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

Abstract

Chebyshev polynomials of the first and the second kind in n n variables z 1 , z 2 , … , z n {z_1},{z_2}, \ldots ,{z_n} are introduced. The variables z 1 , z 2 , … , z n {z_1},{z_2}, \ldots ,{z_n} are the characters of the representations of S L ( n + 1 , C ) 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 , … , z n {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.