Polyharmonic polynomials and mixed boundary value problems in the Heisenberg group $\mathbb H_n$

We give an existence proof for a polynomial solution of the Poisson equation $L_0 u=q$ where $q$ is a polynomial in the one dimensional Heisenberg Group. All the polynomial solutions of the polyharmonic equation $L_0^m u=0$ in terms of harmonic polynomials are determined. In addition, we also discuss the polyharmonic Neumann and mixed boundary value problems on the Kor\'anyi ball in the Heisenberg group $\H_n$ by inductive method. Some necessary and sufficient solvability conditions are obtained for the nonhomogeneous polyharmonic Neumann problem.