The Lauricella hypergeometric function $F_D^{(N)}$, the Riemann–Hilbert problem, and some applications

The problem of analytic continuation is considered for the Lauricella function $F_D^{(N)}$, a generalized hypergeometric functions of $N$ complex variables. For an arbitrary $N$ a complete set of formulae is given for its analytic continuation outside the boundary of the unit polydisk, where it is defined originally by an $N$-variate hypergeometric series. Such formulae represent $F_D^{(N)}$ in suitable subdomains of $\mathbb{C}^N$ in terms of other generalized hypergeometric series, which solve the same system of partial differential equations as $F_D^{(N)}$. These hypergeometric series are the $N$-dimensional analogue of Kummer's solutions in the theory of Gauss's classical hypergeometric equation. The use of this function in the theory of the Riemann–Hilbert problem and its applications to the Schwarz–Christoffel parameter problem and problems in plasma physics are also discussed.
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