Cumulants and convolutions via Abel polynomials

TL;DR

This paper introduces a unified polynomial framework using Abel polynomials to relate moments and cumulants across classical, boolean, and free probability theories, revealing new connections and constructing generalized cumulants.

Contribution

It provides a symbolic polynomial approach to connect moments and cumulants in multiple probability frameworks and introduces a new class of generalized cumulants.

Findings

Unified polynomial expression for moments and cumulants across theories

Identification of volume polynomial as a key component in free cumulant theory

Explicit connection between boolean and free convolutions via umbral Fourier transform

Abstract

We provide an unifying polynomial expression giving moments in terms of cumulants, and viceversa, holding in the classical, boolean and free setting. This is done by using a symbolic treatment of Abel polynomials. As a by-product, we show that in the free cumulant theory the volume polynomial of Pitman and Stanley plays the role of the complete Bell exponential polynomial in the classical theory. Moreover via generalized Abel polynomials we construct a new class of cumulants, including the classical, boolean and free ones, and the convolutions linearized by them. Finally, via an umbral Fourier transform, we state a explicit connection between boolean and free convolution.