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Harmonic Polynomials Associated With Reflection Groups
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- Journal:
- Canadian Mathematical Bulletin / Volume 43 / Issue 4 / 01 December 2000
- Published online by Cambridge University Press:
- 20 November 2018, pp. 496-507
- Print publication:
- 01 December 2000
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- Article
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We extend Maxwell’s representation of harmonic polynomials to
$h$-harmonics associated to a reflection invariant weight function 
${{h}_{k}}$. Let 
${{\mathcal{D}}_{i}},\,1\,\le \,i\,\le \,d$, be Dunkl’s operators associated with a reflection group. For any homogeneous polynomial 
$P$ of degree 
$n$,we prove the polynomial 
${{\left| x \right|}^{2\gamma +d-2+2n}}P\left( \mathcal{D} \right)\left\{ 1/{{\left| x \right|}^{2\gamma +d-2}} \right\}$ is a $h$-harmonic polynomial of degree $n$, where 
$\gamma \,=\,\sum \,ki$ and 
$\mathcal{D}\,=\,\left( {{\mathcal{D}}_{1}},\ldots ,{{\mathcal{D}}_{d}} \right)$. The construction yields a basis for $h$-harmonics. We also discuss self-adjoint operators acting on the space of $h$-harmonics.
