On the analytic continuation of Lauricella function

One of the generalizations of Gaussian hypergeometric function $F(a, b; c; z)$ to the case of several complex variables $(z_{1}, \dots, z_{N}) =: \mathbf{z}$ is Lauricella function $F_{D}^{(N)}\, (\mathbf{a}; b, c; \mathbf{z}\,)$, which is defined by $N$ -multiple series (see [1], [2])
$$ F_{D}^{(N)}\,(\mathbf{a}; b, c; \mathbf{z}\,)=\sum\limits_{|\bf{k}| = 0}^{\infty} \,\frac{(b)_{|\bf{k}|} (a_{1})_{k_{1}} \cdots (a_{N})_{k_{N}}} {(c)_{|\bf{k}|} k_{1}! \cdots k_{N}!}z_{1}^{k_{1}} \cdots z_{N}^{k_{N}},\, $$
where $b$ and $c \notin \mathbb{Z}^{-}$ are some scalar (complex -valued) parameters, $\mathbf{a} = (a_{1}, \dots, a_{N})$ is some vector-valued parameter and $\mathbf{k} = (k_{1}, \dots, k_{N})$ is multi -index of summation with non-negative components. This Lauricella series converges in the unit polydisk $\mathbb{U}^{N}$.
In the talk, we construct the system of formulae that continue analytically the function $F_{D}^{(N)}$ to $N$–dimensional complex space for an arbitrary number of variables (see [3]).
[1] G. Lauricella, Sulle funzioni ipergeometriche a piu variabili. Rendiconti Circ. math. Palermo. 7 (1893). P. 111 – 158.
[2] H. Exton, Multiple hypergeometric functions and application. N.-Y., J. Willey & Sons inc., 1976.
[3] S.I. Bezrodnykh, Analytic continuation formulas and Jacobi- type relations for Lauricella function. Doklady Math. 93:2 (2016). P. 129 – 134.