3. 3.

    For some aspects of the earlier development including results by the Leuven-group, Zagier, Broadhurst, Vermaseren and the Lille-group, see [63].

  4. 4.

    Here the \(\zeta _{\mathbf{a}}\)-values are defined \(\zeta _{a_{1},\ldots,a_{m}} =\sum _{ n_{1}>n_{2}>\ldots>n_{m}}^{\infty }\prod _{k=1}^{m}n_{k}^{-a_{1}}\).

  5. 5.

    For a detailed proof also in case of generalized harmonic sums see [29].

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Acknowledgements

We would like to thank D. Broadhurst, F. Brown, A. De Freitas, E.W.N. Glover, A. Hasselhuhn, D. Kreimer, C. Raab, C. Schneider, S. Weinzierl, F. Wißbrock, and J. Vermaseren for discussions. This work has been supported in part by DFG Sonderforschungsbereich Transregio 9, Computergestützte Theoretische Teilchenphysik, by the Austrian Science Fund (FWF) grants P20347-N18, P22748-N18, SFB F50 (F5009-N15) and by the EU Network LHCPHENOnet PITN-GA-2010-264564.

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    Jakob Ablinger

  2. Deutsches Elektronen-Synchrotron, DESY, Platanenallee 6, D-14738, Zeuthen, Germany

    Johannes Blümlein

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    Carsten Schneider

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    Johannes Blümlein

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Ablinger, J., Blümlein, J. (2013). Harmonic Sums, Polylogarithms,Special Numbers, and Their Generalizations. In: Schneider, C., Blümlein, J. (eds) Computer Algebra in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1616-6_1

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