∆-cumulants in terms of moments

International audienceThe ∆-convolution of real probability measures, introduced by Bo˙ zejko, generalizes both free and boolean convolutions. It is linearized by the ∆-cumulants, and Yoshida gave a combinatorial formula for moments in terms of ∆-cumulants, that implicitly defines the latter. It relies on the definition of an appropriate weight on noncrossing partitions. We give here two different expressions for the ∆-cumulants: the first one is a simple variant of Lagrange inversion formula, and the second one is a combinatorial inversion of Yoshida's formula involving Schröder trees