Multipole moments of general ellipsoids with two polarized domains

The multipole moments of a homogeneously polarized ellipsoid and a homogeneously charged ellipsoidal disc are calculated. The resulting hypergeometric functions are expressed as finite polynomials of the semi-axes a, b and c of the ellipsoid. The polynomial form exists for any order of the multipole moments. It is shown that the solution also applies to a two-domain ellipsoid with antiparallel polarized domains and to a system with radially changing polarization density. The results allow us to calculate the potential as well as interaction energies within the framework of multipole expansion.