Alternative forms of the Coulomb approximation for bound-bound multipole transitions

We discuss three different formulations of the Coulomb approximation that allow us to avoid numerical difficulties encountered with the standard one in the case of transitions between highly excited states. A common feature of these alternative (though not equivalent) methods is the fact that the radial multipole matrix elements are expressed in terms of standard or generalized hypergeometric functions. The recurrence relations satisfied by the latter can be used to bypass the loss of accuracy resulting from dramatic cancellations among their individual terms.