Free Bessel laws

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 ${\pi }_{s1}={\pi }^{⊠s}$ and ${\pi }_{1t}={\pi }^{⊞t}$. 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.