Copyright © 2010 Guangbin Ren and Helmuth R. Malonek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let Ω be a G-invariant convex domain in ℂN including 0, where G is a complex Coxeter group associated with reduced root system R⊂ℝN. We consider holomorphic functions f defined in Ω which are Dunkl polyharmonic, that is, (Δh)nf=0 for some integer n. Here Δh=∑j=1N𝒟j2 is the complex Dunkl Laplacian, and 𝒟j is the complex Dunkl operator attached to the Coxeter group G, 𝒟jf(z)=(∂f/∂zj)(z)+∑v∈R+κv((f(z)-f(σvz))/〈z,v〉)vj, where κv is a multiplicity function on R and σv is the reflection with respect to the root v. We prove that any complex Dunkl polyharmonic function f has a decomposition of the form f(z)=f0(z)+(∑n=1Nzj2)f1(z)+⋯+(∑n=1Nzj2)n-1fn-1(z), for all z∈Ω, where fj are complex Dunkl harmonic functions, that is, Δhfj=0.