Formulas for computing the Lauricella function in the case of crowding of variables

For the Lauricella function $F^{(N)}_D$, which is a hypergeometric function of several complex variables $z_1,\dots, z_N$, analytic continuation formulas are constructed that correspond to the intersection of an arbitrary number of singular hyperplanes of the form $\{z_j=z_l\}$, $j,l=\overline{1,N}$, $j\ne l$. These formulas give an expression for the considered function in the form of linear combinations of Horn hypergeometric series in $N$ variables satisfying the same system of partial differential equations as the original series defining $F^{(N)}_D$ in the unit polydisk. By applying these formulas, the function $F^{(N)}_D$ and Euler-type integrals expressed in terms of $F^{(N)}_D$ can be efficiently computed (with the help of exponentially convergent series) in the entire complex space $\mathbb{C}^N$ in the complicated cases when the variables form one or several groups of “very close” quantities. This situation is referred to as crowding, with the term taken from works concerned with conformal maps.