Multipole matrix elements for hydrogen atom

The orthogonality relations of the eigenfunctions in the discrete and continuum states for the hydrogen-like atom are proved. The multipole matrix elements ⟨ n, l | r a | m, j ⟩ are given as the analytic expressions which are represented by Appell's hypergeometric function of two variables. For the particular case, the electric dipole and electric quadrupole matrix elements ⟨ n, l | r | m, l − 1⟩, ⟨ n, l | r 2 | m, l ⟩ and ⟨ n, l | r 2 | m, l − 2⟩ are represented as the sum of the hypergeometric functions and the electric dipole matrix elements ⟨ n, l | r | m, l − 1⟩ are consistent with the results obtained by Gordon.