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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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Orthogonal polynomials on the sphere with octahedral symmetry
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by Charles F. Dunkl
Trans. Amer. Math. Soc. 282 (1984), 555-575
DOI: https://doi.org/10.1090/S0002-9947-1984-0732106-7

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 \[ p(x) = \sum \limits _{j = 0}^{[n/2]} {|x{|^{2j}}{p_{n - 2j}}(x),} \] 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 \[ h(x) = {({x_1}{x_2} \cdots {x_N})^\alpha }{\left ( {\prod \limits _{i < j} {(x_i^2 - x_j^2)} } \right )^\beta }.\] 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 $(\alpha + 2,\beta + 1)$.
References
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Bibliographic Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 282 (1984), 555-575
  • MSC: Primary 33A65; Secondary 33A45
  • DOI: https://doi.org/10.1090/S0002-9947-1984-0732106-7
  • MathSciNet review: 732106