

Complex Approximations, Orthogonal Polynomials and Applications Workshop
June 8, 2021 09:30–10:10, Sochi






Analytic continuation of the multiple hypergeometric functions
S. I. Bezrodnykh^{} ^{} Federal Research Center "Computer Science and Control" of Russian Academy of Sciences, Moscow

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Abstract:
A wide class of hypergeometric functions in several variables
$\mathbf{z} = (z_1, z_2, …, z_N) \in \mathbb{C}^N$
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 $\mathbf{k} = (k_1, k_2, …, k_N)$ is the multiindices,
$\mathbf{z}^\mathbf{k} := z_1^{k_1} z_2^{k_2}\cdots z_N^{k_N}$,
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 $k_1, k_2, …, k_N$ and
$\mathbf{e}_j = (0,…,1,…,0)$ 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
$\mathbb{U}^N:=\{z_j < 1, j = \overline{1, N} \}$,
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 $(a_1, …, a_N)=: \mathbf{a}$, $b$,
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:
$$
\begin{split}
&z_j (1  z_j) \frac{\partial^2 u}{\partial{z_j}^2}
+ (1  z_j) \sideset {'}\sum\nolimits_{k = 1}^N
z_k \frac{\partial^2u}{\partial z_j \partial z_k} +
+ [c  (1 + a_j& + b) z_j]
\frac{\partial u}{\partial z_j}
 a_j \sideset {'}\sum\nolimits_{k = 1}^N z_k
\frac{\partial u}{\partial z_k} 
a_j b u = 0,\qquad j = \overline{1,N},
\end{split}
$$
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. 190700750.
Language: English
Website:
https://us02web.zoom.us/j/8618528524?pwd=MmxGeHRWZHZnS0NLQi9jTTFTTzFrQT09
^{*} Zoom conference ID: 861 852 8524 , password: caopa

