Journal of Mathematical Physics· 2013Q2
Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms
- 169citations
- Q2SCImago
- 2013year
Short summary
New algorithms and analytic continuations for generalized harmonic sums (S-sums) and polylogarithms, crucial for high-precision QCD calculations, are presented, enabling compactification and evaluation of these complex mathematical objects.
AI-generated from the title and abstract; the full text is not read.
Key points
- Introduces systematic algorithmic and analytic properties of generalized harmonic sums (S-sums).
- Derives explicit analytic continuations for S-sums using asymptotic expansions.
- Establishes algebraic and structural relations for compactifying S-sum expressions.
- Calculates algebraic relations for infinite S-sums and generalized harmonic polylogarithms at special values.
- Algorithms and relations are implemented 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 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.
The authors' abstract, as published at the source. Journal of Mathematical Physics, 2013 · DOI ↗
Continue with a free account
Ask the paper: 3 free questions a day about this paper; save it, get its citation, new summaries every day for your field. Takeaways are Premium.
Continue free on the webSign in with Google or Apple; no card needed. You come back to this paper.
On your phone:
Field: Algebra and Number Theory
Algebra and Number TheoryMathematics