[FORTHCOMING] A unified characterization of certain operators related to deformed oscillator algebras via d-orthogonality

In the presentwork,we are interested in the linear operators of the form $$$S= T(a_+)R(a_-)$$$, where $$$a_-$$$ and $$$a_+$$$ are the annihilation and creation operators, respectively defined in irreducible representation of a deformed oscillator algebra and $$$T$$$, $$$R$$$ are analytic functions. We characterize all real sequences $$$(x_k)_{k\geq0}$$$ and functions $$$T$$$ for which the matrix elements associated to the operator $$$S$$$ are expressed in terms of polynomial sets on the discrete variable $$$x_k$$$ and we show when the considered polynomial sets are $$$d$$$-orthogonal. The analytic function $$$R$$$, in most specific cases is expressed in terms of exponential or $$$q$$$-exponential functions. As a consequence, several known results are recovered and extended, including those related to the Heisenberg-Weyl algebra, and $$$q$$$-deformed oscillator algebras. Explicit realizations are given in terms ofMeixner and Charlier-type $$$d$$$-orthogonal polynomials, together with their $$$q$$$-analogues.

In the presentwork,we are interested in the linear operators of the form $$$S= T(a_+)R(a_-)$$$, where $$$a_-$$$ and $$$a_+$$$ are the annihilation and creation operators, respectively defined in irreducible representation of a deformed oscillator algebra and $$$T$$$, $$$R$$$ are analytic functions. We characterize all real sequences $$$(x_k)_{k\geq0}$$$ and functions $$$T$$$ for which the matrix elements associated to the operator $$$S$$$ are expressed in terms of polynomial sets on the discrete variable $$$x_k$$$ and we show when the considered polynomial sets are $$$d$$$-orthogonal. The analytic function $$$R$$$, in most specific cases is expressed in terms of exponential or $$$q$$$-exponential functions. As a consequence, several known results are recovered and extended, including those related to the Heisenberg-Weyl algebra, and $$$q$$$-deformed oscillator algebras. Explicit realizations are given in terms ofMeixner and Charlier-type $$$d$$$-orthogonal polynomials, together with their $$$q$$$-analogues.

Deformed oscillator algebras matrix elements of operators coherent states d-orthogonal polynomial sets generating functions

Deformed oscillator algebras matrix elements of operators coherent states d-orthogonal polynomial sets generating functions