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Harmonic Polynomials Associated With Reflection Groups

  Published:2000-12-01
 Printed: Dec 2000
  • Yuan Xu
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Abstract

We extend Maxwell's representation of harmonic polynomials to $h$-harmonics associated to a reflection invariant weight function $h_k$. Let $\CD_i$, $1\le i \le d$, be Dunkl's operators associated with a reflection group. For any homogeneous polynomial $P$ of degree $n$, we prove the polynomial $|\xb|^{2 \gamma +d-2+2n}P(\CD)\{1/|\xb|^{2 \gamma +d-2}\}$ is a $h$-harmonic polynomial of degree $n$, where $\gamma = \sum k_i$ and $\CD=(\CD_1,\ldots,\CD_d)$. The construction yields a basis for $h$-harmonics. We also discuss self-adjoint operators acting on the space of $h$-harmonics.
Keywords: $h$-harmonics, reflection group, Dunkl's operators $h$-harmonics, reflection group, Dunkl's operators
MSC Classifications: 33C50, 33C45 show english descriptions Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]
33C50 - Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
33C45 - Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]
 

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