Sums of products of the Coulomb wave functions over degenerate manifolds have been obtained in a closed form. These sums appear in many atomic and molecular problems. The sums have been obtained making use of the properties of the Coulomb Green's function G(r, r('),E), in the limit E-->E(n), where E(n) is the eigenenergy of the hydrogenlike atomic ion. The closed Hostler-Pratt form of G in the coordinate representation has been used. The sums calculated are a consequence of the n degeneracy of the Coulomb atomic energy levels. This itself, as is well known, follows from the four-dimensional symmetry of the Coulomb problem for the hydrogen atom.