Serendipitous Functional Relations Deducible from Certain Generalized Triple Hypergeometric Functions

• Journal title : Kyungpook mathematical journal
• Volume 52, Issue 2,  2012, pp.109-136
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2012.52.2.109
Title & Authors
Serendipitous Functional Relations Deducible from Certain Generalized Triple Hypergeometric Functions
Choi, June-Sang; Hasanov, Anvar; Turaev, Mamasali;

Abstract
We aim at presenting certain unexpected functional relations among various hypergeometric functions of one or several variables (for example, see the identities in Corollary 5) by making use of Carlson's method employed in his work (Some extensions of Lardner's relations between $\small{_0F_3}$ and Bessel functions, SIAM J. Math. Anal. 1(2)(1970), 232-242).
Keywords
eneralized hypergeometric series;Pochhammer symbol;Gauss function;Appell-Kamp$\small{\acute{e}}$ de F$\small{\acute{e}}$riet functions;Bessel functions;Kelvin's functions;Srivastava's hypergeometric functions;
Language
English
Cited by
References
1.
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series, 55, National Bureau of Standards, Washington, D. C., 1964; Reprinted by Dover Publications, New York, 1965.

2.
A. Altin, Some expansion formulas for a class of singular partial differential equations, Proc. Amer. Math. Soc., 85(1)(1982), 42-46.

3.
P. Appell and J. Kampe de Feriet, Fonctions Hypergeometriques et Hyperspheriques; Polynomes d'Hermite, Gauthier - Villars, Paris, 1926.

4.
J. Barros-Neto and I. M. Gelfand, Fundamental solutions for the Tricomi operator, Duke Math. J., 98(3)(1999), 465-483.

5.
J. Barros-Neto and I. M. Gelfand, Fundamental solutions for the Tricomi operator II, Duke Math. J., 111(3)(2002), 561-584.

6.
J. Barros-Neto and I. M. Gelfand, Fundamental solutions for the Tricomi operator III, Duke Math. J., 128(1)(2005), 119-140.

7.
L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Wiley, New York, 1958.

8.
B. C. Carlson, Some extensions of Lardner's relations between \$_0F_3\$ and Bessel functions, SIAM J. Math. Anal., 1(2)(1970), 232-242.

9.
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.

10.
A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill Book Company, New York, Toronto and London, 1953.

11.
F. I. Frankl, Selected Works in Gas Dynamics. Nauka, Moscow 1973.

12.
A. J. Fryant, Growth and complete sequences of generalized bi-axially symmetric potentials, J. Diff. Equa., 31(2)(1979), 155-164.

13.
A. Hasanov, Fundamental solutions of generalized bi-axially symmetric Helmholtz equation, Complex Variables and Elliptic Equations 52(8)(2007), 673-683.

14.
A. Hasanov, Some solutions of generalized Rassias's equation, Intern. J. Appl. Math. Stat., 8(M07)(2007), 20-30.

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. Lett., 22(2009), 1828-1832.

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.
A. Hasanov and H. M. Srivastava, Some decomposition formulas associated with the Lauricella Function and other multiple hypergeometric functions, Appl. Math. Lett., 19(2006), 113-121.

19.
A. Hasanov and H. M. Srivastava, Decomposition formulas associated with the Lauricella multivariable hypergeometric functions, Comput. Math. Appl., 53(7)(2007), 1119-1128.

20.
A. Hasanov, H. M. Srivastava, and M. Turaev, Decomposition formulas for some triple hypergeometric functions, J. Math. Anal. Appl., 324(2006), 955-969.

21.
A. Hasanov and M. Turaev, Decomposition formulas for the double hypergeometric G1 and G2 Hypergeometric functions, Appl. Math. Comput., 187(1)(2007), 195-201.

22.
Y. S. Kim, A. K. Rathie and J. Choi, Note on Srivastava's triple hypergeometric series \$H_A\$, Commun. Korean Math. Soc., 18(3)(2003), 581-586.

23.
T. J. Lardner, Relations between \$_0F_3\$ and Bessel functions, SIAM Review, 11(1969), 69-72.

24.
T. J. Lardner and C. R. Steele, Symmetric deformations of circular cylindrical elas- tic shells of exponentially varying thickness, Trans. ASME Ser. E. J. Appl. Mech., 35(1968), 169-170.

25.
G. Lohofer, Theory of an electromagnetically deviated metal sphere. 1: Absorbed power. SIAM J. Appl. Math., 49(1989), 567-581.

26.
P. A. McCoy, Polynomial approximation and growth of generalized axisymmetric potentials, Canad. J. Math., 31(1)(1979), 49-59.

27.
A. W. Niukkanen, Generalized hypergeometric series arising in physical and quantum chemical applications, J. Phys. A: Math. Gen., 16(1983), 1813-1825.

28.
A. K. Rathie and Y. S. Kim, Further results on Srivastava's triple hypergeometric series \$H_A\$ and \$H_C\$, Ind. J. Pure Appl. Math., 35(8)(2004), 991-1002.

29.
M. S. Salakhitdinov and A. Hasanov, A solution of the Neumann-Dirichlet boundary value problem for generalized bi-axially symmetric Helmholtz equation, Complex Variables and Elliptic Equations, 53(4)(2008), 355-364.

30.
H. M. Srivastava, Hypergeometric functions of three variables, Ganita, 15(2)(1964), 97-108.

31.
H. M. Srivastava, Some integrals representing hypergeometric functions, Rend. Circ. Mat. Palermo, 16(2)(1967), 99-115.

32.
H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.

33.
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.

34.
M. Turaev, Decomposition formulas for Srivastava's hypergeometric function on Saran functions, Comput. Appl. Math., 233(2009), 842-846.

35.
G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd Edi., Cambridge University Press, Cambridge, London and New York, 1944.

36.
A. Weinstein, Discontinuous integrals and generalized potential theory, Trans. Amer. Math. Soc., 63(1946), 342-354.

37.
A. Weinstein, Generalized axially symmetric potential theory, Bull. Amer. Math. Soc., 59(1953), 20-38.

38.
R. J. Weinacht, Fundamental solutions for a class of singular equations, Contrib. Diff. Equa., 3(1964), 43-55.