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

作者：Di Nardo E^*; Oliva I

来源：Annali di Matematica Pura ed Applicata, 2013, 192(5): 917-929.

DOI：10.1007/s10231-012-0252-3

• Time-space harmonic polynomial
• Levy process
• Cumulant
• Levy-Sheffer polynomial
• Umbral calculus

By means of a symbolic method, a new family of time-space harmonic polynomials with respect to L,vy processes is given. The coefficients of these polynomials involve a formal expression of L,vy 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,vy-Sheffer polynomials is recovered as a linear combination of these new polynomials, when they are time-space harmonic with respect to L,vy processes of very general form. We show the role played by cumulants of L,vy processes, so that connections with boolean and free cumulants are also stated.

• 出版日期2013-10