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 Zh. Vychisl. Mat. Mat. Fiz., 2017, Volume 57, Number 4, Pages 555–587 (Mi zvmmf10555)

Analytic continuation of the Appell function $F_1$ and integration of the associated system of equations in the logarithmic case

S. I. Bezrodnykhabc

a Dorodnicyn Computing Center, Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, Moscow, Russia
b Sternberg Astronomical Institute, Moscow State University, Moscow, Russia
c RUDN University, Moscow, Russia

Abstract: The Appell function $F_1$ (i.e., a generalized hypergeometric function of two complex variables) and a corresponding system of partial differential equations are considered in the logarithmic case when the parameters of $F_1$ are related in a special way. Formulas for the analytic continuation of $F_1$ beyond the unit bicircle are constructed in which $F_1$ is determined by a double hypergeometric series. For the indicated system of equations, a collection of canonical solutions are presented that are two-dimensional analogues of Kummer solutions well known in the theory of the classical Gauss hypergeometric equation. In the logarithmic case, the canonical solutions are written as generalized hypergeometric series of new form. The continuation formulas are derived using representations of $F_1$ in the form of Barnes contour integrals. The resulting formulas make it possible to efficiently calculate the Appell function in the entire range of its variables. The results of this work find a number of applications, including the problem of parameters of the Schwarz–Christoffel integral.

Key words: hypergeometric functions two variables, system of partial differential equations, Barnes-type integrals, analytic continuation.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation Russian Foundation for Basic Research 16-01-00781_à16-07-01195_à Russian Academy of Sciences - Federal Agency for Scientific Organizations

DOI: https://doi.org/10.7868/S0044466917040044

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English version:
Computational Mathematics and Mathematical Physics, 2017, 57:4, 559–589

Bibliographic databases:

UDC: 519.651

Citation: S. I. Bezrodnykh, “Analytic continuation of the Appell function $F_1$ and integration of the associated system of equations in the logarithmic case”, Zh. Vychisl. Mat. Mat. Fiz., 57:4 (2017), 555–587; Comput. Math. Math. Phys., 57:4 (2017), 559–589

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. I. Bezrodnykh, “Analytic continuation of the Lauricella function $F_D^{(N)}$ with arbitrary number of variables”, Integral Transform. Spec. Funct., 29:1 (2018), 21–42
2. S. I. Bezrodnykh, “The Lauricella hypergeometric function $F_D^{(N)}$, the Riemann–Hilbert problem, and some applications”, Russian Math. Surveys, 73:6 (2018), 941–1031
3. Tarasov O.V., “Functional Reduction of Feynman Integrals”, J. High Energy Phys., 2019, no. 2, 173
4. O. V. Tarasov, “Using functional equations to calculate Feynman integrals”, Theoret. and Math. Phys., 200:2 (2019), 1205–1221
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