Abstract
In this paper, we review the theory of time space-harmonic polynomials
developed by using a symbolic device known in the literature as the classical umbral
calculus. The advantage of this symbolic tool is twofold. First a moment representation
is allowed for a wide class of polynomial stochastic processes involving the L´evy ones in
respect to which they are martingales. This representation includes some well-known
examples such as Hermite polynomials in connection with Brownian motion. As a
consequence, characterizations of many other families of polynomials having the time
space-harmonic property can be recovered via the symbolic moment representation.
New relations with Kailath-Segall polynomials are also stated. Secondly the generalization
to the multivariable framework is straightforward. Connections with cumulants
and Bell polynomials are highlighted both in the univariate case and multivariate one.
Open problems are addressed at the end of the paper.
