Abstract:
While dealing with a class of generalized Bergman spaces on the unit ball, we construct for each of these spaces a set of coherent states to apply a coherent states quantization method. This provides us with another way to recover the Berezin transforms attached to these spaces. Finally, a new formula representing these transforms a functions of the Laplace-Beltrami operator is established in terms ofWilson polynomials by using the Fourier-Helgason transform.

Abstract:
A family of generalized binomial probability distributions attached to Landau levels on the Riemann sphere is introduced by constructing a kind of generalized coherent states. Their main statistical parameters are obtained explicitly. As an application, photon number statistics related to coherent states under consideration are discussed.

Abstract:
We consider the quantum dynamics of a charged particle evolving under the action of a constant homogeneous magnetic field, with emphasis on the discrete subgroups of the Heisenberg group (in the Euclidean case) and of the SL(2, R) group (in the Hyperbolic case). We investigate completeness properties of discrete coherent states associated with higher order Euclidean and hyperbolic Landau levels, partially extending classic results of Perelomov and of Bargmann, Butera, Girardello and Klauder. In the Euclidean case, our results follow from identifying the completeness problem with known results from the theory of Gabor frames. The results for the hyperbolic setting follow by using a combination of methods from coherent states, time-scale analysis and the theory of Fuchsian groups and their associated automorphic forms.

Abstract:
A class of generalized coherent states with a new type of the identity resolution are constructed by replacing the labeling parameter zn/n! of the canonical coherent states by Meixner-Pollaczek polynomials with specific parameters. The constructed coherent states belong to the state Hilbert space of the Gol'dman-Krivchenkov Hamiltonian.

Abstract:
We construct a family of coherent states transforms attached to generalized Bargmann spaces [C.R. Acad.Sci.Paris, t.325,1997] in the complex plane. This constitutes another way of obtaining the kernel of an isometric operator linking the space of square integrable functions on the real line with the true-poly-Fock spaces [Oper.Theory. Adv.Appl.,v.117,2000].

Abstract:
We construct a one parameter family of integral transforms connecting the classical Hardy space with a class of weighted Bergman spaces called Barut-Girardello spaces.

Abstract:
We construct a new one-parameter family of index hypergeometric transforms associated with the relativistic pseudoharmonic oscillator by using coherent states analysis.

Abstract:
When considering a charged particle evolving in the Poincar\'e disk under influence of a uniform magnetic field with a strength proportional to +1, we construct for all hyperbolic Landau level \epsilon^\gamma_$m$ m = 4m(-m), m 2 Z+ \[0, /2] a family of coherent states transforms labeled by (,m) and mapping isometrically square integrable functions on the unit circle with respect to the measure sin^\gamma-2m (\theta/2) d\theta onto spaces of bound states of the particle. These transforms are called circular Bargmann transforms.

Abstract:
We construct coherent states through special superpositions of photon number states of the relativistic isotonic oscillator. In each superposition the coefficients are chosen to be L 2 eingenfunctions of a sigma weight Maass Laplacian on the Poincare disk, which are associated with discrete eigenvalues. For each nonzero m the associated coherent states transform constitutes the m true polyanalytic extension of a relativistic version of the second Bargmann transform, whose integral kernel is expressed in terms of a special Appel Kampe de Feriet hypergeometric function. The obtained results could be used to extend the known semi classical analysis of quantum dynamics of the relativistic isotonic oscillator.

Abstract:
We construct a class of generalized phase coherent states indexed by points of the unit circle and depending on three positive parameters "gamma","alpha" and "epsilon" by replacing the labelling coefficients of the canonical coherent states by circular Jacobi polynomials with parameter "gamma". The special case "gamma" = 0 corresponds to well known phase coherent phase states. The constructed states are superposition of eigenstates of a one-parameter pseudoharmonic oscillator depending on "alpha" and solve the identity of the state Hilbert space at the limit "epsilon"->0+. Closed form for their wavefunctions are obtained in the case "alpha" = "gamma" + 1 and their associated coherent states transform is defined.