Generalizations of the Jacobi identity to the case of the Lauricella function FD(N)

The paper considers the issue of constructing differential relations for the Lauricella hypergeometric function FD(N). The formulas found give explicit expressions for the derivative of the product of the function FD(N) and some binomials through the product of other binomials and a combination of adjacent Lauricella functions whose parameters differ by 1 or −1. The derived differential relations generalize the Jacobi identity and other differential formulas known in the theory of the Gauss hypergeometric function on the multidimensional case. With special parameter values FD(N), the right-hand side of such Jacobi-type formulas are simplified and have the form of a product of binomials and a polynomial. Such formulas can be used to transform a Cauchy-type integral to the form of the Schwartz–Christoffel integral and have an application to the Riemann–Hilbert problem with piecewise constant coefficients.