Propriétés combinatoires et prolongement analytique effectif de polyzêtas de Hurwitz et de leurs homologues@@@Combinatorial properties and effective analytic continuation of Hurwitz polyzetas and their analogous

Combinatorial properties and effective analytic continuation of Hurwitz polyzetas and their analogous. In this work, we study a family of Parametrized Dirichlet generating series which contains coloured polyzeta and Hurwitz polyzeta functions. This family verifies two shuffle relations; and we also include quasi-periodic relations and translational variable relations. Thanks to encoding with iterated integral, we obtain an integral representation of this series and deduce their analytic continuation over C. At the end, we describe an algorithm giving the residues of this continuation.