Free bessel laws

Abstract

We introduce and study a remarkable family of real probabilitymeasures πst that we call free Bessel laws. These are related to the free Poisson law π via the formulae πs1 = πs and π1t = πt. Our study includes definition and basic properties analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.