Abstract
In this article, some kind of hypergeometric functions are considered. For the three variables hypergeometric functions \({{F}_{1c}}\left(x,y,z\right)\), \({{F}_{2c}}\left(x,y,z\right)\), \({{F}_{4b}}\left(x,y,z\right)\), \({{F}_{4c}}\left(x,y,z\right)\), \({{F}_{5b}}\left(x,y,z\right)\), \({{F}_{5c}}\left(x,y,z\right)\) are obtained the new analytic continuation formulas.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
J. Barros-Neto and I. M. Gelfand, ‘‘Fundamental solutions for the Tricomi operator I,’’ Duke Math. J. 98, 465–483 (1999).
J. Barros-Neto and I. M. Gelfand, ‘‘Fundamental solutions for the Tricomi operator II,’’ Duke Math. J. 111, 561–584 (2002).
J. Barros-Neto and I. M. Gelfand, ‘‘Fundamental solutions for the Tricomi operator III,’’ Duke Math. J. 128, 119–140 (2005).
L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics (Wiley, New York, 1958).
A. Hasanov, ‘‘Fundamental solutions of generalized bi-axially symmetric Helmholtz equation,’’ Complex Variab. Ellipt. Equat. 52, 673–683 (2007).
A. Hasanov, ‘‘Some solutions of generalized Rassias’s equation,’’ Int. J. Appl. Math. Stat. 8 (M07), 20–30 (2007).
A. Hasanov, ‘‘The solution of the Cauchy problem for generalized Euler–Poisson–Darboux equation,’’ Int. J. Appl. Math. Stat. 8 (M07), 30–44 (2007).
A. Hasanov, ‘‘Fundamental solutions for degenerated elliptic equation with two perpendicular lines of degeneration,’’ Int. J. Appl. Math. Stat. 13 (8), 41–49 (2008).
A. Hasanov and E. T. Karimov, ‘‘Fundamental solutions for a class of three-dimensional elliptic equations with singular coeffcients,’’ Appl. Math. Lett. 22, 1828–1832 (2009).
A. Hasanov, J. M. Rassias, and M. Turaev, ‘‘Fundamental solution for the generalized elliptic Gellerstedt equation,’’ in Functional Equations, Difference Inequalities and ULAM Stability Notions (Nova Science, New York, 2010), Vol. 6, pp. 73–83.
G. Lohofer, ‘‘Theory of an electro-magnetically deviated metal sphere. 1: Absorbed power,’’ SIAM J. Appl. Math. 49, 567–581 (1989).
A. W. Niukkanen, ‘‘Generalized hypergeometric series arising in physical and quantum chemical applications,’’ J. Phys. A: Math. Gen. 16, 1813–1825 (1983).
F. I. Frankl, Selected Works in Gas Dynamics (Nauka, Moscow, 1973) [in Russian].
R. J. Weinacht, ‘‘Fundamental solutions for a class of singular equations,’’ Contrib. Differ. Equat. 3, 43–551 (1964).
A. Weinstein, ‘‘Discontinuous integrals and generalized potential theory,’’ Contrib. Trans. Am. Math. Soc. 63, 342–354 (1946).
Y. L. Luke, The Special Functions and their Approximations (Academic, New York, 1969), Vols. 1, 2.
A. S. Berdyshev, A. Hasanov, and Zh. A. Abdiramanov, ‘‘Solution of Cauchy problem for the generalized Gellerstedt equation,’’ Lobachevskii J. Math. 41, 1762–1771 (2020).
A. Hasanov and M. Ruzhansky, ‘‘Hypergeometric expansions of solutions of the degenerating model parabolic equations of the third order,’’ Lobachevskii J. Math. 41, 27–31 (2020).
M. Ruzhansky and A. Hasanov, ‘‘Self-similar solutions of some model degenerate partial differential equations of the second, third and fourth order,’’ Lobachevskii J. Math. 41, 1103–1114 (2020).
S. I. Bezrodnykh, ‘‘Analytic continuation formulas and Jacobi-type relations for the Lauricella function,’’ Dokl. Math. 93, 129–134 (2016).
S. I. Bezrodnykh and V. I. Vlasov, ‘‘On a new representation for the solution of the Riemann–Hilbert problem,’’ Math. Notes 99, 932–937 (2016).
S. I. Bezrodnykh, ‘‘Jacobi-type differential relations for the Lauricella function \(F_{D}^{(N)}\),’’ Math. Notes 99, 821–833 (2016).
S. I. Bezrodnykh and V. I. Vlasov, ‘‘The Riemann–Hilbert problem in a complex domain for a model of magnetic reconnection in a plasma,’’ Comput. Math. Math. Phys. 42, 263–298 (2002).
A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.
H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series (Halsted, Ellis Horwood, New York, 1985).
P. Appell and Kampe de Feriets, Fonctions Hypergeometriques et Hyperspheriques; Polynomes d’Hermite (Gauthier-Villars, Paris, 1926).
S. Saran, ‘‘Relations between functions contiguous to certain hypergeometric functions of three variables,’’ Ganita, тДЦ 5, 69–76 (1954).
S. Saran, ‘‘Transformations of certain hypergeometric functions of three variables,’’ Acta Math. 93, 292–312 (1955).
R. C. Pandey, ‘‘On the expansions of hypergeometric functions,’’ Agra Univ. J. Res. Sci. 12, 159–169 (1963).
A. Hasanov and M. Ruzhanskiy, ‘‘Euler-type integral representations for hypergeometric functions in three second-order variables,’’ Bull. Inst. Math. 6, 73–223 (2019).
Author information
Authors and Affiliations
Corresponding authors
Additional information
(Submitted by A. M. Elizarov)
Rights and permissions
About this article
Cite this article
Hasanov, A., Yuldashev, T.K. Analytic Continuation Formulas for the Hypergeometric Functions in Three Variables of Second Order. Lobachevskii J Math 43, 386–393 (2022). https://doi.org/10.1134/S1995080222050146
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080222050146