Analytic continuation of the Horn hypergeometric series with an arbitrary number of variables

The problem of analytic continuation is considered for the general hypergeometric Horn series with an arbitrary number of variables. An approach is proposed that allows one to find formulas for the continuation of such series into the exterior of the set of their convergence in the form of linear combinations of other hypergeometric series. These new hypergeometric series belong also to the Horn class, and are solutions of the same system of partial differential equations. Using this general result, we construct formulas for the analytic continuation for a double hypergeometric series of Horn's type, which plays an important role in the theory of the Appell function F1.