Kleene stars of the plane, polylogarithms and symmetries
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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 ∙ 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.
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