Estimation of the remainder term of the Lauricella series $f^{(n)}_d$

In this paper, new formulas are obtained for estimating the remainder term arising in the summation of the Lauricella hypergeometric series $F_D^{(N)}$. Such formulas allow one to effectively estimate the remainder of the summation when calculating the value of the function $F_D^{(N)}$ in a unit polydisk and have applications to calculating Euler-type integrals and certain solutions of systems of differential equations that the Lauricella function satisfies. The results can be applied to problems in mathematical physics and function theory where the Lauricella function arises, including those which are needed for calculations of conformal mappings of polygons.