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Transactions of the American Mathematical Society

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



Orthogonal polynomials on the sphere with octahedral symmetry

Author: Charles F. Dunkl
Journal: Trans. Amer. Math. Soc. 282 (1984), 555-575
MSC: Primary 33A65; Secondary 33A45
MathSciNet review: 732106
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Abstract: For any finite reflection group $ G$ acting on $ {{\mathbf{R}}^N}$ there is a family of $ G$-invariant measures ( $ ({h^2}d\omega $, where $ h$ is a certain product of linear functions whose zero-sets are the reflecting hyperplanes for $ G$) on the unit sphere and an associated partial differential operator ( $ {L_h}f: = \Delta (fh) - f\Delta h $; $ \Delta $ is the Laplacian). Analogously to spherical harmonics, there is an orthogonal (with respect to $ {h^2}d\omega $) decomposition of homogeneous polynomials, that is, if $ p$ is of degree $ n$ then

$\displaystyle p(x) = \sum\limits_{j = 0}^{[n/2]} {\vert x{\vert^{2j}}{p_{n - 2j}}(x),} $

where $ {L_h}{p_i} = 0$ and $ {\operatorname{deg}}{p_i} = i$ for each $ i$, but with the restriction that $ p$ and $ {p_i}$ must all be $ G$-invariant. The main topic is the hyperoctahedral group with

$\displaystyle h(x) = {({x_1}{x_2} \cdots {x_N})^\alpha }{\left( {\prod\limits_{i < j} {(x_i^2 - x_j^2)} } \right)^\beta }.$

The special case $ N = 2$ leads to Jacobi polynomials. A detailed study of the case $ N = 3$ is made; an important result is the construction of a third-order differential operator that maps polynomials associated to $ h$ with indices $ (\alpha ,\beta )$ to those associated with $ (\alpha + 2,\beta + 1)$.

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Article copyright: © Copyright 1984 American Mathematical Society

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