Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Generalized complex Hermite polynomials with associated nonlinear coherent states

Khalid Ahbli, Fouzia El Wassouli, Zouhaı̈r Mouayn

Published Jul 1, 2024• 0 citations
Publisher

Abstract

While dealing with a class of generalized complex Hermite polynomials, we discuss some of their basic properties and we give operational formulae of Burchnall-type. New results, including a Nielsen identity, a generating function and a Runge addition formula are derived. These polynomials may also be used to define a set of nonlinear coherent states for the harmonic oscillator. They may also be viewed as eigenfunctions of a Landau-type operator.

DOI
10.1063/5.0194370

Related papers

New nonlinear coherent states based on hypergeometric-type operators

Alireza Dehghani, B. Mojaveri

The main goal of this paper is to present an alternative method to construct new kinds of nonlinear coherent states. To do this, we first establish a class of hypergeometric type of generalized displacement operators, 1 F r ([0], [0, 1, ..., r − 1], za †), act on the vacuum state of the harmonic oscillator and generate normalized quantum states of the Fock space which admit a resolution of the identity through a positive definite measure on the complex plane. Furthermore, realization of the compact form of these states, as functions of the position coordinate x for r = 2, leads to a generating function of the Hermite polynomials in terms of the modified Bessel function. Finally, studying some statistical characters reveals that they have indeed non-classical features such as squeezing, an anti-bunching effect and sub-Poissonian statistics, too.

Journal of Physics A Mathematical and Theoretical•Feb 20, 2012•18 citations

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

Вадим Васильевич Борзов

For the oscillator-like systems, connected with q-Hermite polynomials, coherent states of Barut-Girardello type are defined. The well-known Arik-Coon oscillator naturally arose in the framework of suggested approach as oscillator, connected with the Rogers q-Hermite polynomials, in the same way as usual oscillator with standard Hermite polynomials. The results about the coherent states for discrete q-Hermite polynomials of II type are quite new.

arXiv:math/0307356•Days on Diffraction•Jan 1, 2003•12 citations

Generalized coherent states for the harmonic oscillator by the J -matrix method with an extension to the Morse potential

Hashim A. Yamani, Zouhaı̈r Mouayn

While dealing with the J -matrix method for the harmonic oscillator to write down its tridiagonal matrix representation in an orthonormal basis of L 2 ( R ) , we rederive a set of generalized coherent states (GCS) of Perelomov type labelled by points z of the complex plane C and depending on an integer number m ∈ Z + . The number states expansion of these GCS gives rise to coefficients that are complex Hermite polynomials and whose linear superpositions provide eigenfunctions for the two-dimensional magnetic Laplacian, associated with the m th Landau level. We further generalize this construction to the Morse oscillator by introducing a new set of Glauber-type GCS.

Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Aug 1, 2025•1 citation

New generalized Hermite polynomials with three variables obtained via quantum optics method and their applications

An-peng Wang, Yi-xing Wang

Special polynomials (e.g. Hermite polynomials) are very important for the development of physics and mathematics. As a further extension of ordinary Hermite polynomials, we introduce new generalized Hermite polynomials with three variables and find their generating functions using the operator ordering method in quantum optics. Also, some new operator identities and integral formulas are obtained. As applications, the normalization, Wigner functions and evolutions for certain quantum states are analytically presented. These analytical results can provide conveniences for numerically studying the properties and applications of such quantum states.

Lithuanian Journal of Physics•Aug 12, 2025•0 citations

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

Won Sang Chung, Mahouton Norbert Hounkonnou, Sama Arjika

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 three-term 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.

arXiv:1310.1280•Oct 4, 2013•1 citation

Generalized q-deformed oscillators, q-Hermite polynomials, generalized coherent states

Burban, I. M.

The aim of this paper is to study generalized q-analogs of the well-known q-deformed harmonic oscillators and to connect them with q-Hermite polynomials. We give a construction of the appropriate oscillator-like algebras and show that corresponding Hermite polynomials are generalization of the discrete q-Hermite I and the discrete q-Hermite II polynomials. We also construct generalized coherent states of Barut-Girardello type for oscillator-like systems connected with these polynomials.

arXiv:math-ph/0607045•Jul 20, 2006•1 citation
While dealing with a class of generalized complex Hermite polynomials, we discuss some of their basic properties and we give operational formulae of Burchnall-type. New results, including a Nielsen identity, a generating function and a Runge addition formula are derived. These polynomials may also be used to define a set of nonlinear coherent states for the harmonic oscillator. They may also be viewed as eigenfunctions of a Landau-type operator.
The main goal of this paper is to present an alternative method to construct new kinds of nonlinear coherent states. To do this, we first establish a class of hypergeometric type of generalized displacement operators, 1 F r ([0], [0, 1, ..., r − 1], za †), act on the vacuum state of the harmonic oscillator and generate normalized quantum states of the Fock space which admit a resolution of the identity through a positive definite measure on the complex plane. Furthermore, realization of the compact form of these states, as functions of the position coordinate x for r = 2, leads to a generating function of the Hermite polynomials in terms of the modified Bessel function. Finally, studying some statistical characters reveals that they have indeed non-classical features such as squeezing, an anti-bunching effect and sub-Poissonian statistics, too.
While dealing with the J -matrix method for the harmonic oscillator to write down its tridiagonal matrix representation in an orthonormal basis of L 2 ( R ) , we rederive a set of generalized coherent states (GCS) of Perelomov type labelled by points z of the complex plane C and depending on an integer number m ∈ Z + . The number states expansion of these GCS gives rise to coefficients that are complex Hermite polynomials and whose linear superpositions provide eigenfunctions for the two-dimensional magnetic Laplacian, associated with the m th Landau level. We further generalize this construction to the Morse oscillator by introducing a new set of Glauber-type GCS.
Special polynomials (e.g. Hermite polynomials) are very important for the development of physics and mathematics. As a further extension of ordinary Hermite polynomials, we introduce new generalized Hermite polynomials with three variables and find their generating functions using the operator ordering method in quantum optics. Also, some new operator identities and integral formulas are obtained. As applications, the normalization, Wigner functions and evolutions for certain quantum states are analytically presented. These analytical results can provide conveniences for numerically studying the properties and applications of such quantum states.
New q−q-q−Hermite polynomials: characterization, operator algebra and associated coherent states
This paper addresses a construction of new q−q-q−Hermite polynomials with a full characterization of their main properties and corresponding raising and lowering operator algebra. The three-term 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.