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In order to extend the Sch\"utzenberger's factorization, the combinatorialHopf algebra of the $q$-stuffles product is developed systematically in aparallel way with that of the shuffle product and and in emphasizing the Lieelements as studied by Ree. In particular, we will give here an effectiveconstruction of pair of bases in duality.

Sch\"utzenberger's factorization on the (completed) Hopf algebra of $q-$stuffle product