Polynômes de Tchébychev et de Jacobi dans un espace euclidien de dimension p. (French) Zbl 0557.33004

Debiard, Amédée

C. R. Acad. Sci., Paris, Sér. I 296, 529-532 (1983).

This announces some results on classes of orthogonal polynomials in several variables, which are analogues of Chebyshev and Gegenbauer polynomials. The domain and weight function are constructed by means of the root system \(A_ p\) (in \({\mathbb{R}}^ p\), for \(p=1,2,3,...)\). Accordingly the symmetric group \(S_{p+1}\) acts naturally on the domain. The two variable case was introduced by T. H. Koornwinder [Indagationes 36, 357-369, 370-381 (1974; Zbl 0291.33013)]. In this situation the domain of orthogonality is a three-sided curvilinear figure, known as Steiner’s hypocycloid.
The author also discusses partially differential operators for which the polynomials are eigenfunctions. For certain parameter values the polynomials are spherical functions on homogeneous spaces.

Reviewer: Ch.F.Dunkl

MSC:

33C50	Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
43A90	Harmonic analysis and spherical functions
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

Chebyshev polynomials; Gegenbauer polynomials; Steiner’s hypocycloid; eigenfunctions

Citations:

Zbl 0291.33013