On generalized Cauchy-Stieltjes transforms of some Beta distributions

Details On generalized Cauchy-Stieltjes transforms of some Beta distributions On generalized Cauchy-Stieltjes transforms of some Beta distributions

2009-01-31

arxiv.org/abs/0902.0054v1

Mathematics

Probability

Scientific paper

We express generalized Cauchy-Stieltjes transforms of some particular Beta distributions (of ultraspherical type generating functions for orthogonal polynomials) as a powered Cauchy-Stieltjes transform of some measure. For suitable values of the power parameter, the latter measure turns out to be a probability measure and its density is written down using Markov transforms. The discarded values give a negative answer to a deformed free probability unless a restriction on the power parameter is made. A particular symmetric distribution interpolating between Wigner and arcsine distributions is obtained. Its moments are expressed through a terminating hypergeometric series interpolating between Catalan and shifed Catalan numbers. for small values of the power parameter, the free cumulants are computed. Interesting opne problems related to a deformed representation theory of the infinite symmetric group and to a deformed Bozejko's convolution are discussed.

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