Raney distributions and random matrix theory

Recent works have shown that the family of probability distributions with moments given by the Fuss-Catalan numbers permit a simple parameterized form for their density. We extend this result to the Raney distribution which by definition has its moments given by a generalization of the Fuss-Catalan numbers. Such computations begin with an algebraic equation satisfied by the Stieltjes transform, which we show can be derived from the linear differential equation satisfied by the characteristic polynomial of random matrix realizations of the Raney distribution. For the Fuss-Catalan distribution, an equilibrium problem characterizing the density is identified. The Stieltjes transform for the limiting spectral density of the singular values squared of the matrix product formed from $q$ inverse standard Gaussian matrices, and $s$ standard Gaussian matrices, is shown to satisfy a variant of the algebraic equation relating to the Raney distribution. Supported on $(0,\infty)$, we show that it too permits a simple functional form upon the introduction of an appropriate choice of parameterisation. As an application, the leading asymptotic form of the density as the endpoints of the support are approached is computed, and is shown to have some universal features.