We are concerned with a phase-space probability distribution which is known as Husimi Q-function of a density operator with respect to a set of coherent states vertical bar(kappa) over tilde (Z,B,R,m)> attached to an mth hyperbolic Landau level and labeled by points z of an open disk of radius R, where B > 0 is proportional to a magnetic field strength. For a density operator representing a projector on a Fock state vertical bar j > we obtain the Q(j) distribution and discuss some of its basic properties such as its characteristic function and its main statistical parameters. We achieve the same program for the thermal density operator (mixed states) of the isotonic oscillator for which we establish a lower bound for the associated thermodynamic potential. We recover most of the results of the Euclidean setting (flat case) as the parameter R goes to infinity by making appeal to asymptotic formulas involving orthogonal polynomials and special functions. As a tool, we establish a summation formula for the special Kampe de Feriet function F-2:0:0(1:2:2).

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. Our construction leads to a new q-deformation of the m-true- polyanalytic Bargmann transform on the complex plane. In the analytic case m = 0, the obtained coherent states transform can be associated with the Arik-Coon oscillator for q(0) = q(-1) > 1. These result may be used to introduce a q-deformed Ginibre-type point process.

We discuss a model of a q-harmonic oscillator based on Rogers-Szego functions. We combine these functions with a class of q-analogs of complex Hermite polynomials to construct a new set of coherent states depending on a nonnegative integer parameter m. Our construction leads to a new q-deformation of the m-true-polyanalytic Bargmann transform whose range defines a generalization of the Arik-Coon space. We also give an explicit formula for the reproducing kernel of this space. The obtained results may be exploited to define a q-deformation of the Ginibre-m-type process on the complex plane.

We construct coherent states for each eigenspace of a magnetic Laplacian on the complex projective n-space in order to apply a quantization-dequantization method. Doing so allows to define the Berezin transform for these spaces. We then establish a formula for this transform as a function of the Fubini-Study Laplacian in a closed form involving of a terminating Kampe de Feriet function. For the lowest spherical Landau level on the Riemann sphere the obtained formula reduces to the one derived by Berezin himself.

While dealing with a Hamiltonian with a continuous spectrum, we use a tridiagonal method involving orthogonal polynomials to construct a set of coherent states obeying a Glauber-type condition. We perform a Bayesian decomposition of the weight function of the orthogonality measure to show that the obtained coherent states can be recast in the Gazeau-Klauder approach. The Hamiltonian of the l-wave free particle is treated as an example to illustrate the method. Published under an exclusive license by AIP Publishing.