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Free Bessel Laws

Published online by Cambridge University Press:  20 November 2018

T. Banica ,
S. T. Belinschi ,
M. Capitaine  and
B. Collins
T. Banica
Affiliation:
Department of Mathematics, Toulouse 3 University, Toulouse, France email: banica@picard.ups-tlse.frcapitain@cict.fr
S. T. Belinschi
Affiliation:
Department of Mathematics, University of Saskatchewan, Saskatoon, SK email: belinsch@math.usask.ca
M. Capitaine
Affiliation:
Department of Mathematics, Toulouse 3 University, Toulouse, France email: banica@picard.ups-tlse.frcapitain@cict.fr
B. Collins
Affiliation:
Department of Mathematics, Lyon 1 University, France and University of Ottawa, Ottawa, ON email: collins@math.univ-lyon1.fr
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Abstract

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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 ${{\text{ }\!\!\pi\!\!\text{ }}_{s1}}={{\text{ }\!\!\pi\!\!\text{ }}^{\boxtimes s}}$ and $\text{ }\pi {{\text{ }}_{1t}}=\text{ }\pi {{\text{ }}^{\boxtimes }}^{t}$. 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

46L5415A5216W30Poisson lawBessel functionWishart matrixquantum group
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 1 , 01 February 2011 , pp. 3 - 37
Copyright
Copyright © Canadian Mathematical Society 2011

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