· 2013
Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
- 225citations
- 2013year
Short summary
This paper systematically explores the algorithmic and analytic properties of generalized harmonic sums (S-sums), which appear in high-energy physics calculations and combinatorial problems. It establishes a connection between S-sums and generalized harmonic polylogarithms via Mellin transforms.
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Key points
- Establishes a Mellin transform connection between generalized harmonic sums (S-sums) and generalized harmonic polylogarithms.
- Provides explicit analytic continuations of S-sums using asymptotic expansions.
- Derives algebraic and structural relations for simplifying S-sum expressions.
- Calculates algebraic relations for infinite S-sums and generalized harmonic polylogarithms at special values.
- Implements these algorithms and relations in the HarmonicSums computer algebra package.
AI-generated from the title and abstract; the full text is not read.
Abstract
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 w.r.t. 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.
The authors' abstract, as published at the source., 2013 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics