A new family of time-space harmonic polynomials with respect to Lévy processes

Details A new family of time-space harmonic polynomials with respect to Lévy processes A new family of time-space harmonic polynomials with respect to Lévy processes

2011-08-03

arxiv.org/abs/1108.0788v3

Mathematics

Probability

Ann. Mat. Pura Appl. In press

Scientific paper

By means of a symbolic method, a new family of time-space harmonic polynomials with respect to L\'evy processes is given. The coefficients of these polynomials involve a formal expression of L\'evy processes by which many identities are stated. We show that this family includes classical families of polynomials such as Hermite polynomials. Poisson-Charlier polynomials result to be a linear combinations of these new polynomials, when they have the property to be time-space harmonic with respect to the compensated Poisson process. The more general class of L\'evy-Sheffer polynomials is recovered as a linear combination of these new polynomials, when they are time-space harmonic with respect to L\'evy processes of very general form. We show the role played by cumulants of L\'evy processes so that connections with boolean and free cumulants are also stated.

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