Densities of the raney distributions

Abstract

We prove that if p ≥ 1 and 0 < r ≤ p then the sequence (mp+rm) rmp+r 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 < r ≤ 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.