We obtain necessary and sufficient conditions for the solvability of the Riquier–Neumann problem for the inhomogeneous polyharmonic equation in the unit ball.

A new basis of harmonic polynomials is given. Proposed polynomials are orthogonal on the unit sphere and each term of this basis consists of monomials not present in the others.

An explicit representation of the Green's function of the Dirichlet problem for the biharmonic equation in the unit ball is given. Expansion of the constructed Green's function in the complete system of homogeneous harmonic polynomials that are orthogonal on the unit sphere is obtained. Solution of the homogeneous Dirichlet problem with a simple right-hand side of the biharmonic equation is found.

An algorithm is proposed for the analytical construction of a polynomial solution to Dirichlet problem for an inhomogeneous polyharmonic equation with a polynomial right-hand side and polynomial...