We construct analytic continuation formulas for two families of bivariate hypergeometric series, which, as special cases, include the classical series H₃ and G₃ occurring in Horn’s list of second-order hypergeometric functions. The derived formulas give expressions for the functions under consideration in the form of linear combinations of basis functions of solution spaces of systems of partial differential equations satisfied by the original series. For the domains into which the analytic continuations are carried out, we provide explicit representations that are advantageous from a practical point of view.
Alekseev et al. (Mon,) studied this question.