# \Delta-cumulants in terms of moments

## TL;DR

This paper explores -cumulants related to -convolution, providing new formulas for -cumulants using Lagrange inversion and Schr46der trees, enhancing understanding of their combinatorial structure.

## Contribution

It introduces two novel formulas for -cumulants, expanding the combinatorial and analytical tools available for studying -convolution.

## Key findings

- Derived a simple Lagrange inversion formula for -cumulants.
- Developed a combinatorial inversion involving Schr46der trees.
- Provided explicit expressions linking moments and -cumulants.

## Abstract

The \Delta-convolution of real probability measures, introduced by Bo\.zejko, generalizes both free and boolean convolutions. It is linearized by the \Delta-cumulants, and Yoshida gave a combinatorial formula for moments in terms of \Delta-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 \Delta-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.