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 ${}_s{F}_{s-1}$. 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.

• Received 31 March 2011

DOI:https://doi.org/10.1103/PhysRevE.83.061118