We introduce and study a remarkable family of real probability measures $\pi_{ st}$, that we call free Bessel laws. These are related to the free Poisson law $\pi$ via the formulae $\pi_{ s1}=\pi^{ \boxtimes s}$ and $\pi_{ 1t}=\pi^{ \boxplus t}$. 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.
Categories: math.PR math.QA