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Uspekhi Mat. Nauk, 2018, Volume 73, Issue 6(444), Pages 3–94 (Mi umn9841)  

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

S. I. Bezrodnykhab

a Dorodnicyn Computing Centre of Russian Academy of Sciences
b Peoples' Friendship University of Russia

Abstract: 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.
Bibliography: 163 titles.

Keywords: multivariate hypergeometric functions, systems of partial differential equations, analytic continuation, Riemann–Hilbert problem, Schwarz–Christoffel integral, crowding problem, magnetic reconnection phenomenon.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 5-100
Russian Foundation for Basic Research 16-01-00781
16-07-01195
This research was carried out with the support of the Peoples' Friendship University of Russia, programme “5-100”, and the Russian Foundation for Basic Research (grant nos. 16-01-00781 and 16-07-01195).


DOI: https://doi.org/10.4213/rm9841

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English version:
Russian Mathematical Surveys, 2018, 73:6, 941–1031

Bibliographic databases:

Document Type: Article
UDC: 517.5
MSC: Primary 33C65, 30E25, 30C20; Secondary 82D10, 85A15
Received: 18.07.2018

Citation: S. I. Bezrodnykh, “The Lauricella hypergeometric function $F_D^{(N)}$, the Riemann–Hilbert problem, and some applications”, Uspekhi Mat. Nauk, 73:6(444) (2018), 3–94; Russian Math. Surveys, 73:6 (2018), 941–1031

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