Polynomial Solutions of Poisson and Polyharmonic Equations in Rn

We give an inductive existence proof for a polynomial solution of the Poisson equation Δu = q where q is a polynomial. We then find all the polynomial solutions of the polyharmonic equation Δmu = 0 for m ≥ 2 in terms of harmonic polynomials. Based on the harmonic polynomials computed by a recently found algorithm we obtain generating polyharmonic polynomials in Rn for n ≥ 2. Some examples of polyharmonic polynomials are considered at the end