Densities of the Raney distributions
Karol A. Penson
We prove that if and then the sequence is positive definite. More precisely, it is the moment sequence of a probability measure with compact support contained in . This family of measures encompasses the multiplicative free powers of the Marchenko-Pastur distribution as well as the Wigner's semicircle distribution centered at . We show that if is a rational number and then is absolutely continuous and its density can be expressed in terms of the generalized hypergeometric functions. In some cases, including the multiplicative free square and the multiplicative free square root of the Marchenko-Pastur measure, turns out to be an elementary function.
Cite this article
Wojciech Mlotkowski, Karol A. Penson, Karol Życzkowski, Densities of the Raney distributions. Doc. Math. 18 (2013), pp. 1573–1596DOI 10.4171/DM/437