On calculation of generating functions of Chebyshev polynomials in several variables

Abstract

We propose a new method of calculation of generating functions of Chebyshev polynomials in several variables associated with root systems of simple Lie algebras. We obtain the generating functions of the polynomials in two variables corresponding to the Lie algebras C 2 and G 2 .K 00 = 8, K 10 = −6x, K 20 = 4y + 8,A few polynomials calculated using (44) with the normalization (41) are listed belowThe recurrence relations for the polynomials under consideration can be obtained by the multiplication rules (7) putting s = (m, n) and k = (1, 0) in the first relation and k = (0, 1) in the second one. As a result we finally obtain where the polynomials P 1 , P 2 are given by the relations (62), (63), and the coefficients K ij have the formK 01 = −10y, K 11 = 9xy + 2y − 2y 2 + 4x + 12, K 21 = 40xy − 8y 4 + 25xy 2 − 2x 2 + 72y 2 − 6x + 74y, K 31 = 6x 2 y − 12y 4 + 35xy 2 + 2x 2 + 72xy + 108y 2 + 6x + 118y, K 41 = −4y 4 + 12xy 2 + 21xy + 38y 2 − 4x + 34y − 12, K 51 = 2xy − 2y, K 02 = 8x + 8y + 24, K 12 = −13xy − 8x 2 − 16y + 2y 3 − 28x − 12, K 22 = 6y 4 + 7xy 3 − 21x 2 y − 18xy 2 − 40x 2 − 166xy + 24y 3 − 56y 2 − 192x − 268y − 216, K 32 = 13xy 3 −44x 2 y−6x 3 +10y 4 −92x 2 −29xy 2 +36y 3 −285xy−90y 2 −422y−338x−348, K 42 = −12x 2 y + 4xy 3 − 13xy 2 + 4y 4 − 92x − 20x 2 − 128y − 86xy − 36y 2 + 10y 3 − 96, K 52 = −2xy − 2x 2 + 2y 2 − 2y − 6x, K 03 = −6y 2 + 12x + 24, K 13 = 6xy 2 − 2xy − 12x 2 − 6y − 26x, K 23 = −6y 5 − 35x 2 y + 30xy 3 + 31xy 2 − 62x 2 − 187xy + 82y 3 + 54y 2 − 254x − 246y − 252, K 33 = 5x 2 y 2 − 10y 5 − 10x 3 + 60xy 2 + 50xy 3 − 62x 2 y − 330xy + 138y 3 − 144x 2 + + 96y 2 − 432y − 480x − 456, K 43 = −4y 5 + 20xy 3 − 23x 2 y + 21xy 2 − 127xy + 56y 3 − 42x 2 + 36y 2 − 174y − 178x − 180, K 53 = 2xy 2 − 2xy − 4x 2 − 6y − 10x, K 04 = 4x + 4y + 12,

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

“…As it is shown in [1], this form is closely connected with the polynomials P 1 , P 2 , standing in the denominator of the generating function. Namely, the polynomials P 1 , P 2 are the characteristic polynomials of matrices M The polynomials P 1 , P 2 are the minimal polynomials for the matrices M x , M y .…”

“…So obviously, one need an effective method of direct computation of Chebyshev polynomials in several variables. In the works [1], [2] we propose a new method for calculation of generating functions for Chebyshev polynomials of any kind associated with any simple Lie algebra. Let us describe it briefly.…”

“…The present work finishes the series of the articles ( [1], [2], [3]) which were initiated by P. Kulish. In these works we propose the method of constructing of generating functions for Chebyshev polynomials in several variables (of the first and second kinds), associated with simple Lie algebras. The method was tested on examples of polynomials related to Lie algebras A 2 , C 2 , A 3 and G 2 (in the last case only for the polynomials of the first kind).…”

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

3Dedicated to the memory of our dear friend Peter KulishThe generating function of the second kind bivariate Chebyshev polynomials associated with the simple Lie algebra G 2 is constructed by the method proposed in [1] and [2].

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“…The generating function of the second kind bivariate Chebyshev polynomials associated with the simple Lie algebra G 2 is constructed by the method proposed in [1] and [2].…”