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

By R. J. Beerends


Abstract. Chebyshev polynomials of the first and the second kind in n variables z., Zt,..., z „ are introduced. The variables z, , z-,..... z „ are the characters of the representations of SL(n + 1, 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 zi, z2,..., zn 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. 1

Year: 1991
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