The package HarmonicSums: Computer Algebra and Analytic aspects of Nested Sums

TL;DR

The paper presents HarmonicSums, a Mathematica package designed to facilitate analytic and algebraic work on nested sums and iterated integrals arising in high-order perturbative calculations. It details core features including Mellin transforms, sum-integral transformations, algebraic relations, basis reduction, and series/asymptotic expansions, supported by a comprehensive set of commands (e.g., TransformToSSums, ReduceToBasis, Mellin, HarmonicSumsSeries). The work emphasizes interoperability between sum-based and integral-based representations (harmonic, S-, cyclotomic sums and polylogarithms) and showcases extensive tooling for simplification, expansion, and symbolic manipulation. This enables more efficient derivations, simplifications, and analytic continuations in quantum field theory computations, with precomputed tables and online resources for broader use.

Abstract

This paper summarizes the essential functionality of the computer algebra package HarmonicSums. On the one hand HarmonicSums can work with nested sums such as harmonic sums and their generalizations and on the other hand it can treat iterated integrals of the Poincare and Chen-type, such as harmonic polylogarithms and their generalizations. The interplay of these representations and the analytic aspects are illustrated by concrete examples.