Densities of the Raney distributions

  • Wojciech Mlotkowski

  • Karol A. Penson

  • Karol Życzkowski

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We prove that if p1p\ge1 and 0<rp0< r\le p then the sequence (mp+rm)rmp+r\binom{mp+r}m\frac{r}{mp+r} is positive definite. More precisely, it is the moment sequence of a probability measure μ(p,r)\mu(p,r) with compact support contained in [0,+)[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=2x=2. We show that if p>1p>1 is a rational number and 0<rp0<r\le p then μ(p,r)\mu(p,r) is absolutely continuous and its density Wp,r(x)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, Wp,r(x)W_{p,r}(x) turns out to be an elementary function.

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Wojciech Mlotkowski, Karol A. Penson, Karol Życzkowski, Densities of the Raney distributions. Doc. Math. 18 (2013), pp. 1573–1596

DOI 10.4171/DM/437