Squared singular values of a product of s square random Ginibre matrices are asymptotically characterized by probability distributions Ps(x), such that their moments are equal to the Fuss\char21{}Catalan numbers of order s. We find a representation of the Fuss-Catalan distributions Ps(x) in terms of a combination of s hypergeometric functions of the type sFs-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.

We construct a large number of coherent states satisfying the resolution of unity with a positive weight function, obtained through analytic solutions of the Stieltjes moment problem (coherent states on a plane) and the Hausdorff moment problem (coherent states on a disk). These solutions are obtained through the method of inverse Mellin transform. In addition, these coherent states induce a deformation of the metric that has been calculated analytically.

We study functions g$\alpha${}(x) which are one-sided, heavy-tailed L\'evy stable probability distributions of index $\alpha${}, 0<$\alpha${}<1, of fundamental importance in random systems, for anomalous diffusion and fractional kinetics. We furnish exact and explicit expressions for g$\alpha${}(x), 0$\le${}x<$\infty${}, for all $\alpha${}=l/k4, all expressed in terms of known functions. This will allow a ``fine-tuning'' of $\alpha${} in order to adapt g$\alpha${}(x) to a given experimental situation.

We investigate functions g($\alpha${},$\beta${};x) which are heavy-tailed L\'evy stable probability distributions of index 0<$\alpha${}$\leqslant${}2 and appropriate asymmetry parameter $\beta${}. They are of central importance in physics of amorphous and disordered systems, econophysics, geology, hydrology, internet traffic, dynamics of human relations, etc. We present an ensemble of exact and explicit solutions of g($\alpha${},$\beta${};x) for all admissible rational values of $\alpha${} and $\beta${}. We reproduce all the previously known cases and furnish distributions for values $\alpha${} and $\beta${} which were previously inaccessible. We point out many instances of experimental and statistical data that could be described by our results.

We obtain exact results for fractional equations of Fokker-Planck type using the evolution operator method. We employ exact forms of one-sided L\'evy stable distributions to generate a set of self-reproducing solutions. Explicit cases are reported and studied for various fractional order of derivatives, different initial conditions, and for different versions of Fokker-Planck operators.