Transactions of the American Mathematical Society

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



Differential-difference operators associated to reflection groups

Author: Charles F. Dunkl
Journal: Trans. Amer. Math. Soc. 311 (1989), 167-183
MSC: Primary 33A45; Secondary 20H15, 33A65, 42C10, 51F15
MathSciNet review: 951883
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Abstract: There is a theory of spherical harmonics for measures invariant under a finite reflection group. The measures are products of powers of linear functions, whose zero-sets are the mirrors of the reflections in the group, times the rotation-invariant measure on the unit sphere in $ {{\mathbf{R}}^n}$. A commutative set of differential-difference operators, each homogeneous of degree $ -1$, is the analogue of the set of first-order partial derivatives in the ordinary theory of spherical harmonics. In the case of $ {{\mathbf{R}}^2}$ and dihedral groups there are analogues of the Cauchy-Riemann equations which apply to Gegenbauer and Jacobi polynomial expansions.

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Keywords: Orthogonal polynomials in several variables, reflection groups, spherical harmonics, Gegenbauer polynomials, Jacobi polynomials
Article copyright: © Copyright 1989 American Mathematical Society