On free stable distributions

Abstract

We investigate the analytical properties of free stable distributions and discover many connections with the classical stable distributions. Our main results include an explicit formula for the Mellin transform, series representations for the characteristic function and for the density of the free stable distribution; all of these explicit formulas exhibit close resemblance to the corresponding expressions for classical stable distributions. As applications of these results we obtain a new derivation of the duality law obtained in [1] and a factorization of a classical stable random variable into the independent (in the classical sense) product of a free stable random variable and a power of Gamma(2) random variable.Comment: 12 page