Complex Hermite Polynomials Research Articles

Generalized complex Hermite polynomials with associated nonlinear coherent states

Generalized complex Hermite polynomials with associated nonlinear coherent states

What hide the Hermite coherent states?

For the single mode the Hermite coherent states are wave packets analogous to the standard harmonic oscillator coherent states but constructed with complex Hermite polynomials substituting traditionally used complex monomials zn . Hermite coherent states unify the standard coherent and squeezed states within one class of quantum states whose representatives share all properties customarily required from the standard coherent states, in particular (over)completeness. Generalization of the Hermite coherent/squeezed states to the multimode case, illustrated on the bipartite states, shows that such states exhibit entanglement which presence or absence depends on the nature of squeezing but also on the choice of description method being used.

Hörmander’s L^2-Method, {overline{partial }}-Problem and Polyanalytic Function Theory in One Complex Variable

In this paper we consider the classical {overline{partial }}-problem in the case of one complex variable both for analytic and polyanalytic data. We apply the decomposition property of polyanalytic functions in order to construct particular solutions of this problem and obtain new Hörmander type estimates using suitable powers of the Cauchy-Riemann operator. We also compute particular solutions of the {overline{partial }}-problem for specific polyanalytic data such as the Itô complex Hermite polynomials and polyanalytic Fock kernels.

A note on partially degenerate Hermite-Bernoulli polynomials of the first kind

In this paper, we introduce a new class of partially degenerate Hermite-Bernoulli polynomials of the first kind and generalized Gould-Hopper-partially degenerate Bernoulli polynomials of the first kind and present some properties and identities of these polynomials. A new class of polynomials generalizing different classes of Hermite polynomials such as the real Gould-Hopper, as well as the 1-d and 2-d holomorphic, ternary and polyanalytic complex Hermite polynomials and their relationship to the partially degenerate Hermite-Bernoulli polynomials of the first kind are also discussed.

Complex Hermite polynomials: their generating functions via Lie algebra representation and operational methods

Complex Hermite polynomials: their generating functions via Lie algebra representation and operational methods

On λ-Changhee–Hermite polynomials

Abstract In this paper, we introduce a new class of λ-analogues of the Changhee–Hermite polynomials and generalized Gould–Hopper–Appell type λ-Changhee polynomials, and present some properties and identities of these polynomials. A new class of polynomials generalizing different classes of Hermite polynomials such as the real Gould–Hopper, as well as the 1D and 2D holomorphic, ternary and polyanalytic complex Hermite polynomials and their relationship to the Appell type λ-Changhee polynomials are also discussed.

Multivariate holomorphic Hermite polynomials

We introduce holomorphic Hermite polynomials in n complex variables that generalize the Hermite polynomials in n real variables introduced by Hermite in the late 19th century. We discuss cases in which these polynomials are orthogonal and construct a reproducing kernel Hilbert space related to one such orthogonal family. We also introduce a multivariate analog of the Ito polynomials. We show how these multivariate polynomials generalize the univariate complex Hermite and Ito polynomials. Generating functions, orthogonality relations, Rodrigues formulas, recurrence and linearization relations, and operator formulas are also derived for these multivariate holomorphic Hermite and Ito polynomials. A Kibble–Slepian formula and a Mehler-type formula for the multivariate Ito polynomials are established.

On the Bochner Laplacian operator on theta line bundle over quasi-tori

In this paper, we consider the Laplcian operator on theta line bundle over the quasi-torus, which is called the Bochner Laplacian. This operator has a canonical realization as a magnetic Laplacian acting on complex valued functions satisfying a functional equation. We study the spectral properties of such Laplacian and we show that its spectrum is reduced to eigenvalues πm; Then, we give a concrete description of each eigenspace in terms of Hermite and complex Hermite polynomials. In particular, an explicit description of the L2-holomorphic sections on the above line bundle is presented as the eigenspace of the magnetic Laplacian corresponding to the least eigenvalue. Also by using the periodization principle, the associated invariant integral operators are discussed.

On the complex Hermite polynomials

In this paper we use a set of partial differential equations to prove an expansion theorem for multiple complex Hermite polynomials. This expansion theorem allows us to develop a systematic and completely new approach to the complex Hermite polynomials. Using this expansion, we derive the Poisson Kernel, the Nielsen type formula, the addition formula for the complex Hermite polynomials with ease. A multilinear generating function for the complex Hermite polynomials is proved.

Non-trivial 1d and 2d Segal–Bargmann transforms

ABSTRACTSpecial generating functions and Mehler's formula for the univariate complex Hermite polynomials are obtained and next employed to introduce and study some one- and two-dimensional integral transforms of Segal–Bargmann type in the framework of some specific functional Hilbert spaces; including the so-called generalized Bargmann–Fock spaces that are realized as -eigenspaces of a special magnetic Schrödinger operator.

Quantum Hydrodynamics: Kirchhoff Equations

In this paper, we show that the Kirchhoff equations are derived from the Schrodinger equation by assuming the wave function to be a polynomial like solution. These Kirchhoff equations describe the evolution of n point vortices in hydrodynamics. In two dimensions, Kirchhoff equations are used to demonstrate the solution to single particle Laughlin wave function as complex Hermite polynomials. We also show that the equation for optical vortices, a two dimentional system, is derived from Kirchhoff equation by using paraxial wave approximation. These Kirchhoff equations satisfy a Poisson bracket relationship in phase space which is identical to the Heisenberg uncertainty relationship. Therefore, we conclude that being classical equations, the Kirchhoff equations, describe both a particle and a wave nature of single particle quantum mechanics in two dimensions.

Harmonic Analysis in Phase Space and Finite Weyl\u2013Heisenberg Ensembles

Weyl–Heisenberg ensembles are translation-invariant determinantal point processes on mathbb {R}^{2d} associated with the Schrödinger representation of the Heisenberg group, and include as examples the Ginibre ensemble and the polyanalytic ensembles, which model the higher Landau levels in physics. We introduce finite versions of the Weyl–Heisenberg ensembles and show that they behave analogously to the finite Ginibre ensembles. More specifically, guided by the observation that the Ginibre ensemble with N points is asymptotically close to the restriction of the infinite Ginibre ensemble to the disk of area N, we define finite WH ensembles as adequate finite approximations of the restriction of infinite WH ensembles to a given domain Omega . We provide a precise rate for the convergence of the corresponding one-point intensities to the indicator function of Omega , as Omega is dilated and the process is rescaled proportionally (thermodynamic regime). The construction and analysis rely neither on explicit formulas nor on the asymptotics for orthogonal polynomials, but rather on phase-space methods. Second, we apply our construction to study the pure finite Ginibre-type polyanalytic ensembles, which model finite particle systems in a single Landau level, and are defined in terms of complex Hermite polynomials. On a technical level, we show that finite WH ensembles provide an approximate model for finite polyanalytic Ginibre ensembles, and we quantify the corresponding deviation. By means of this asymptotic description, we derive estimates for the rate of convergence of the one-point intensity of polyanalytic Ginibre ensembles in the thermodynamic limit.

On some analytic properties of slice poly‐regular Hermite polynomials

We consider a quaternionic analogue of the univariate complex Hermite polynomials and study some of their analytic properties in some detail. We obtain their integral representation as well as the operational formulas of exponential and Burchnall types they obey. We show that they form an orthogonal basis of the slice Hilbert space of all quaternionic‐valued functions defined the whole quaternions space and subject to norm boundedness with respect to the Gaussian measure on a given slice as well as of the full left quaternionic Hilbert space of square integrable functions on quaternions with respect to the Gaussian measure on the whole . We also provide different types of generating functions. Remarkable identities, including quadratic recurrence formulas of Nielsen type, are also derived.

Analytic and arithmetic properties of the $$(\Gamma ,\chi )$$ ( Γ , χ ) -automorphic reproducing kernel function and associated Hermite–Gauss series

We consider the reproducing kernel function of the theta Bargmann–Fock Hilbert space associated with given full-rank lattice and pseudo-character, and we deal with some of its analytical and arithmetical properties. Specially, the distribution and the discreteness of its zeros are examined. The analytic sets of zeros of the theta Bargmann–Fock space inside a given fundamental cell is characterized and shown to be finite and of cardinal less or equal to its dimension. Moreover, we obtain some remarkable lattice sums by evaluating the so-called complex Hermite–Gauss coefficients. Some of them generalize some of the arithmetic identities given by Perelomov in the framework of coherent states for the specific case of von Neumann lattice. Such complex Hermite–Gauss coefficients are nontrivial examples of the so-called lattice’s functions according the Serre terminology. The perfect use of the basic properties of the complex Hermite polynomials is crucial in this framework.

Mehler's formulas for the univariate complex Hermite polynomials and applications

We give 2 widest Mehler's formulas for the univariate complex Hermite polynomials , by performing double summations involving the products and . They can be seen as the complex analogues of the classical Mehler's formula for the real Hermite polynomials. The proof of the first one is based on a generating function giving rise to the reproducing kernel of the generalized Bargmann space of level m. The second Mehler's formula generalizes the one appearing as a particular case of the so‐called Kibble‐Slepian formula. The proofs we present here are direct and more simpler. Moreover, direct applications are given and remarkable identities are derived.

Complex Hermite functions as Fourier–Wigner transform

ABSTRACTWe prove that the complex Hermite polynomials on the complex plane can be realized as the Fourier–Wigner transform of the well-known real Hermite functions on the real line . This reduces considerably Wong's proof [Wong MW. Weyl transforms. New York: Universitext. Springer-Verlag; 1998. Chapter 21] giving the explicit expression of in terms of the Laguerre polynomials. Moreover, we derive some new integral identities for the classical real Hermite polynomials .

D-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization

The ${\mathcal D}$-pseudo-boson formalism is illustrated with two examples. The first one involves deformed complex Hermite polynomials built using finite-dimensional irreducible representations of the group ${\rm GL}(2,{\mathbb C})$ of invertible $2 \times 2$ matrices with complex entries. It reveals interesting aspects of these representations. The second example is based on a pseudo-bosonic generalization of operator-valued functions of a complex variable which resolves the identity. We show that such a generalization allows one to obtain a quantum pseudo-bosonic version of the complex plane viewed as the canonical phase space and to understand functions of the pseudo-bosonic operators as the quantized versions of functions of a complex variable.

Analytic properties of complex Hermite polynomials

We study the complex Hermite polynomials { H m , n ( z , z ¯ ) } \{H_{m,n}(z, \bar z)\} in some detail, establish operational formulas for them and prove a Kibble-Slepian type formula, which extends the Poisson kernel for these polynomials. Positivity of the associated kernels is discussed. We also give an infinite family of integral operators whose eigenfunctions are { H m , n ( z , z ¯ ) } \{H_{m,n}(z,\bar z)\} . Some inverse relations are also given. We give a two dimensional moment representation for H m , n ( z , z ¯ ) H_{m,n}(z,\bar z) and evaluate several related integrals. We also introduce bivariate Appell polynomials and prove that { H m , n ( z , z ¯ ) } \{H_{m,n}(z, \bar z)\} are the only bivariate orthogonal polynomials of Appell type.

Complex Hermite polynomials: Their combinatorics and integral operators

Complex Hermite polynomials: Their combinatorics and integral operators

Asymptotics for Laguerre polynomials with large order and parameters

We study the asymptotic behavior of Laguerre polynomials Ln(αn)(z) as n→∞, where αn/n has a finite positive limit or the limit is +∞. Applying the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems, we derive the uniform asymptotics of such polynomials, which improves on the results of Bosbach and Gawronski (1998). In particular, our theorem is useful to obtain the asymptotics of complex Hermite polynomials and related double integrals.