Multivariate time-space harmonic polynomials: a symbolic approach

TL;DR

This paper introduces a symbolic method for multivariate time-space harmonic polynomials, enabling easier analysis of multivariate Lévy processes as martingales and providing explicit formulas for classical polynomial families.

Contribution

It develops a new symbolic framework for multivariate harmonic polynomials, simplifying their analysis and computation, and introduces multivariate Lévy-Sheffer systems.

Findings

Explicit closed-form expressions for multivariate classical polynomials

Simplified symbolic representation reduces complexity

Introduction of multivariate Lévy-Sheffer systems

Abstract

By means of a symbolic method, in this paper we introduce a new family of multivariate polynomials such that multivariate L\'evy processes can be dealt with as they were martingales. In the univariate case, this family of polynomials is known as time-space harmonic polynomials. Then, simple closed-form expressions of some multivariate classical families of polynomials are given. The main advantage of this symbolic representation is the plainness of the setting which reduces to few fundamental statements but also of its implementation in any symbolic software. The role played by cumulants is emphasized within the generalized Hermite polynomials. The new class of multivariate L\'evy-Sheffer systems is introduced.