  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.
