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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Chebyshev polynomials in several variables and the radial part of the Laplace-Beltrami operator
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by R. J. Beerends PDF
Trans. Amer. Math. Soc. 328 (1991), 779-814 Request permission

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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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 328 (1991), 779-814
  • MSC: Primary 33C45; Secondary 22E30, 33C80, 43A85
  • DOI: https://doi.org/10.1090/S0002-9947-1991-1019520-3
  • MathSciNet review: 1019520