Harmonic Polylogarithms

TL;DR

This paper introduces harmonic polylogarithms, a generalization of Nielsen's polylogarithms, exploring their algebraic properties, transformations, and connections to harmonic sums and Mellin transforms.

Contribution

It defines harmonic polylogarithms, demonstrates their algebraic structure, and explores their transformations and relationships to harmonic sums and Mellin transforms.

Findings

Harmonic polylogarithms form a closed set under argument transformations.

Their coefficients relate to harmonic sums.

They satisfy a product algebra similar to Nielsen's polylogarithms.

Abstract

The harmonic polylogarithms (hpl's) are introduced. They are a generalization of Nielsen's polylogarithms, satisfying a product algebra (the product of two hpl's is in turn a combination of hpl's) and forming a set closed under the transformation of the arguments x=1/z and x=(1-t)/(1+t). The coefficients of their expansions and their Mellin transforms are harmonic sums.