Classical Orthogonal Polynomials

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

doi:10.1007/978-3-642-74748-9_1

Cited by 9 publications

(61 citation statements)

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“…The orthonormalized total wave function Ψ nm (τ 1 , τ 2 ) is given by (7), (10) and (12). The symmetry operator describing this coordinate system is…”

Section: Solutions Of the Schrödinger Equationmentioning

confidence: 99%

Superintegrability on the two-dimensional hyperboloid. II

Kalnins

Miller

Hakobyan

et al. 1999

Journal of Mathematical Physics

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Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform This work is devoted to the investigation of the quantum mechanical systems on the two-dimensional hyperboloid which admits separation of variables in at least two coordinate systems. Here we consider two potentials introduced in a paper of C. P. Boyer, E. G. Kalnins, and P. Winternitz ͓J. Math. Phys. 24, 2022 ͑1983͔͒, which have not yet been studied. We give an example of an interbasis expansion and work out the structure of the quadratic algebra generated by the integrals of motion.

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“…The orthonormalized total wave function Ψ nm (τ 1 , τ 2 ) is given by (7), (10) and (12). The symmetry operator describing this coordinate system is…”

Section: Solutions Of the Schrödinger Equationmentioning

confidence: 99%

Superintegrability on the two-dimensional hyperboloid. II

Kalnins

Miller

Hakobyan

et al. 1999

Journal of Mathematical Physics

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“…HPs satisfies the following orthogonality relation [ 23 ]: where δ nm denotes the delta of Kronecker, ω ( x ) is the weight function, and represents the square norm function. These functions are defined as follows [ 29 ]: …”

Section: Conventional and Proposed Computation Of Hpsmentioning

confidence: 99%

“…It is worth mentioning that the computation time of HPs is significantly reduced when using the following HPs symmetry relation defined for α = β [ 29 ]: …”

Section: Conventional and Proposed Computation Of Hpsmentioning

confidence: 99%

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Fast and stable computation of higher-order Hahn polynomials and Hahn moment invariants for signal and image analysis

Daoui

Karmouni

Sayyouri

et al. 2021

Multimed Tools Appl

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This article presents, on the one hand, new algorithms for the fast and stable computation of discrete orthogonal Hahn polynomials of high order (HPs) based on the elimination of all gamma and factorial functions that cause the numerical fluctuations of HPs, and based on the use of appropriate stability conditions. On the other hand, a new method for the fast and numerically stable computation of Hahn moment invariants (HMIs) is also proposed. This method is mainly based on the use of new recursive relations of HPs and of matrix multiplications when calculating HMIs. To validate the efficiency of the algorithms proposed for the calculation of HPs, several signals and large images (≥4000 × 4000) are reconstructed by Hahn moments (HMs) up to the last order with a reconstruction error tending towards zero ( MSE ≃ 10 −10 ). The efficiency of the proposed method for calculating HMIs is demonstrated on large medical images (2048 × 2048) with a very low relative error ( RE ≃ 10 −10 ). Finally, comparisons with some recent work in the literature are provided.

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“…Recently, much attention has been given to the so-called quantum groups and quantum algebras [1] (or to be more precise, to quantum deformations of universal enveloping algebras for classical Lie algebras), which give natural examples of noncommutative and noncocommutative Hopf algebras.There exists a close relations.hip [2] between these objects and the difference special functions [3] of q-analysis [4,5].…”

Section: Introductionmentioning

confidence: 99%