Classical Orthogonal Polynomials of a Discrete Variable

Nikiforov, A. F.; Uvarov, V.B.; Suslov, Sergeĭ K.

doi:10.1007/978-3-642-74748-9_2

Cited by 417 publications

(590 citation statements)

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“…from the differencial equation and Rodrigues formula for the Laguerre polynomials [4] we deduce the following properties for the Laguerre functions (25) i) differential equation…”

Section: Wave Equation For the Hidrogen Atom With Discrete Variablesmentioning

confidence: 96%

“…We have worked out general formulas for the differential/difference equation, recurrence relations, raising and lowering operators, commutation and anticommutation relations. The starting point is the general properties of classical orthogonal polynomials of continuous and discrete variable [4] of hypergeometric type, in particular the Rodrigues formula from which the raising and lowering operators are derived. Similar results were obtained with more sofisticated method using the factorization of the hamiltonian [6] [7].…”

Section: Introductionmentioning

confidence: 99%

“…Now we make connection between the Kravchuk function and the Wigner functions that appear in the generalized spherical functions [4] …”

Section: Introductionmentioning

confidence: 99%

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Continuous vs. discrete models for the quantum harmonic oscillator and the hydrogen atom

Lorente

2001

Physics Letters A

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The Kravchuk and Meixner polynomials of discrete variable are introduced for the discrete models of the harmonic oscillator and hydrogen atom. Starting from Rodrigues formula we construct raising and lowering operators, commutation and anticommutation relations. The physical properties of discrete models are figured out through the equivalence with the continuous models obtained by limit process. PACS: 02.20.+b, 03.65.Bz, 03.65.Fd

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“…from the differencial equation and Rodrigues formula for the Laguerre polynomials [4] we deduce the following properties for the Laguerre functions (25) i) differential equation…”

Section: Wave Equation For the Hidrogen Atom With Discrete Variablesmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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Continuous vs. discrete models for the quantum harmonic oscillator and the hydrogen atom

Lorente

2001

Physics Letters A

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“…One underlying motivation of this series of works is to inaugurate a new theory for the Askey scheme of hypergeometric orthogonal polynomials [7][8][9][10] within the framework of QM. The abundant concepts and methods of QM accumulated over 80 years after the construction of QM would be available for the research of orthogonal polynomials.…”

Section: Introductionmentioning

confidence: 99%

Exactly and quasi-exactly solvable ‘discrete’ quantum mechanics

Sasaki

2011

Phil. Trans. R. Soc. A.

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A brief introduction to discrete quantum mechanics is given together with the main results on various exactly solvable systems. Namely, the intertwining relations, shape invariance, Heisenberg operator solutions, annihilation/creation operators and dynamical symmetry algebras, including the q-oscillator algebra and the AskeyWilson algebra. A simple recipe to construct exactly and quasi-exactly solvable (QES) Hamiltonians in one-dimensional 'discrete' quantum mechanics is presented. It reproduces all the known Hamiltonians whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable. Several new exactly and QES Hamiltonians are constructed. The sinusoidal coordinate plays an essential role.

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“…When the polynomial sequence (P n ) n is classical discrete [28,29], i.e. orthogonal with respect to a positive weight function r defined on the set I ¼ {a; a þ 1; .…”

Section: Introductionmentioning

confidence: 99%

On Fourth-order Difference Equations for Orthogonal Polynomials of a Discrete Variable: Derivation, Factorization and Solutions

Foupouagnigni

Koepf

Ronveaux

2003

Journal of Difference Equations and Applications

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We derive and factorize the fourth-order difference equations satisfied by orthogonal polynomials obtained from some modifications of the recurrence coefficients of classical discrete orthogonal polynomials such as: the associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified classical orthogonal polynomials. Moreover, we find four linearly independent solutions of these fourth-order difference equations, and show how the results obtained for modified classical discrete orthogonal polynomials can be extended to modified semi-classical discrete orthogonal polynomials. Finally, we extend the validity of the results obtained for the associated classical discrete orthogonal polynomials with integer order of association from integers to reals.

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Classical Orthogonal Polynomials of a Discrete Variable

Cited by 417 publications

References 0 publications

Continuous vs. discrete models for the quantum harmonic oscillator and the hydrogen atom

Continuous vs. discrete models for the quantum harmonic oscillator and the hydrogen atom

Exactly and quasi-exactly solvable ‘discrete’ quantum mechanics

On Fourth-order Difference Equations for Orthogonal Polynomials of a Discrete Variable: Derivation, Factorization and Solutions

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