By using a symbolic technique known in the literature as the classical umbral
calculus, we characterize two classes of polynomials related to L\'evy
processes: the Kailath-Segall and the time-space harmonic polynomials. We
provide the Kailath-Segall formula in terms of cumulants and we recover simple
closed-forms for several families of polynomials with respect to not centered
L\'evy processes, such as the Hermite polynomials with the Brownian motion, the
Poisson-Charlier polynomials with the Poisson processes, the actuarial
polynomials with the Gamma processes, the first kind Meixner polynomials with
the Pascal processes, the Bernoulli, Euler and Krawtchuk polynomials with
suitable random walks
