We extend the definition and construct several bases for polylogarithms LiT, where T are some series, recognizable by a finite state (multiplicity) automaton of alphabet1 X={x0,x1}. The kernel of this new “polylogarithmic map” Li• is also characterized and provides a rewriting process which terminates to a normal form. We concentrate on algebraic and analytic aspects of this extension allowing index polylogarithms at non-positive multi-indices, by rational series and regularize polyzetas at non-positive multi-indices2.

By an effective construction of pairs of bases in duality in shuffle and quasi-shuffle algebras, we identify the local coordinates of noncommutative generating series of polyzetas which are group-like. Algorithms lead to the ideal of homogeneous polynomial relations, in weight, among polyzetas and their explicit representation on irreducible elements.