Constructing basises in solution space of the system of equations for the Lauricella Function FD (N)

Abstract

The paper considers the issue of constructing basises in the solution space of the system of partial differential equations, which is satisfied by the Lauricella hypergeometric function , depending on N complex variables and having complex parameters , b, c. For an arbitrary number N of variables, we have obtained explicit representations for such basis functions in the vicinity of points and in terms of the Horn type hypergeometric series in N variables. For some of these functions we have obtained formulas of analytic continuation. The found continuation formulas are important for calculating the solution of the Riemann – Hilbert problem with piecewise constant coefficients and studying its geometrical meaning. Besides, these formulas are effective for solving the parameters problem for the Schwarz – Christoffel integral and calculating conformal mapping of complex-shaped polygons.