Abstract:
Squared singular values of a product of s square random Ginibre matrices are asymptotically characterized by probability distribution P_s(x), such that their moments are equal to the Fuss-Catalan numbers or 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 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.

Abstract:
We study the probability measure $\mu_{0}$ for which the moment sequence is $\binom{3n}{n}\frac{1}{n+1}$. We prove that $\mu_{0}$ is absolutely continuous, find the density function and prove that $\mu_{0}$ is infinitely divisible with respect to the additive free convolution.

Abstract:
We prove that if $p\geq 1$ and $-1\leq r\leq p-1$ then the binomial sequence $\binom{np+r}{n}$, $n=0,1,...$, is positive definite and is the moment sequence of a probability measure $\nu(p,r)$, whose support is contained in $\left[0,p^p(p-1)^{1-p}\right]$. If $p>1$ is a rational number and $-11$ the measures $\nu(p,-1)$ and $\nu(p,0)$ are certain free convolution powers of the Bernoulli distribution. Finally we prove that the binomial sequence $\binom{np+r}{n}$ is positive definite if and only if either $p\geq 1$, $-1\leq r\leq p-1$ or $p\leq 0$, $p-1\leq r \leq 0$. The measures corresponding to the latter case are reflections of the former ones.

Abstract:
We prove that if $p\ge 1$ and $0< r\le p$ then the sequence $\binom{mp+r}{m}\frac{r}{mp+r}$, $m=0,1,2,...$, is positive definite, more precisely, is the moment sequence of a probability measure $\mu(p,r)$ with compact support contained in $[0,+\infty)$. 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, $0

Abstract:
We consider the Hubbard model and its extensions on bipartite lattices. We define a dynamical group based on the $\eta$-pairing operators introduced by C.N.Yang, and define coherent pairing states, which are combinations of eigenfunctions of $\eta$-operators. These states permit exact calculations of numerous physical properties of the system, including energy, various fluctuations and correlation functions, including pairing ODLRO to all orders. This approach is complementary to BCS, in that these are superconducting coherent states associated with the exact model, although they are not eigenstates of the Hamiltonian.

Abstract:
We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobinski relations. These new combinatorial numbers are shown to be coherent state matrix elements of powers of the monomials in question. It is further demonstrated that such Bell-type numbers, when considered as power moments, give rise to positive measures on the positive half-axis, which in many cases can be written in terms of known functions.

Abstract:
Conventional Bell and Stirling numbers arise naturally in the normal ordering of simple monomials in boson operators. By extending this process we obtain generalizations of these combinatorial numbers, defined as coherent state matrix elements of arbitrary monomials, as well as the associated Dobinski relations. These Bell-type numbers may be considered as power moments and give rise to positive measures which allow the explicit construction of new classes of coherent states.

Abstract:
We construct coherent states using sequences of combinatorial numbers such as various binomial and trinomial numbers, and Bell and Catalan numbers. We show that these states satisfy the condition of the resolution of unity in a natural way. In each case the positive weight functions are given as solutions of associated Stieltjes or Hausdorff moment problems, where the moments are the combinatorial numbers.

Abstract:
We study a two-parameter family $a_{n}(p,t)$ of deformations of the Fuss numbers. We show a sufficient condition for positive definiteness of $a_n(p,t)$ and prove that some of the corresponding probability measures are infinitely divisible with respect to the additive free convolution.

Abstract:
We investigate the level density for several ensembles of positive random matrices of a Wishart--like structure, $W=XX^{\dagger}$, where $X$ stands for a nonhermitian random matrix. In particular, making use of the Cauchy transform, we study free multiplicative powers of the Marchenko-Pastur (MP) distribution, ${\rm MP}^{\boxtimes s}$, which for an integer $s$ yield Fuss-Catalan distributions corresponding to a product of $s$ independent square random matrices, $X=X_1\cdots X_s$. New formulae for the level densities are derived for $s=3$ and $s=1/3$. Moreover, the level density corresponding to the generalized Bures distribution, given by the free convolution of arcsine and MP distributions is obtained. We also explain the reason of such a curious convolution. The technique proposed here allows for the derivation of the level densities for several other cases.