Generalization of a Transformation Formula for the Exton's Triple Hypergeometric Series X12 and X17

• Journal title : Kyungpook mathematical journal
• Volume 54, Issue 4,  2014, pp.677-684
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2014.54.4.677
Title & Authors
Generalization of a Transformation Formula for the Exton's Triple Hypergeometric Series X12 and X17
Choi, Junesang; Rathie, Arjun K.;

Abstract
In the theory of hypergeometric functions of one or several variables, a remarkable amount of mathematicians's concern has been given to develop their transformation formulas and summation identities. Here we aim at generalizing the following transformation formula for the Exton's triple hypergeometric series $\small{X_{12}}$ and $\small{X_{17}}$: $\small{(1+2z)^{-b}X_{17}\;\left(a,b,c_3;\;c_1,c_2,2c_3;\;x,{\frac{y}{1+2z}},{\frac{4z}{1+2z}}\right)\\{\hfill{53}}=X_{12}\;\left(a,b;\;c_1,c_2,c_3+{\frac{1}{2}};\;x,y,z^2\right).}$ The results are derived with the help of two general hypergeometric identities for the terminating $\small{_2F_1(2)}$ series which were very recently obtained by Kim et al. Four interesting results closely related to the Exton's transformation formula are also chosen, among ten, to be derived as special illustrative cases of our main findings. The results easily obtained in this paper are simple and (potentially) useful.
Keywords
Hypergeometric functions of several variables;Multiple Gaussian hypergeometric series;Exton's triple hypergeometric series;Gauss's hypergeometric functions;
Language
English
Cited by
References
1.
J. Choi, Notes on formal manipulations of double series, Commun. Korean Math. Soc., 18(4)(2003), 781-789.

2.
J. Choi, A. Hasanov and M. Turaev, Decomposition formulas and integral representations for some Exton typergeometric functions, J. Chungcheong Math. Soc., 24(4)(2011), 745-758.

3.
J. Choi, A. Hasanov and M. Turaev, Linearly independent solutions for the hypergeometric Exton functions $X_1$ and $X_2$, Honam Math. J., 32(2)(2010), 223-229.

4.
J. Choi, A. Hasanov and M. Turaev, Certain integral representations of Euler type for the Exton function $X_5$, Honam Math. J., 32(3)(2010), 389-397.

5.
J. Choi, A. Hasanov and M. Turaev, Certain integral representations of Euler type for the Exton function $X_2$, J. Korean Soc. Math. Edu. Ser. B: Pure Appl. Math., 17(4)(2010), 347-354.

6.
H. Exton, Hypergeometric functions of three variables, J. Indian Acad. Math., 4(1982), 113-119.

7.
Y. S. Kim, M. A. Rakha and A. K. Rathie, Generalization of Kummer's second summation theorem with applications, Comput. Math. Math. Phys., 50(3)(2010), 387-402.

8.
Y. S. Kim, J. Choi and A. K. Rathie, Remark on two results by Padmanabham for Exton's triple triple hypergeometric series $X_8$, Honam Math. J., 27(4)(2005), 603-608.

9.
Y. S. Kim and A. K. Rathie, On an extension formulas for the triple hypergeometric X8 due to Exton, Bull. Korean Math. Soc., 44(4)(2007), 743-751.

10.
Y. S. Kim, A. K. Rathie and J. Choi, Another method for Padmanabham's transformation formula for Exton's triple triple hypergeometric series X$X_8$, Commun. Korean Math. Soc., 24(4)(2009), 517-521.

11.
S. W. Lee and Y. S. Kim, An extension of the triple hypergeometric series by Exton, Honam Math. J., 31(1)(2009), 61-71.

12.
E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.

13.
H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985.