A generating integral for the matrix elements of the Coulomb Green's function with the Coulomb wave functions

We analytically evaluate the generating integral $K_{nl}(β,β') = \int_{0}^{\infty}\int_{0}^{\infty} e^{-βr - β' r'}G_{nl} r^{q} r'^{q'} dr dr'$ and integral moments $J_{nl}(β, r) = \int_{0}^{\infty} dr' G_{nl}(r,r') r'^{q} e^{-βr'}$ for the reduced Coulomb Green's 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.

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