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Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms

Jakob Ablinger, Johannes Blümlein, Carsten Schneider

TL;DR

The paper formalizes generalized harmonic sums ($S$-sums) and their deep connections to generalized harmonic polylogarithms via Mellin transforms. It develops comprehensive analytic and algorithmic machinery—Mellin and inverse Mellin transforms, analytic continuations, asymptotic expansions, and a rich set of algebraic relations (quasi-shuffle, differential, and duplication)—and implements these in the HarmonicSums package. By linking $S$-sums with Poincaré iterated integrals and polylogarithms, it enables transformation in both directions, series expansions, and systematic simplifications relevant to high-order QCD calculations. The work provides explicit procedures for handling infinite sums, special alphabets, and asymptotics, with concrete QCD applications and extensive tool support for symbolic manipulation. Overall, it equips theorists with a robust framework to manage and simplify the advanced nested sums arising in multi-loop computations.

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 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 {\tt HarmonicSums}.

Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms

TL;DR

The paper formalizes generalized harmonic sums (-sums) and their deep connections to generalized harmonic polylogarithms via Mellin transforms. It develops comprehensive analytic and algorithmic machinery—Mellin and inverse Mellin transforms, analytic continuations, asymptotic expansions, and a rich set of algebraic relations (quasi-shuffle, differential, and duplication)—and implements these in the HarmonicSums package. By linking -sums with Poincaré iterated integrals and polylogarithms, it enables transformation in both directions, series expansions, and systematic simplifications relevant to high-order QCD calculations. The work provides explicit procedures for handling infinite sums, special alphabets, and asymptotics, with concrete QCD applications and extensive tool support for symbolic manipulation. Overall, it equips theorists with a robust framework to manage and simplify the advanced nested sums arising in multi-loop computations.

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 -sums) arise. They are characterized by rational (or real) numerator weights also different from . 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 -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 -sum expressions. Finally, we calculate algebraic relations for infinite -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}.

Paper Structure

This paper contains 35 sections, 26 theorems, 228 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Basic properties of $S$-Sums and generalized polylogarithms
  3. Identities between Multiple Polylogarithms of Related Arguments
  4. $x+b\rightarrow x$
  5. $b-x\rightarrow x$
  6. $\frac{1-x}{1+x} \rightarrow x$
  7. $k x\rightarrow x$
  8. $\frac{1}{x}\rightarrow x$
  9. Power Series Expansion of Multiple Polylogarithms
  10. Values of Multiple Polylogarithms Expressed by $S$-Sums at Infinity
  11. Analytic Continuation of $S$-sums
  12. The Mellin Transform and Inverse Mellin Transform Between Multiple Polylogarithms and $S$-sums
  13. Calculating the Mellin Transform in terms of $S$-sums
  14. Calculating the Inverse Mellin Transform in terms of generalized polylogarithms
  15. Differentiation of $S$-Sums
  16. ...and 20 more sections

Key Result

Theorem 2.1

Let $a_1, a_2, \ldots a_k \in \mathbb N$ and $x_1, x_2, \ldots x_k \in \mathbb R^*$ for $k \in \mathbb N.$ The $S$-sum $\textnormal{S}_{a_1,a_2,\ldots,a_k}\space\left(x_1,x_2,\ldots,x_k;n\right)$ is absolutely convergent, when $n\rightarrow \infty$, if and only if one of the following conditions hol In addition the $S$-sum is conditional convergent (convergent but not absolutely convergent) if and

Figures (1)

  • Figure 1: $\textnormal{S}_{1,2,2}\space\left(-1,2,-\frac{1}{3};n\right)$ for $n\in \{0,1,\ldots,11\}$, and $f_e\left(\frac{n}{2}\right)$, $f_o\left(\frac{n-1}{2}\right)$ for $n\in (-\frac{1}{2},11)$.

Theorems & Definitions (38)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Remark 3.5
  • Lemma 3.6
  • proof
  • ...and 28 more