Geodesic
Densities of the Raney distributions
Documenta mathematica, Tome 18 (2013), pp. 1573-1596
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We prove that if p≥1 and 0≤p then the sequence (mmp+r​)mp+rr​ is positive definite. More precisely, it is the moment sequence of a probability measure μ(p,r) with compact support contained in [0,+∞). 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≤p then μ(p,r) is absolutely continuous and its density Wp,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, Wp,r​(x) turns out to be an elementary function.
DOI : 10.4171/dm/437
Classification : 33C20, 44A60
Mots-clés : free convolution, Mellin convolution, meijer G-function, generalized hypergeometric function 
@article{10_4171_dm_437,
     author = {Karol A. Penson and Wojciech Mlotkowski and Karol \.Zyczkowski},
     title = {Densities of the {Raney} distributions},
     journal = {Documenta mathematica},
     pages = {1573--1596},
     year = {2013},
     volume = {18},
     doi = {10.4171/dm/437},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/437/}
}
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Karol A. Penson; Wojciech Mlotkowski; Karol Życzkowski. Densities of the Raney distributions. Documenta mathematica, Tome 18 (2013), pp. 1573-1596. doi: 10.4171/dm/437

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