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Differential-difference operators associated to reflection groups. (English) Zbl 0652.33004

Summary: 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 \({\mathbb{R}}^ n\). 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 \({\mathbb{R}}^ 2\) and dihedral groups there are analogues of the Cauchy-Riemann equations which apply to Gegenbauer and Jacobi polynomial expansions.

MSC:

33C55 Spherical harmonics
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
20F55 Reflection and Coxeter groups (group-theoretic aspects)
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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References:

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