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

DOI:
https://doi.org/10.1090/S0002-9947-1988-0946432-3

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 -dimensional Euclidean space, we construct a sequence of polynomials so that the mapping is conjugate to stretching the figure by a factor and folding it back onto itself. If and the figure is the interval, this construction yields the Chebyshev polynomials (up to conjugation). The polynomials 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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Additional Information

DOI:
https://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