In recent three-loop calculations of massive Feynman integrals within Quantum Chromodynamics (QCD) and, e.g., in recent combinatorial problems the so-called generalized harmonic sums (in short S-sums) arise. They are characterized by rational (or real) numerator weights also different from ±1. In this article we explore the algorithmic and analytic properties of these sums systematically. We work out the Mellin and inverse Mellin transform which connects the sums under consideration with the associated Poincaré iterated integrals, also called generalized harmonic polylogarithms. In this regard, we obtain explicit analytic continuations by means of asymptotic expansions of the S-sums which started to occur frequently in current QCD calculations. In addition, we derive algebraic and structural relations, like differentiation with respect to the external summation index and different multi-argument relations, for the compactification of S-sum expressions. Finally, we calculate algebraic relations for infinite S-sums, or equivalently for generalized harmonic polylogarithms evaluated at special values. The corresponding algorithms and relations are encoded in the computer algebra package HarmonicSums.
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In general, relations containing sums of logarithmic growth such as S1(n) can be also formally utilized in asymptotic expansions. However, in the following only convergent sums are considered.
144.
In http://www.risc.jku.at/research/combinat/software/HarmonicSums/ all relations are available up to weight four. The most involved calculation is the alphabet xi ∈ {1, −1, 1/2, −1/2, 2, −2}; the relations are obtained executing
$\texttt {ComputeSSumInfBasis}\left[4, \lbrace 1, -1, 1/2, -1/2, 2, -2\rbrace , \texttt {ExtendAlphabet}\rightarrow \texttt {True}\right]$
in about 36 hours.145.
Obviously, the xi must be chosen from a subfield of
$\mathbb {R}$
which is computable, in particular which can be treated within the computer algebra system Mathematica.© 2013 AIP Publishing LLC.
2013
AIP Publishing LLC
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