Analytic continuation of the multiple hypergeometric functions

A wide class of hypergeometric functions in several variables $\mathbf{z} = (z_1, z_2, \dots, z_N) \in \mathbb{C}^N$ is defined with the help of the Horn series [1–3], which has the form:
$$\Phi^{(N)} (\mathbf{z}) = \sum\nolimits_{\mathbf{k} \in \mathbb{Z}^N} \Lambda(\mathbf{k}) \mathbf{z}^\mathbf{k}; \,(1)$$
here $\mathbf{k} = (k_1, k_2, \dots, k_N)$ is the multi-indices, $\mathbf{z}^\mathbf{k} := z_1^{k_1} z_2^{k_2}\cdots z_N^{k_N}$, and the coefficients $\Lambda(\mathbf{k})$ are such that the ratio of any two adjacent is a rational function of the components of the summation index. In other words, for all $j = \overline{1,N}$ the relations are fulfilled: $\Lambda(\mathbf{k} + \mathbf{e}_j) / \Lambda(\mathbf{k}) = P_j (\mathbf{k}) / Q_j(\mathbf{k})$, $j = \overline{1,N}$, where $P_j$ and $Q_j$ are some polynomials in the $N$ variables $k_1, k_2, \dots, k_N$ and $\mathbf{e}_j = (0,\dots,1,\dots,0)$ denote the vectors with $j$th component equal to 1.

The talk describes the approach proposed in [4] for deriving formulas for the analytic continuation of series (1) with respect to the variables $\mathbf{z}$ into the entire complex space $\mathbb{C}^N$ in the form of linear combinations $\Phi^{(N)} (\mathbf{z}) = \sum_m A_m u_m (\mathbf{z})$, where $u_m (\mathbf{z})$ are hypergeometric series of the Horn type satisfying the same system of partial differential equations as the series (1) and $A_m$ are some coefficients. The implementation of this approach is demonstrated by the example of the Lauricella hypergeometric function $F_D^{(N)}$. In the unit polydisk $\mathbb{U}^N:=\big\{|z_j| < 1,\, j = \overline{1, N}\,\big\}$, this function is defined by the following series, see [5], [6]:
$$F_D^{(N)}\, (\mathbf{a}; b, \, c;\, \mathbf{z})\,:=\,\sum_{|\mathbf{k}| = 0}^{\infty}\,\frac{(b)_{|\mathbf{k}|} (a_1)_{k_1} \cdots (a_N)_{k_N}}{(c)_{|\mathbf{k}|} k_1! \cdots k_N!}\,\mathbf{z}^\mathbf{k}\,; \,(2)$$
here the complex values $(a_1, \dots, a_N)=: \mathbf{a}$, $b$, and $c$ play the role of parameters, $c \notin \mathbb{Z}^-$, $|\mathbf{k}| := \sum_{j = 1}^N k_j$, and $k_j \geq 0$, $j = \overline{1, N}$, the Pochhammer symbol is defined as $(a)_m := {\Gamma (a + m)}/{\Gamma (a)} = a (a + 1) \cdots (a + m - 1)$.

In [7], we have constructed a complete set of formulas for the analytic continuation of series (1) for an arbitrary $N$ into the exterior of the unit polydisk. Such formulas represent the function $F_D^{(N)}$ in suitable subdomains of $\mathbb{C}^N$ as linear combinations of hypergeometric series that are solutions of the following system [5], [6] of partial differential equations:
$$\begin{split} &z_j (1 - z_j) \frac{\partial^2 u}{\partial{z_j}^2} + (1 - z_j) \sideset{}{'}\sum\nolimits_{k = 1}^N z_k \frac{\partial^2u}{\partial z_j \partial z_k}\, +\\ + \Big[c - (1 + a_j& + b) z_j\Big] \frac{\partial u}{\partial z_j} -\, a_j \sideset{}{'}\sum\nolimits_{k = 1}^N z_k\, \frac{\partial u}{\partial z_k}\, -\, a_j b\, u = 0,\qquad j = \overline{1,N}, \end{split}$$
which the function $F_D^{(N)}$ satisfies; here a prime on a summation sign means that the sum is taken for $k \not = j$. The convergence domains of the found continuation formulas together cover $\mathbb{C}^N \setminus\mathbb{U}^N$.

We give an application of the obtained results on the analytic continuation of the Lauricella function $F_D^{(N)}$ to effective computation of conformal map of polygonal domains in the crowding situation.

The work is financially supported by RFBR, proj. 19-07-00750.