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Chebyshev polynomials in several variables and the radial part of the Laplace-Beltrami operator

Author: R. J. Beerends
Journal: Trans. Amer. Math. Soc. 328 (1991), 779-814
MSC: Primary 33C45; Secondary 22E30, 33C80, 43A85
MathSciNet review: 1019520
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Abstract: Chebyshev polynomials of the first and the second kind in $ n$ variables $ {z_1},{z_2}, \ldots,{z_n}$ are introduced. The variables $ {z_1},{z_2}, \ldots,{z_n}$ are the characters of the representations of $ SL(n + 1,{\mathbf{C}})$ corresponding to the fundamental weights. The Chebyshev polynomials are eigenpolynomials of a second order linear partial differential operator which is in fact the radial part of the Laplace-Beltrami operator on certain symmetric spaces. We give an explicit expression of this operator in the coordinates $ {z_1},{z_2}, \ldots,{z_n}$ and then show how many results in the literature on differential equations satisfied by Chebyshev polynomials in several variables follow immediately from well-known results on the radial part of the Laplace-Beltrami operator. Related topics like orthogonality, symmetry relations, generating functions and recurrence relations are also discussed. Finally we note that the Chebyshev polynomials are a special case of a more general class of orthogonal polynomials in several variables.

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