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Search: MSC category 33C50 ( Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable )

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1. CJM 1998 (vol 50 pp. 193)

Xu, Yuan
Intertwining operator and $h$-harmonics associated with reflection groups
We study the intertwining operator and $h$-harmonics in Dunkl's theory on $h$-harmonics associated with reflection groups. Based on a biorthogonality between the ordinary harmonics and the action of the intertwining operator $V$ on the harmonics, the main result provides a method to compute the action of the intertwining operator $V$ on polynomials and to construct an orthonormal basis for the space of $h$-harmonics.

Keywords:$h$-harmonics, intertwining operator, reflection group
Categories:33C50, 33C45

2. CJM 1997 (vol 49 pp. 373)

Stokman, Jasper V.; Koornwinder, Tom H.
Limit transitions for BC type multivariable orthogonal polynomials
Limit transitions will be derived between the five parameter family of Askey-Wilson polynomials, the four parameter family of big $q$-Jacobi polynomials and the three parameter family of little $q$-Jacobi polynomials in $n$ variables associated with root system $\BC$. These limit transitions generalize the known hierarchy structure between these families in the one variable case. Furthermore it will be proved that these three families are $q$-analogues of the three parameter family of $\BC$ type Jacobi polynomials in $n$ variables. The limit transitions will be derived by taking limits of $q$-difference operators which have these polynomials as eigenfunctions.

Categories:33D45, 33C50

3. CJM 1997 (vol 49 pp. 175)

Xu, Yuan
Orthogonal Polynomials for a Family of Product Weight Functions on the Spheres
Based on the theory of spherical harmonics for measures invariant under a finite reflection group developed by Dunkl recently, we study orthogonal polynomials with respect to the weight functions $|x_1|^{\alpha_1}\cdots |x_d|^{\alpha_d}$ on the unit sphere $S^{d-1}$ in $\RR^d$. The results include explicit formulae for orthonormal polynomials, reproducing and Poisson kernel, as well as intertwining operator.

Keywords:Orthogonal polynomials in several variables, sphere, h-harmonics
Categories:33C50, 33C45, 42C10

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