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# Orthogonal Polynomials for a Family of Product Weight Functions on the Spheres

Published:1997-02-01
Printed: Feb 1997
• Yuan Xu
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## Abstract

Based on the theory of spherical harmonics for measures invariant under a finite reflection group developed by Dunkl recently, we study orthogonal polynomials with respect to the weight functions $|x_1|^{\alpha_1}\cdots |x_d|^{\alpha_d}$ on the unit sphere $S^{d-1}$ in $\RR^d$. The results include explicit formulae for orthonormal polynomials, reproducing and Poisson kernel, as well as intertwining operator.
 Keywords: Orthogonal polynomials in several variables, sphere, h-harmonics
 MSC Classifications: 33C50 - Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable 33C45 - Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 42C10 - Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

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