When , , , and ,
where the branches of the square roots have their principal values when and are continuous when . For this and similar results see Erdélyi et al. (1953a, §3.11).
where and are ellipses with foci at , being properly interior to . The series converges uniformly for outside or on , and within or on . For a generalization in terms of Gegenbauer polynonials see Theorem 2.1 in Cohl (2013).