Transactions of the American Mathematical Society

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

 

 

Generalized Chebyshev polynomials associated with affine Weyl groups


Authors: Michael E. Hoffman and William Douglas Withers
Journal: Trans. Amer. Math. Soc. 308 (1988), 91-104
MSC: Primary 33A65; Secondary 42C05, 58F13
MathSciNet review: 946432
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Abstract: We begin with a compact figure that can be folded into smaller replicas of itself, such as the interval or equilateral triangle. Such figures are in one-to-one correspondence with affine Weyl groups. For each such figure in $ n$-dimensional Euclidean space, we construct a sequence of polynomials $ {P_k}:{{\mathbf{R}}^n} \to {{\mathbf{R}}^n}$ so that the mapping $ {P_k}$ is conjugate to stretching the figure by a factor $ k$ and folding it back onto itself. If $ n = 1$ and the figure is the interval, this construction yields the Chebyshev polynomials (up to conjugation). The polynomials $ {P_k}$ are orthogonal with respect to a suitable measure and can be extended in a natural way to a complete set of orthogonal polynomials.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0946432-3
Keywords: Chebyshev polynomials, affine Weyl groups, orthogonal polynomials, reflection groups, root systems
Article copyright: © Copyright 1988 American Mathematical Society