Abstract
For the Lauricella function \(F_{D}^{{(N)}}\), which is a hypergeometric function of several complex variables \({{z}_{1}}, \ldots ,{{z}_{N}}\), analytic continuation formulas are constructed that correspond to the intersection of an arbitrary number of singular hyperplanes of the form \(\{ {{z}_{j}} = {{z}_{l}}\} \), \(j,l = \overline {1,N} \), \(j \ne l.\) These formulas give an expression for the considered function in the form of linear combinations of Horn hypergeometric series in \(N\) variables satisfying the same system of partial differential equations as the original series defining \(F_{D}^{{(N)}}\) in the unit polydisk. By applying these formulas, the function \(F_{D}^{{(N)}}\) and Euler-type integrals expressed in terms of \(F_{D}^{{(N)}}\) can be efficiently computed (with the help of exponentially convergent series) in the entire complex space \({{\mathbb{C}}^{N}}\) in the complicated cases when the variables form one or several groups of “very close” quantities. This situation is referred to as crowding, with the term taken from works concerned with conformal maps.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
H. Exton, Multiple Hypergeometric Functions and Application (Wiley, New York, 1976).
I. M. Gel’fand, M. I. Graev, and V. S. Retakh, “General hypergeometric systems of equations and series of hypergeometric type,” Russ. Math. Surv. 47 (4), 1–88 (1992).
K. Iwasaki, H. Kimura, Sh. Shimomura, and M. Yoshida, From Gauss to Painlevé: A Modern Theory of Special Functions (Friedrich Vieweg & Sohn, Braunschweig, 1991).
K. Aomoto and M. Kita, Theory of Hypergeometric Functions (Springer, Tokyo, 2011).
N. Akerblom and M. Flohr, “Explicit formulas for the scalar modes in Seiberg–Witten theory with an application to the Argyres–Douglas point,” J. High Energy Phys. 2 (057), 24 (2005).
R.-P. Holzapfel, A. M. Uludag, and M. Yoshida, Arithmetic and Geometry around Hypergeometric Functions (Birkhäuser, Basel, 2007).
O. V. Tarasov, “Using functional equations to calculate Feynman integrals,” Theor. Math. Phys. 200, 1205–1221 (2019).
S. I. Bezrodnykh, “The Lauricella hypergeometric function \(F_{D}^{{(N)}}\), the Riemann–Hilbert problem, and some applications,” Russ. Math. Surv. 73 (6), 941–1031 (2018).
Yu. A. Brychkov and N. V. Savischenko, “Application of hypergeometric functions of two variables in wireless communication theory,” Lobachevskii J. Math. 40 (7), 938–953 (2019).
J. Bergé, R. Massey, Q. Baghi, and P. Touboul, “Exponential shapelets: Basis functions for data analysis of isolated feature,” Mon. Not. R. Astron. Soc. 486 (1), 544–559 (2019).
S. I. Bezrodnykh and V. I. Vlasov, “Asymptotics of the Riemann–Hilbert problem for the Somov model of magnetic reconnection of long shock waves,” Math. Notes 110 (6), 853–871 (2021).
V. I. Vlasov and S. L. Skorokhodov, “Analytical solution for the cavitating flow over a wedge II,” Comput. Math. Math. Phys. 61 (11), 1834–1854 (2021).
G. Lauricella, “Sulle funzioni ipergeometriche a piu variabili,” Rend. Circ. Math. Palermo 7, 111–158 (1893).
Higher Transcendental Functions (Bateman Manuscript Project), Ed. by A. Erdélyi (McGraw-Hill, New York, 1953), Vol. 1.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge Univ. Press, Cambridge, 1996), Vol. 2.
S. I. Bezrodnykh, “Analytic continuation of the Lauricella function \(F_{D}^{{(N)}}\) with arbitrary number of variables,” Integral Transforms Spec. Funct. 29 (1), 21–42 (2018).
S. I. Bezrodnykh, “Analytic continuation of Lauricella’s function \(F_{D}^{{(N)}}\) for large in modulo variables near hyperplanes \(\{ {{z}_{j}} = {{z}_{l}}\} \),” Integral Transforms Spec. Funct. (2021). https://doi.org/10.1080/10652469.2021.1929206
S. I. Bezrodnykh, “Analytic continuation of Lauricella’s function \(F_{D}^{{(N)}}\) for variables close to unit near hyperplanes \(\{ {{z}_{j}} = {{z}_{l}}\} \),” Integral Transforms Spec. Funct. (2021). https://doi.org/10.1080/10652469.2021.1939329
P. Henrici, Applied and Computational Complex Analysis (Wiley, New York, 1991), Vols. 1–3.
L. N. Trefethen, “Numerical construction of conformal maps,” Appendix to E. B. Saff and A. D. Snider, Fundamentals of Complex Analysis for Mathematics, Science, and Engineering (Prentice Hall, New York, 1993).
P. K. Kythe, Computational Conformal Mapping (Birkhäuser, Basel, 1998).
V. I. Vlasov and S. L. Skorokhodov, “Multipole method for the Dirichlet Problem on doubly connected domains of complex geometry: A general description of the method,” Comput. Math. Math. Phys. 40 (11), 1567–1581 (2000).
S. I. Bezrodnykh and V. I. Vlasov, “The Riemann–Hilbert problem in a complicated domain for the model of magnetic reconnection in plasma,” Comput. Math. Math. Phys. 42 (3), 263–298 (2002).
L. N. Trefethen and T. A. Driscoll, Schwarz–Christoffel Transformation (Cambridge Univ. Press, Cambridge, 2005).
L. Banjai, “Revisiting the crowding phenomenon in Schwarz–Christoffel mapping,” SIAM J. Sci. Comput. 30 (2), 618–636 (2008).
N. Papamichael and N. Stylianopoulos, Numerical Conformal Mapping: Domain Decomposition and the Mapping of Quadrilaterals (World Scientific, Hackensack, NJ, 2010).
T. M. Sadykov and A. K. Tsikh, Hypergeometric and Algebraic Functions of Several Variables (Nauka, Moscow, 2014) [in Russian].
S. I. Bezrodnykh, “Analytic continuation of the Horn hypergeometric series with an arbitrary number of variables,” Integral Transforms Spec. Funct. 31 (10), 788–803 (2020).
F. Fox, “The asymptotic expansion of hypergeometric functions,” Proc. London Math. Soc. 27 (2), 389–400 (1928).
E. M. Wright, “The asymptotic expansion of hypergeometric functions,” Proc. London Math. Soc. 10 (4), 286–293 (1935).
S. I. Bezrodnykh, “Analytic continuation of the Kampé de Fériet function and the general double Horn series,” Integral Transforms Spec. Funct. (2022). https://doi.org/10.1080/10652469.2022.2056601
S. I. Bezrodnykh, “Formulas for analytic continuation of Horn functions of two variables,” Comput. Math. Math. Phys. 62 (6), 884–903 (2022).
Yu. A. Brychkov and N. V. Savischenko, “On some formulas for the Horn functions \({{H}_{5}}(a,b;c;w,z)\) and \(H_{5}^{c}(a,c;w,z)\),” Integral Transforms Spec. Funct. (2021). https://doi.org/10.1080/10652469.2021.1938026
Yu. A. Brychkov and N. V. Savischenko, “On some formulas for the Horn functions \({{H}_{6}}(a,b,b',w,z)\) and \(H_{8}^{{(c)}}(a,b;w,z)\),” Integral Transforms Spec. Funct. (2021). https://doi.org/10.1080/10652469.2021.2017427
B. Ananthanarayan, S. Beraay, S. Friot, O. Marichev, and T. Pathak, “On the evaluation of the Appell F 2 double hypergeometric function” (2021). arXiv:2111.05798v1
Yu. A. Brychkov and N. V. Savischenko, “On some formulas for the Horn function \({{H}_{7}}(a,b,b';c;w,z)\),” Integral Transforms Spec. Funct. (2021). https://doi.org/10.1080/10652469.2022.2056600
M. Kalmykov, V. Bytev, B. Kniehl, S.-O. Moch, B. Ward, and S. Yost, “Hypergeometric functions and Feynman diagrams,” in Anti-Differentiation and the Calculation of Feynman Amplitudes, Ed. by J. Blümlein and C. Schneider (Springer, Cham, 2021).
Funding
This work was financially supported by the Russian Science Foundation, grant no. 22-21-00727, https://rscf.ru/project/22-21-00727.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author declares that he has no conflicts of interest.
Additional information
Translated by I. Ruzanova
Rights and permissions
About this article
Cite this article
Bezrodnykh, S.I. Formulas for Computing the Lauricella Function in the Case of Crowding of Variables. Comput. Math. and Math. Phys. 62, 2069–2090 (2022). https://doi.org/10.1134/S0965542522120041
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542522120041