Analytic continuation of the Lauricella function with arbitrary number of variables

The Lauricella function FD(N), which is a generalized hypergeometric function of N variables, and a corresponding system of partial differential equations are considered. For an arbitrary N, we give a complete collection of analytic continuation formulas of FD(N). This formulas give representation of the Lauricella function outside the polydisk in the form of a linear combination of other generalized hypergeometric series that are solutions of the same system of partial differential equations, which is also satisfied by the function FD(N). The obtained hypergeometric series are N-dimensional analogues of the Kummer solutions well known in the theory of the classical hypergeometric Gauss equation. The obtained analytic continuation formulas provide an effective algorithm for computation of the Lauricella function FD(N).