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Generalized Chebyshev polynomials associated with affine Weyl groups. (English) Zbl 0681.33020

With the help of geometry the author describes generalized Chebyshev polynomials. They are connected with a geometric figure F in \({\mathbb{R}}^ n\) which can be mapped by reflections in hyperplanes into smaller replicas of itself. It is shown that such figures are in one-to-one correspondence with the well known affine Weyl groups of root systems. A function h: \({\mathbb{R}}^ n\to {\mathbb{R}}^ n\), which generalizes the cosine is associated with a figure F. The generalized Chebyshev polynomials are defined as \(P_ k(h(x))=h(kx)\), where \(k>1\) and \(x\in {\mathbb{R}}^ n\). It is shown that properties of \(P_ k\) generalize those for classical Chebyshev polynomials of the first kind.
Reviewer: A.Klimyk

MSC:

33E99 Other special functions
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
17B20 Simple, semisimple, reductive (super)algebras
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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