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.
Let be a -invariant convex domain in including , where is a complex Coxeter group associated with reduced root system . We consider holomorphic functions defined in which are Dunkl polyharmonic, that is, for some integer . Here is the complex Dunkl Laplacian, and is the complex Dunkl operator attached to the Coxeter group , where is a multiplicity function on and is the reflection with respect to the root . We prove that any complex Dunkl polyharmonic function has a decomposition of the form , for all , where are complex Dunkl harmonic functions, that is, .