Complex Hermite polynomials: Their combinatorics and integral operators

Ismail, Mourad E. H.; Simeonov, Plamen

doi:10.1090/s0002-9939-2014-12362-8

Cited by 40 publications

(30 citation statements)

References 19 publications

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“…for q = x q + q I q ∈ Ω ∖ R. The following product rule (9) for the slice derivative follows immediately from (8), which can be obtained easily by direct computation starting from (4). It will be used systematically in the next sections.…”

Section: The Slice Derivativementioning

confidence: 99%

“…Such polynomials have found several interesting applications in various branches of mathematics, physics, and technology. They are used as basic tools in studying the complex Markov process, 1 the nonlinear analysis of travelling wave tube amplifiers, 2 the singular values of the Cauchy transform, 3 the coherent states, 4-6 combinatory, 7,8 the poly-analytic functions, and signal processing. 9,10 A natural extension of H m,n (z,z) to the quaternionic setting is defined by Thirulogasanthar and Twareque 6 :…”

Section: Introductionmentioning

confidence: 99%

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On some analytic properties of slice poly‐regular Hermite polynomials

Hamyani

Ghanmi

2018

Math Methods in App Sciences

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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 L2false(LI;e−false|q|2dλIfalse) 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 L2false(double-struckH;e−false|q|2dλfalse) of square integrable functions on quaternions with respect to the Gaussian measure on the whole double-struckH≡R4. We also provide different types of generating functions. Remarkable identities, including quadratic recurrence formulas of Nielsen type, are also derived.

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Section: The Slice Derivativementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On some analytic properties of slice poly‐regular Hermite polynomials

Hamyani

Ghanmi

2018

Math Methods in App Sciences

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“…They studied basic properties of {ψ s k (z)} ∞ k=0 including its orthonormality in χ s (C). After that, the function space X s (C) and the system of holomorphic Hermite functions {ψ s k } ∞ k=0 have been applied to studying quantization on C and related problems (see [8,12,13]), combinatorics and counting (see [10]) and etc. We call ψ s 0 (z) = π √ se −z 2 /2 /(1−s) the generator of the family {ψ s n } ∞ n=0 .…”

Section: See [7]mentioning

confidence: 99%

Holomorphic Hermite Functions in Segal–Bargmann Spaces

Chihara

2018

Complex Anal. Oper. Theory

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We study systems of holomorphic Hermite functions in the Segal-Bargmann spaces, which are Hilbert spaces of entire functions on the complex Euclidean space, and are determined by the Bargmann-type integral transform on the real Euclidean space. We prove that for any positive parameter which is strictly smaller than the minimum eigenvalue of the positive Hermitian matrix associated with the transform, one can find a generator of holomorphic Hermite functions whose anihilation and creation operators satisfy canonical commutation relations. In other words, we find the necessary and sufficient conditions so that some kinds of entire functions can be such generators. Moreover, we also study the complete orthogonality, the eigenvalue problems and the Rodrigues formulas.2000 Mathematics Subject Classification. Primary 33C45; Secondary 46E20, 46E22, 35S30.

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“…By mean from (38) and (59), the constants c ββ , b ββ , f 0 and g 0 can be related to the initial standard deviations σ q j (0) ≡ σ q j and σ π j (0) ≡ σ π j in the form…”

Section: A Standard Deviations and Uncertainty Relationsmentioning

confidence: 99%

Coherent and Semiclassical States of a Charged Particle in Electromagnetic Fields

Pereira

2018

Braz J Phys

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In the present article we extend our study (BJP 45 (2015) 369) of generalized coherent states (GCS) of a one-dimensional particle considering such important physical system as a 3-dimensional charged particle in electric and magnetic fields. Constructing GCS in many-dimensional case, we meet nontrivial technical complications that make the consideration nontrivial and instructive.The GCS of a system under consideration are constructed. We study properties of these GCS such as completeness relations, minimization of uncertainty relations and so on. We point out which family of the obtained GCS of a charged particle in magnetic field is related with the CS constructed first by Malkin and Man'ko. We obtain conditions under which some of the GCS can be considered as semiclassical states. * Electronic address: albertoufcg@hotmail.com arXiv:1711.07406v2 [quant-ph]

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Complex Hermite polynomials: Their combinatorics and integral operators

Cited by 40 publications

References 19 publications

On some analytic properties of slice poly‐regular Hermite polynomials

On some analytic properties of slice poly‐regular Hermite polynomials

Holomorphic Hermite Functions in Segal–Bargmann Spaces

Coherent and Semiclassical States of a Charged Particle in Electromagnetic Fields

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