Harmonic sums and polylogarithms at non-positive multi-indices

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.