Multivariate Bernoulli and Euler polynomials via L\'evy processes

TL;DR

This paper introduces multivariate Bernoulli and Euler polynomials using Le9vy processes, providing a symbolic framework that simplifies their properties and reveals a direct relation between the two families.

Contribution

It presents a novel symbolic method to define multivariate Bernoulli and Euler polynomials via Le9vy processes, enabling straightforward property derivation.

Findings

Properties of the polynomials are easily derived from the representation.

A simple relation between Bernoulli and Euler polynomials is established.

The method is suitable for implementation in symbolic computation systems.

Abstract

By a symbolic method, we introduce multivariate Bernoulli and Euler polynomials as powers of polynomials whose coefficients involve multivariate L\'evy processes. Many properties of these polynomials are stated straightforwardly thanks to this representation, which could be easily implemented in any symbolic manipulation system. A very simple relation between these two families of multivariate polynomials is provided.