A new analytic continuation of Appell’s hypergeometric series F2

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Abstract
The doubly infinite series for Appell’s function F2(α,a1,a2,b1,b2;x,y) is written in terms of four of Appell’s F3 functions. Analytic continuations are given for the F3 series, thereby allowing one to obtain a new analytic continuation for the F2 series. The new doubly infinite series are all absolutely convergent if their variables satisfy ‖x‖<1 and ‖y‖<1, whereas the F2 series is absolutely convergent only in the domain ‖x‖+‖y‖<1. The analytic continuations given here are very useful for evaluating the Appell F2 series when one of the variables is near unity. In particular, our results are useful for calculating radial matrix elements over products of Dirac–Coulomb functions and the electromagnetic interaction Green’s function.

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