Lauricella hypergeometric function and its application

The problem of analytical continuation of the Lauricella function, a generalized hypergeometric function of N variables, is considered. For an arbitrary N, a complete set of formulas for the analytical continuation of this function beyond the boundary of a single polycircle is specified, where it is initially defined as an N-fold hypergeometric series. Such formulas represent the Lauricella function in suitable subdomains of an N-dimensional complex space in the form of other generalized hypergeometric series that are solutions of the same system of partial differential equations that the Lauricella function satisfies. The paper also discusses the application of the obtained results to the problem of parameters of the Christoffel–Schwartz integral.