A new family of time-space harmonic polynomials with respect to L\'evy processes

TL;DR

This paper introduces a new family of time-space harmonic polynomials related to Lévy processes, unifying classical polynomials and exploring their connections with cumulants, boolean, and free probability.

Contribution

It presents a novel family of polynomials with harmonic properties for Lévy processes, encompassing classical and Lévy-Sheffer polynomials, and highlights their connections with cumulants.

Findings

Includes classical polynomial families like Hermite and Poisson-Charlier.

Shows that Lévy-Sheffer polynomials are linear combinations of the new family.

Establishes links between cumulants of Lévy processes and boolean/free cumulants.

Abstract

By means of a symbolic method, a new family of time-space harmonic polynomials with respect to L\'evy processes is given. The coefficients of these polynomials involve a formal expression of L\'evy processes by which many identities are stated. We show that this family includes classical families of polynomials such as Hermite polynomials. Poisson-Charlier polynomials result to be a linear combinations of these new polynomials, when they have the property to be time-space harmonic with respect to the compensated Poisson process. The more general class of L\'evy-Sheffer polynomials is recovered as a linear combination of these new polynomials, when they are time-space harmonic with respect to L\'evy processes of very general form. We show the role played by cumulants of L\'evy processes so that connections with boolean and free cumulants are also stated.