Densities of the Raney distributions

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, $0<r\le p$, then $\mu(p,r)$ is absolutely continuous and its density $W_{p,r}(x)$ can be expressed in terms of the Meijer and the generalized hypergeometric functions. In some cases, including the multiplicative free square and the multiplicative free square root of the Marchenko-Pastur measure, $W_{p,r}(x)$ turns out to be an elementary function

Coherent States from Combinatorial Sequences

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.Comment: 4 pages, Latex; Conference 'Quantum Theory and Symmetries 2', Krakow, Poland, July 200

Densities of the Raney distributions

We prove that if $p _{-}^{> } 1$ and $0 1 is a rational number,$0 < r _{-}^{< } p, then \mu \left ( p,r \right )$is absolutely continuous and its density$W_{p,r}(x)$can be expressed in terms of the Meijer and the generalized hypergeometric functions. In some cases, including the multiplicative free square and the multiplicative free square root of the Marchenko-Pastur measure,$W_{p,r}(x)$ turns out to be an elementary function

Rational Hadamard products via Quantum Diagonal Operators

We use the remark that, through Bargmann-Fock representation, diagonal operators of the Heisenberg-Weyl algebra are scalars for the Hadamard product to give some properties (like the stability of periodic fonctions) of the Hadamard product by a rational fraction. In particular, we provide through this way explicit formulas for the multiplication table of the Hadamard product in the algebra of rational functions in \C[[z]]

The probability measure corresponding to 2-plane trees

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