Coulomb Green's function and an addition formula for the Whittaker functions

Abstract

A series of the form $\sum_{\ell=0}^{\infty}c(\kappa,\ell)\,M_{\kappa,\ell+1/2}(r_{0})W_{\kappa,\ell+1/2}(r)P_{\ell}(\cos(\gamma))$ is evaluated explicitly where $c(\kappa,\ell)$ are suitable complex coefficients, $M_{\kappa,\mu}$ and $W_{\kappa,\mu}$ are the Whittaker functions, $P_{\ell}$ are the Legendre polynomials, $r_{0}<r$ are radial variables, $\gamma$ is an angle and $\kappa$ is a complex parameter. The sum depends, as far as the radial variables and the angle are concerned, on their combinations $r+r_{0}$ and $(r^{2}+r_{0}^{\,2}-rr_{0}\cos(\gamma))^{1/2}$. This addition formula generalizes in some respect Gegenbauer's Addition Theorem and follows rather straightforwardly from some already known results, particularly from Hostler's formula for Coulomb Green's function. In addition, several complementary summation formulas are derived. They suggest that a further extension of this addition formula may be possible.