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# Free Bessel Laws

Published:2010-07-06
Printed: Feb 2011
• T. Banica,
Department of Mathematics, Toulouse 3 University, Toulouse, France
• S. T. Belinschi,
• M. Capitaine,
Department of Mathematics, Toulouse 3 University, Toulouse, France
• B. Collins,
Department of Mathematics, Lyon 1 University, France
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## Abstract

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.
 Keywords: Poisson law, Bessel function, Wishart matrix, quantum group
 MSC Classifications: 46L54 - Free probability and free operator algebras 15A52 - Random matrices16W30 - Coalgebras, bialgebras, Hopf algebras (See also 16S40, 57T05); rings, modules, etc. on which these act
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