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Reflection groups and orthogonal polynomials on the sphere. (English) Zbl 0616.33005

A method for constructing families of orthogonal polynomials in several variables is presented. Special cases include Gegenbauer and Jacobi polynomials, and the spherical harmonics. Each family is determined by a finite reflection group and a weight function which is a product of linear functions and is invariant under the group. In close analogy to spherical harmonics, the polynomials satisfy orthogonality relations on the surface of the sphere, while being annihilated by a second-order differential-difference operator and obeying a maximum principle in the interior. The invariant and representation theories of the reflection group are important components of the structure. Orthogonal polynomials associated to a version of Selberg’s multi-variable beta integral are covered by this theory.

MSC:

33C55 Spherical harmonics
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
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References:

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