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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Differential-difference operators associated to reflection groups
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by Charles F. Dunkl
Trans. Amer. Math. Soc. 311 (1989), 167-183
DOI: https://doi.org/10.1090/S0002-9947-1989-0951883-8

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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Bibliographic Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 311 (1989), 167-183
  • MSC: Primary 33A45; Secondary 20H15, 33A65, 42C10, 51F15
  • DOI: https://doi.org/10.1090/S0002-9947-1989-0951883-8
  • MathSciNet review: 951883