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Let &#x003a9; be a G-invariant convex domain in &#x2102;N including 0, where G is a complex Coxeter group associated with reduced root system R&#x02282;&#x211D;N. We consider holomorphic functions f defined in &#x003a9; which are Dunkl polyharmonic, that is, (&#x00394;h)nf=0 for some integer n. Here &#x00394;h=&#x02211;j=1N&#x1D49F;j2 is the complex Dunkl Laplacian, and &#x1D49F;j is the complex Dunkl operator attached to the Coxeter group G, &#x1D49F;jf(z)=(&#x02202;f/&#x02202;zj)(z)+&#x02211;v&#x02208;R+&#x003ba;v((f(z)-f(&#x003c3;vz))/&#x02329;z,v&#x0232a;)vj, where &#x003ba;v is a multiplicity function on R and &#x003c3;v 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)+(&#x02211;n=1Nzj2)f1(z)+&#x022ef;+(&#x02211;n=1Nzj2)n-1fn-1(z), for all z&#x02208;&#x003a9;, where fj are complex Dunkl harmonic functions, that is, &#x00394;hfj=0

Decomposition of Polyharmonic Functions with Respect to the Complex Dunkl Laplacian