Harmonic sums and polylogarithms at non-positive multi-indices

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