A generating integral for the matrix elements of the Coulomb Greens function with the Coulomb wave functions

We analytically evaluate the generating integral $K_{nl}(beta,beta) = int_{0}^{infty}int_{0}^{infty} e^{-beta r - beta r}G_{nl} r^{q} r^{q} dr dr$ and integral moments $J_{nl}(beta, r) = int_{0}^{infty} dr G_{nl}(r,r) r^{q} e^{-beta r}$ for the reduced Coulomb Greens function $G_{nl}(r,r)$ for all values of the parameters $q$ and $q$, when the integrals are convergent. These results can be used in second-order perturbation theory to analytically obtain the complete energy spectra and local physical characteristics such as electronic densities of multi-electron atoms or ions.