Product of Ginibre matrices: Fuss-Catalan and Raney distributions

Squared singular values of a product of s square random Ginibre matrices are asymptotically characterized by probability distributions P_s(x), such that their moments are equal to the Fuss-Catalan numbers of order s. We find a representation of the Fuss-Catalan distributions P_s(x) in terms of a combination of s hypergeometric functions of the type _sF_s-1. The explicit formula derived here is exact for an arbitrary positive integer s, and for s=1 it reduces to the Marchenko-Pastur distribution. Using similar techniques, involving the Mellin transform and the Meijer G function, we find exact expressions for the Raney probability distributions, the moments of which are given by a two-parameter generalization of the Fuss-Catalan numbers. These distributions can also be considered as a two-parameter generalization of the Wigner semicircle law.