On the computation of classical, boolean and free cumulants

TL;DR

This paper presents a unified, efficient algorithm for converting between moments and cumulants across classical, boolean, and free cases, utilizing umbral calculus and polynomial representations.

Contribution

It introduces a novel, unified algorithm based on umbral calculus for computing moments and cumulants, applicable to multiple types including classical, boolean, and free.

Findings

Algorithm is simple and computationally efficient

Provides a single formula for different cumulant types

Includes MAPLE implementation and practical examples

Abstract

This paper introduces a simple and computationally efficient algorithm for conversion formulae between moments and cumulants. The algorithm provides just one formula for classical, boolean and free cumulants. This is realized by using a suitable polynomial representation of Abel polynomials. The algorithm relies on the classical umbral calculus, a symbolic language introduced by Rota and Taylor in 1994, that is particularly suited to be implemented by using software for symbolic computations. Here we give a MAPLE procedure. Comparisons with existing procedures, especially for conversions between moments and free cumulants, as well as examples of applications to some well-known distributions (classical and free) end the paper.