Chebyshev-Koornwinder oscillator

Borzov, V. V.; Damaskinsky, E. V.

doi:10.1007/s11232-013-0062-9

Cited by 13 publications

(27 citation statements)

References 5 publications

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“…In [18] we have been generalized two-parameter deformed oscillator algebra [11] by introducing additional parameters α, γ, l in the structure function. This is a generalization of the well-known deformations [6,4,5,11,14,15,23] of these algebras. The generators {I, a, a + , N }, structure function (7) and the defining relations…”

Section: Some Aspects Of Generalized (P Q; α γ L)-deformed Oscillamentioning

confidence: 82%

“…On the level of oscillator algebra, the study of multi-parameter deformations of the oscillator algebra was continued in the works [14,15,16,17,18].…”

Section: Introductionmentioning

confidence: 99%

“…Naturally, the increasing of the number of the deformation parameters makes the application of the deformed oscillator algebras more flexible in the practical applications. In the framework of one-parameter deformation this scheme combined with scheme of construction of the generalized deformed oscillator algebras [22] has given possibility to unify in unified framework [14,15,23,16,18] the well-known deformations of the oscillator algebra [6,4,5].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we propose the unified deformation of the oscillator algebra which envelop as particular cases the well known deformations [6,4,5,11,14,15]. The structure function ("generalized (p, q; α, γ, l)-number") of this deformation…”

Section: Introductionmentioning

confidence: 99%

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Unified -deformation of oscillator algebra and two-dimensional conformal field theory

Burban

2013

Physics Letters A

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Section: Some Aspects Of Generalized (P Q; α γ L)-deformed Oscillamentioning

confidence: 82%

“…On the level of oscillator algebra, the study of multi-parameter deformations of the oscillator algebra was continued in the works [14,15,16,17,18].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Unified -deformation of oscillator algebra and two-dimensional conformal field theory

Burban

2013

Physics Letters A

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“…In contrast to the case of Chebyshev polynomials in two variables associated with the Lie algebra A 2 [25], the transition from relations (18) to (19) is not obvious.…”

Section: Description Of the Methodsmentioning

confidence: 96%

The generating function of bivariate Chebyshev polynomials associated with the Lie algebra G 2

Damaskinsky¹,

Sokolov²

2017

Theor Math Phys

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Generating Functions of Chebyshev Polynomials in Three Variables

Sokolov

2016

J Math Sci

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In this paper, we obtain generating functions of three-variable Chebyshev polynomials (of the first as well as of the second type) associated with the root system of the A 3 Lie algebra. Bibliography: 21 titles.To Petr Petrovich Kulish on the occasion of his 70th birthday 1. The aim of the present paper is to obtain generating functions of Chebyshev polynomials in three variables. Chebyshev polynomials in several variables associated with the root systems of simple Lie algebras were intensively studied in the last decades [1][2][3][4][5][6]. Such polynomials are applied in various areas of mathematics, as well as in physics. Examples of such applications with references can be found in the papers [7][8][9][10][11][12][13][14][15][16].Multivariate Chebyshev polynomials are natural generalizations of the classical ones. The classical Chebyshev polynomials T n (x) of the first kind are defined by the following formula (see, for example, [17]):( 1 ) they satisfy the following three-term recurrence relation:The classical Chebyshev polynomials U n (x) of the second kind are defined by the formulawhich is different from (1), but they satisfy the same recurrence relation (2). The initial conditions for the above polynomials are T 0 (x) = 1, T 1 (x) = x, U 0 (x) = 1, and U 1 (x) = 2x;( 3 ) together with the recurrence relation (2), they determine the polynomials uniquely without reference to (1) and (2). It is known that the function cos nφ can be treated as the invariant mean of the exponential function over the Weyl group of the A 1 algebra root system:Transition to a function that is the invariant mean over the Weyl group W of another root system leads us to generalization of the classical Chebyshev polynomials to polynomials in several variables [1][2][3][4]. Let us shortly recall the way in which such W -invariant functions can be constructed. Let R(L) be a reduced root system of any simple Lie algebra L. Such a system is a set of vectors in the d-dimensional Euclidean space E d supplied with scalar product (., .). The system R is uniquely determined by the base of simple roots α i , i = 1, . . . , d, and by a finite group W (R) which is generated by base reflections. This group is called the Weyl group.

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Chebyshev-Koornwinder oscillator

Cited by 13 publications

References 5 publications

Unified -deformation of oscillator algebra and two-dimensional conformal field theory

Unified -deformation of oscillator algebra and two-dimensional conformal field theory

The generating function of bivariate Chebyshev polynomials associated with the Lie algebra G 2

Generating Functions of Chebyshev Polynomials in Three Variables

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