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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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\textit{C. F. Dunkl}, Trans. Am. Math. Soc. 311, No. 1, 167--183 (1989; Zbl 0652.33004)

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### References:

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