A q deformation of true-polyanalytic Bargmann transforms when q^{-1}> 1

We combine continuous $q^{-1}$-Hermite Askey polynomials with new $2D$ orthogonal polynomials introduced by Ismail and Zhang as $q$-analogs for complex Hermite polynomials to construct a new set of coherent states depending on a nonnegative integer parameter $m$. In the analytic case corresponding to $m=0$, we recover a known result on the Ar\"{\i}k-Coon oscillator for $q'=q^{-1}>1$. Our construction leads to a new $q$-deformation of the $m$-true-polyanalytic Bargmann transform on the complex plane. The obtained result may be used to introduce a $q$-deformed Ginibre-type point process.Comment: 15 page

Maximal univalent disks of real rational functions and Hermite-Biehler polynomials

The well-known Hermite-Biehler theorem claims that a univariate monic polynomial s of degree k has all roots in the open upper half-plane if and only if s=p+iq where p and q are real polynomials of degree k and k-1 resp. with all real, simple and interlacing roots, and q has a negative leading coefficient. Considering roots of p as cyclically ordered on RP^1 we show that the open disk in CP^1 having a pair of consecutive roots of p as its diameter is the maximal univalent disk for the function R=\frac{q}{p}. This solves a special case of the so-called Hermite-Biehler problem.Comment: 10 pages, 4 figure

Gaussian Limits and Polynomials on High Dimensional Spheres

We show in detail that the limit of spherical surface integrals taken over slices of a high dimensional sphere is a Gaussian integral on an affine plane of finite codimension in infinite dimensional space. We then utilize these ideas to show that a natural class of orthogonal polynomials on high dimensional spheres limit to Hermite polynomials