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Abstract. This work is concerned with a quantization of the Pais-Uhlenbeck oscillators from the point of view of their multi-Hamiltonian structures. It is shown that the 2n-th order oscillator with a simple spectrum is equivalent to the usual anisotropic n -dimensional oscillator.

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The spectral measure of the position (momentum) operator X for q-deformed oscillator is calculated in the case of the indetermine Hamburger moment problem. The exposition is given for concrete choice of generators for q-oscillator algebra, although developed technique apply for every other cases with indetermine moment problem. The Stieltjes transformation m(z) of spectral measure is expressed in terms of the entries of Jacobi matrix X only. The direct connection between values of parameters labeling the spectral measures and related selfadjoint extensions of X is established.

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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,

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