Geodesic
Densities of the Raney distributions
Documenta mathematica, Tome 18 (2013), pp. 1573-1596.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We prove that if $p\ge1$ and $0 r\le p$ then the sequence $\binom{mp+r}m\frac{r}{mp+r}$ is positive definite. More precisely, it is the moment sequence of a probability measure $\mu(p,r)$ with compact support contained in $[0,+\infty)$. This family of measures encompasses the multiplicative free powers of the Marchenko-Pastur distribution as well as the Wigner's semicircle distribution centered at $x=2$. We show that if $p>1$ is a rational number and $0\le p$ then $\mu(p,r)$ is absolutely continuous and its density $W_{p,r}(x)$ 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, $W_{p,r}(x)$ turns out to be an elementary function.
Classification : 44A60, 33C20
Keywords: Mellin convolution, free convolution, meijer G-function, generalized hypergeometric function 
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     author = {Mlotkowski, Wojciech and Penson, Karol A. and \.Zyczkowski, Karol},
     title = {Densities of the {Raney} distributions},
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     pages = {1573--1596},
     publisher = {mathdoc},
     volume = {18},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2013__18__a1/}
}
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Mlotkowski, Wojciech; Penson, Karol A.; Życzkowski, Karol. Densities of the Raney distributions. Documenta mathematica, Tome 18 (2013), pp. 1573-1596. http://geodesic.mathdoc.fr/item/DOCMA_2013__18__a1/