The aim of this paper is to review our results on the description of multiparameter deformed oscillators and their oscillator algebras. We define generalized (q; α, β, γ, ν)-deformed oscillator algebras and study their irreducible representations. The Arik–Coon oscillator with the main relation aa+ – qa+a = 1, where q >1, is embedded in this framework. We have found the connection of this oscillator with the Askey q–1-Hermite polynomials. We construct a family of generalized coherent states associated with these polynomials and give their explicit expression in terms of standard special functions. By means of the solution of the appropriate classical Stieltjes moment problem, we prove the property of (over)completeness of these states.
In this paper we study the Arik–Coon oscillator with the main relation aa + − qa + a = 1, where q > 1, in the framework of the unified ( q ; α, β, γ; ν)-deformed oscillator algebra. This oscillator is connected with the Askey q −1 -Hermite polynomials. We construct a family of the generalized coherent states associated with these polynomials and give their explicit expression in terms of standard special functions. By means of the solution of the appropriate classical moment problem, we prove the over-completeness of this family of coherent states.
The aim of this paper is the study of the generalized deformed quantum oscillators in the framework (q; α, β, γ; ν)-deformed of oscillator algebra. By selecting the special values of deformation parameters, we have separated a generalized deformed oscillator connected with generalized discrete Hermite II polynomials. By these means we have constructed Barut–Girardello type coherent states of this oscillator. We have found the conditions on the (q; α, β, γ; ν)-deformation parameters at which the free (q; α, β, γ; ν)-deformed oscillator approximate the usual anharmonic oscillator in the homogeneous Kerr medium.
The Heisenberg algebra is first deformed with the set of parameters
q,l,λ to generate a new family of generalized coherent states. In this framework, the matrix elements of relevant operators are exactly computed. A proof on sub-Poissonian character of the statistics of the main deformed states is provided. This property is used to determine a generalized metric. A unified method of calculating structure functions from commutation relations of deformed single-mode oscillator algebras is then presented. A natural approach to building coherent states associated to deformed algebras is deduced. Known deformed algebras are given as illustration. Futhermore, we generalize a class of two-parameter deformed Heisenberg algebras related to meromorphic functions, called
R(p,q)-deformed algebra. Relevant families of coherent states maps are probed and their corresponding hypergeometric series are computed. The latter generalizes known hypergeometric series and gives to a generalization of the binomial theorem. The involved notions of differentiation and integration generalize the usual
q- and
(p,q)-differentiation and integration. A Hopf algebra structure compatible with the
R(p,q)-algebra is deduced. We succeed in giving a new characterization of Rogers- Szegö polynomials, called
R(p,q)-deformed Rogers-Szegö polynomials, by their three-term recursion relations and the associated quantum algebra built with corresponding creation and annihilation operators. Continuous
R(p,q)-deformed Hermite polynomials and their recursion relation are also deduced. Novel algebraic relations are provided and discussed. The whole formalism is performed in a unified way, generalizing known relevant results which are straightforwardly derived as particular cases.