Addition theorems for solid harmonics and the second Born amplitudes

A simple but novel method is described to prove the addition theorems for solid harmonics. This method gives new insight into the richness of the contents of the expansions of plane waves and the Coulomb potential between two point particles in terms of complete set of spherical harmonics. As a concrete application of the addition theorem for the regular solid harmonic, an integral containing the spherical harmonic Yml which frequently occurs in the second Born term is evaluated in a compact form.