Counting factorizations of Coxeter elements into products of reflections

In this paper, we count factorizations of Coxeter elements in wellgenerated complex reflection groups into products of reflections. We obtain a simple product formula for the exponential generating function of such factorizations, which is expressed uniformly in terms of natural parameters of the group. In the case of factorizations of minimal length, we recover a formula due to P. Deligne, J. Tits and D. Zagier in the real case and to D. Bessis in the complex case. For the symmetric group, our formula specializes to a formula of B. Shapiro, M. Shapiro and A. Vainshtein.