The Lauricella function is defined as \[ F_{D}^{(N)}(\mathbf{a} ; b, c ; \mathbf{z}):=\sum_{|\mathbf{k}|=0}^{\infty} \frac{(b)_{|\mathbf{k}|}\left(a_{1}\right)_{k_{1}} \cdots\left(a_{N}\right)_{k_{N}}}{\left.(c)_{|\mathbf{k}|}\right|^{k_{1}} ! \cdots k_{N} !} z_{1}^{k_{1}} \cdots z_{N}^{k_{N}}, \] on the polydisk \(\mathbb{U}^{N}:=\left\{\mathbf{z} \in \mathbb{C}^{N}:\left|z_{j}\right|<1, j=1, \ldots, N\right\}\). (The sum is taken on the multi-index set \((k_1,\dots,k_n\ge0\), with \(|\mathbf{k}|=\sum_i k_i\).)
The author first addresses the question of analytic continuation from \(\mathbb{U}^{N}\) to larger domains. He presents a complete set of formulae for analytic continuation of the Lauricella function \(F_{D}^{(N)}\) of an arbitrary number \(N\) of variables. The subdomains of \(\mathbb{C}^N\) where these formulae hold totally cover \(\mathbb{C}^N\), away from certain hyperplanes. The formula extending \(F_{D}^{(N)}\) is asserted to be in the form \[ F_{D}^{(N)}(\mathbf{a} ; b, c ; \mathbf{z})=\sum_{j=0}^{N} \lambda_{j} u_{j}(\mathbf{a} ; b, c ; \mathbf{z}), \quad \mathbf{z} \notin \mathbb{U}^{N}, \] where \(\lambda_j\)’s are constants not all of them simultaneously vanishing, and the functions \(u_j(a; b, c; z)\) are generalized hypergeometric series which satisfy the same system of PDEs as \(F_{D}^{(N)}\) itself.
The Jacobi identity \[ \frac{d}{d z}\left[z^{c-1}(1-z)^{a+b-c} F(a, b ; c ; z)\right]=(c-1) z^{c-2}(1-z)^{a+b-c-1} F(a-1, b-1 ; c-1 ; z) \]
for the Gauss hypergeometric function is well known. The author finds a version generalized to the Lauricella function. This is needed in finding the solution of the Riemann-Hilbert problem in terms of a Schwarz-Christoffel integral.
Section 4 contains an application from plasma physics. In stellar physics problems often a rarefied plasma is considered, when magnetic forces dominate other forces, and large amounts of energy are released as a result of the magnetic reconnection phenomenon (this is a fundamental change in the configuration of the magnetic field). The central mathematical problem in the investigation of such processes is often an effective calculation of the magnetic field for this or that plasma configuration. The author presents a solution of the Riemann-Hilbert problem in a polygonal domain which arises in modelling magnetic reconnection near a disintegrating current layer in the corona of the Sun.
In the final section of the paper, a deeper study of the connection between the Lauricella function and the Schwarz-Christoffel parameter problem is presented.