Estimation of the remainder term of the hypergeometric series $G_D^{(N,j)}$

In this paper, new formulas are obtained for estimating the remainder term arising from the summation of the hypergeometric series $G_D^{(N,j)}$. Such formulas allow one to effectively estimate the remainder of the summation when calculating the value of the function $G_D^{(N,j)}$ in the unit polydisk. The obtained formulas can be used for calculation of the analytic continuation of the Lauricella function.