Jacobi-Type Differential Relations for the Lauricella Function $F_D^{(N)}$

For the generalized Lauricella hypergeometric function $F_D^{(N)}$, Jacobi-type differential relations are obtained and their proof is given. A new system of partial differential equations for the function $F_D^{(N)}$ is derived. Relations between associated Lauricella functions are presented. These results possess a wide range of applications, including the theory of Riemann–Hilbert boundary-value problem.