-cumulants in terms of moments
Abstract
The -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 Schr\"oder trees.
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