Harmonic sums and polylogarithms at non-positive multi-indices

TL;DR

This paper explores the combinatorial structure of harmonic sums and polylogarithms at non-positive multi-indices, introducing renormalization techniques using shuffle Hopf algebras to handle divergences.

Contribution

It extends classical formulas and develops a combinatorial framework for renormalizing divergent polyzetas at non-positive multi-indices.

Findings

Provides a new combinatorial approach to harmonic sums and polylogarithms.

Introduces a renormalization process using shuffle Hopf algebras.

Enhances understanding of the structure of polylogarithms at non-positive indices.

Abstract

Extending Eulerian polynomials and Faulhaber's formula 1, we study several combi-natorial aspects of harmonic sums and polylogarithms at non-positive multi-indices as well as their structure. Our techniques are based on the combinatorics of non-commutative generating series in the shuffle Hopf algebras giving a global process to renormalize the divergent polyzetas at non-positive multi-indices.