Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms

TL;DR

This paper systematically studies generalized harmonic sums and polylogarithms, focusing on their analytic properties, transformations, and algebraic relations, with applications in high-order QCD calculations and combinatorics.

Contribution

It develops algorithms and analytic methods for generalized harmonic sums, including Mellin transforms, asymptotic expansions, and algebraic relations, implemented in the HarmonicSums software.

Findings

Explicit analytic continuations of S-sums via asymptotic expansions

Derivation of algebraic and structural relations for S-sums

Algorithms for infinite S-sums and harmonic polylogarithms at special values

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 $\pm 1$. 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 Poincar\'{e} 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 {\tt HarmonicSums}.