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1. CJM 1998 (vol 50 pp. 193)
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)
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)
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 |