  We extend the definition and construct several bases for polylogarithms Li T
, where T are some series, recognizable by a finite state (multiplicity)
automaton of alphabet 4 X = {x 0 , x 1 }. The kernel of this new
"polylogarithmic map" Li $\bullet$ 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-indices.
