# LINEARLY INDEPENDENT SOLUTIONS FOR THE HYPERGEOMETRIC EXTON FUNCTIONS X1 AND X2

• Choi, June-Sang (Department of Mathematics, Dongguk University) ;
• Hasanov, Anvar (Department of Mathematics, Dongguk University) ;
• Turaev, Mamasali (Department of Mathematics, Dongguk University)
• Accepted : 2011.04.16
• Published : 2011.06.25
• 91 9

#### Abstract

In investigation of boundary-value problems for certain partial differential equations arising in applied mathematics, we often need to study the solution of system of partial differential equations satisfied by hypergeometric functions and find explicit linearly independent solutions for the system. Here we choose the Exton functions $X_1$ and $X_2$ among his twenty functions to show how to find the linearly independent solutions of partial differential equations satisfied by these functions $X_1$ and $X_2$.

#### Keywords

Exton hypergeometric functions;System of partial differential equations;Linearly independent solutions;Global solution of differential equation

#### References

1. P. Appell and J. Kampe de Feriet, Fonctions Hypergeometriques et Hyperspheriques; Polynomes d'Hermite, Gauthier - Villars, Paris, 1926.
2. L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Wiley, New York, 1958.
3. F.I. Frankl, Selected Works in Gas Dynamics, Nauka, Moscow, 1973.
4. A.W. Niukkanen, Generalized hypergeometric series arising in physical and quantum chemical applications, J. Phys. A: Math. Gen. 16 (1983) 1813-1825. https://doi.org/10.1088/0305-4470/16/9/007
5. G. Lohofer, Theory of an electromagnetically deviated metal sphere. 1: Absorbed power, SIAM J. Appl. Math. 49 (1989), 567-581. https://doi.org/10.1137/0149032
6. A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw-Hill Book Company, New York, Toronto and London, 1953.
7. J. Barros-Neto and I.M. Gelfand, Fundamental solutions for the Tricomi operator, Duke Math. J. 98(3) (1999), 465-483. https://doi.org/10.1215/S0012-7094-99-09814-9
8. J. Barros-Neto and I.M. Gelfand, Fundamental solutions for the Tricomi operator II, Duke Math. J. 111(3) (2002), 561-584. https://doi.org/10.1215/S0012-7094-02-11137-5
9. J. Barros-Neto and I.M. Gelfand, Fundamental solutions for the Tricomi operator III, Duke Math. J. 128(1) (2005), 119-140. https://doi.org/10.1215/S0012-7094-04-12815-5
10. H. Exton, Hypergeometric functions of three variables, J. Indian Acad. Math. 4 (1982), 113-119.
11. A.J. Fryant, Growth and complete sequences of generalized bi-axially symmetric potentials, J. Differential Equations 31(2) (1979), 155-164. https://doi.org/10.1016/0022-0396(79)90141-4
12. A. Hasanov, Fundamental solutions of generalized bi-axially symmetric Helmholtz equation, Complex Variables and Elliptic Equations 52(8) (2007), 673-683. https://doi.org/10.1080/17476930701300375
13. A. Hasanov, Some solutions of generalized Rassias's equation, Intern. J. Appl. Math. Stat. 8(M07) (2007), 20-30.
14. A. Hasanov, The solution of the Cauchy problem for generalized Euler-Poisson-Darboux equation. Intern. J. Appl. Math. Stat. 8 (M07) (2007), 30-44.
15. A. Hasanov, Fundamental solutions for degenerated elliptic equation with two perpendicular lines of degeneration. Intern. J. Appl. Math. Stat. 13(8) (2008), 41-49.
16. A. Hasanov and E.T. Karimov, Fundamental solutions for a class of three-dimensional elliptic equations with singular coefficients. Appl. Math. Letters 22 (2009), 1828-1832. https://doi.org/10.1016/j.aml.2009.07.006
17. A. Hasanov, J.M. Rassias , and M. Turaev, Fundamental solution for the generalized Elliptic Gellerstedt Equation, Book: "Functional Equations, Difference Inequalities and ULAM Stability Notions", Nova Science Publishers Inc. NY, USA, 6 (2010), 73-83.
18. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), Wiley, New York, Chichester, Brisbane, and Toronto, 1985.
19. R.J. Weinacht, Fundamental solutions for a class of singular equations, Contrib. Differential Equations 3 (1964), 43-55.
20. A. Weinstein, Discontinuous integrals and generalized potential theory, Trans. Amer. Math. Soc. 63 (1946), 342-354.

#### Cited by

1. Relations between Lauricella’s triple hypergeometric function F A ( 3 ) ( x , y , z ) and Exton’s function X 8 vol.2013, pp.1, 2013, https://doi.org/10.1186/1687-1847-2013-34