Generalized coherent states for q-oscillator connected with q-Hermite polynomials

Borzov, V. V.

doi:10.1109/dd.2003.238130

Cited by 8 publications

(5 citation statements)

References 45 publications

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“…Original Paper 5 Coherent states Definition 3. The coherent states of the Barut-Girardello type for the algebra (76) in the Fock space H F are defined as [14] |z…”

Section: Progress Of Physicsmentioning

confidence: 99%

“…In 1996, Ismail and Rahman [21] derived the raising and lowering operators for orthogonal polynomials on the unit circle introduced by Szegö and for their four parameter generalization to 4 φ 3 biorthogonal rational functions on the unit circle. Later, in 2003, Borsov and Damaskinsky [14] constructed generalized coherent states of Barut-Girardello type for q-oscillator with q-Hermite polynomials. The well-known Arik-Coon oscillator naturally arose in the framework of their approach as oscillator, connected with the Rogers q-Hermite polynomials, in the same way as usual oscillator with standard Hermite polynomials.…”

Section: Introductionmentioning

confidence: 99%

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New q -Hermite polynomials: characterization, operators algebra and associated coherent states

Chung

Hounkonnou

2015

Fortschr. Phys.

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This paper addresses a construction of new q-Hermite polynomials with a full characterization of their main properties and corresponding raising and lowering operator algebra. The threeterm recursive relation as well as the second-order differential equation obeyed by these new polynomials are explicitly derived. Relevant operator actions, including the eigenvalue problem of the deformed oscillator and the self-adjointness of the related position and momentum operators, are investigated and analyzed. The associated coherent states are constructed and discussed with an explicit resolution of the induced moment problem. The phase collapse in a q-deformed boson system is studied. Fortschritte der Physik Progress of PhysicsOriginal Paper lowering operators from the three term recursive relation for Rogers Szegö polynomials and the Jackson q-derivative, and furnished new realizations of the q-deformed algebra associated with the q-deformed harmonic oscillator. For more details on q-calculus and hypergeometric orthogonal polynomials and their q-analogue, see nice compilations performed by Ernst [5] and Koekoek and Swarttouw [6]. See also [22] for information on classical and quantum orthogonal polynomials in one variable.The present work aims at providing a new q-Hermite polynomials with a full characterization of their main properties. Corresponding raising and lowering operators are deduced. The three-term recursion relation as well as the second-order differential equation satisfied by the new polynomials are explicitly derived. Relevant operator actions, including the eigenvalue problem of the deformed oscillator and the self-adjointness of the related position and momentum operators, are investigated and discussed. The coherent states are constructed and discussed; the induced moment problem is solved. The phase collapse in a q-deformed boson system is examined.The paper is organized as follows. As a matter of clarity, Section 2 gives a brief review of known results on Hermite polynomials and q-Rogers Szegö polynomials. In Section 3, we introduce a new family of q-Hermite polynomials and discuss the associated deformed oscillator algebra. In Section 4, we give the relevant operator properties and solve the eigenvalue problem. The selfadjointness of the related position and momentum operators is examined. In Section 5, we construct the associated coherent states. In Section 6, we discuss the phase collapse in a q-deformed boson system in the framework of the constructed q-deformed coherent states. Concluding remarks are highlighted in Section 7.2 Quick overview on the Hermite polynomials and q-Rogers Szegö polynomials der Physik Progress of Physics

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“…Original Paper 5 Coherent states Definition 3. The coherent states of the Barut-Girardello type for the algebra (76) in the Fock space H F are defined as [14] |z…”

Section: Progress Of Physicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

New q -Hermite polynomials: characterization, operators algebra and associated coherent states

Chung

Hounkonnou

2015

Fortschr. Phys.

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“…where n 2 N 0 : Discrete version of the Hermite polynomials is also important and has enormous applications in several problems on theoretical and mathematical physics, e.g., in the continued fractions, Eulerian series [16], algebras and quantum groups [21,22,31], discrete mathematics, algebraic combinatorics (coding theory, design theory, various theories of group representation) [8], q-Schrödinger equation and q-harmonic oscillators [3,4,5,6,7,9,11,23].…”

Section: Introductionmentioning

confidence: 99%

On the limit of discrete q-Hermite I polynomials

Adıgüzel¹,

Alwhishi²,

Turan³

2019

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

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The main purpose of this paper is to introduce the limit relations between the discrete q-Hermite I and Hermite polynomials such that the orthogonality property and the three-terms recurrence relations remain valid. The discrete q-Hermite I polynomials are the q-analogues of the Hermite polynomials which form an important class of the classical orthogonal polynomials. The q-di¤erence equation of hypergeometric type, Rodrigues formula and generating function are also considered in the limiting case.

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“…In the present paper, we consider the monic GHP H (µ) n , n ≥ 0 introduced by Szego [5], p. 380, Problem 25, as a set of real polynomials orthogonal with respect to the weight |x| 2µ e −x 2 , µ > − 1 2 , x ∈ R. These polynomials were then investigated by Chihara in his Ph.D. thesis [6] and further studied by Rosenblum in [7]. These monic polynomials H (µ) n of degree equal to n are defined by [8][9][10][11][12][13][14][15][16][17]…”

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confidence: 99%

On Spectral Vectorial Differential Equation of Generalized Hermite Polynomials

Atia

Benabdallah

2022

Axioms

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In this paper, we first give some results on monic generalized Hermite polynomials (GHP) {Hn(μ)(x)}n≥0, orthogonal with respect to the positive weight |x|2μe−x2,μ>−12,x∈R, which will lead to the formulation of the second-order spectralvectorial differential equation (SVDE) that the GHP satisfies. This SVDE differs from the one given in G. Szego (problem 25. p. 380), which is a pseudo-spectral equation. Second, we give the SVDE, as conjecture, satisfied by the generalized Jacobi polynomials Jn(α,α+1)(x,μ), orthogonal with respect to the positive weight w(x,α;μ)=|x|−μ(1−x2)α(1−x), μ<1, α>−1 on [−1,1].

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Generalized coherent states for q-oscillator connected with q-Hermite polynomials

Cited by 8 publications

References 45 publications

New q -Hermite polynomials: characterization, operators algebra and associated coherent states

New q -Hermite polynomials: characterization, operators algebra and associated coherent states

On the limit of discrete q-Hermite I polynomials

On Spectral Vectorial Differential Equation of Generalized Hermite Polynomials

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