TWO RESULTS FOR THE TERMINATING 3F2(2) WITH APPLICATIONS

Title & Authors
TWO RESULTS FOR THE TERMINATING 3F2(2) WITH APPLICATIONS
Kim, Yong-Sup; Choi, June-Sang; Rathie, Arjun K.;

Abstract
By establishing a new summation formula for the series $\small{_3F_2(\frac{1}{2})}$, recently Rathie and Pogany have obtained an interesting result known as Kummer type II transformation for the generalized hypergeometric function $\small{_2F_2}$. Here we aim at deriving their result by using a very elementary method and presenting two elegant results for certain terminating series $\small{_3F_2(2)}$. Furthermore two interesting applications of our new results are demonstrated.
Keywords
generalized hypergeometric series;Gausss summation theorem;Gausss second theorem;Kummer`s first and second theorems;quadratic transformation;Kummer type I and II transformations;Kamp$\small{\acute{e}}$ de F$\small{\acute{e}}$riet function;
Language
English
Cited by
1.
On an Extension of Kummer's Second Theorem, Abstract and Applied Analysis, 2013, 2013, 1
2.
On a reducibility of the Kampé de Fériet function, Mathematical Methods in the Applied Sciences, 2015, 38, 12, 2600
References
1.
G. E. Andrews, Applications of basic hypergeometric functions, SIAM Rev. 16 (1974), no. 4, 441-484.

2.
P. Appell and J. Kampe de Feriet, Functions hypergeometiques et hypersperique, Gauthier Villars, Paris, 1926.

3.
W. N. Bailey, Products of generalized hypergeometric series, Proc. London Math. Soc. 28 (1928), no. 2, 242-254.

4.
J. L. Burchnall and T. W. Chaundy, Expansions of Appell's double hypergeometric functions, Quart. J. Math. Oxford Ser. 11 (1940), 249-270.

5.
J. L. Burchnall and T. W. Chaundy, Expansions of Appell's double hypergeometric functions (II), Quart. J. Math. Oxford Ser. 12 (1941), 112-128.

6.
R. G. Bushman and H. M. Srivastava, Some identities and reducibility of Kampe de Feriet functions, Math. Proc. Cambridge Philos. Soc. 91 (1982), 435-440.

7.
J. Choi, A. K. Rathie, and H. Harsh, A note on Reed Dawson identities, Korean J. Math. Sci. 11 (2004), no. 2, 1-4.

8.
H. Exton, Multiple Hypergeometric Functions, Ellis Horwood, Chichester, U. K., 1976.

9.
H. Exton, On the reducibility of the Kampe de Feriet function, J. Comput. Appl. Math. 83 (1997), no. 1, 119-121.

10.
P. W. Karlsson, Some reduction formulas for power series and Kampe de Feriet functions, Proc. A. Kon. Nedel. Akad. Weten. 87 (1984), 31-36.

11.
Y. S. Kim, On certain reducibility of Kampe de Feriet functions, Honam Math. J. 31 (2009), no. 2, 167-176.

12.
Y. S. Kim, M. A. Rakha, and A. K. Rathie, Extensions of certain classical summation theorems for the series $_{2}F_{1}$ and $_{3}F_{2}$ with applications in Ramanujan's summations, Int. J. Math. Math. Sci. 2010 (2010), Art. ID 309503, 26 pp.

13.
Y. S. Kim, M. A. Rakha, and A. K. Rathie, Generalizations of Kummer's second theorem with applications, Comput. Math. Math. Phys. 50 (2010), no. 3, 387-402.

14.
E. D. Krupnikov, A register of computer oriented reduction of identities for Kampe de Feriet functions, Novosibirsk, Russia, 1996.

15.
E. E. Kummer, Uber die hypergeometrische Reihe $1+\frac{\alpha{\cdot}\beta}{1{\cdot}\gamma}x+\frac{\alpha(\alpha+1)\beta(\beta+1)}{1{\cdot}2{\gamma}{\cdot}(\gamma+1)}x^{2}+{\cdots}$ , J. Reine Angew Math. 15 (1836), 39-83; 127-172.

16.
E. E. Kummer, Collected Papers, Vol. 2, Springer Verlag, Berlin, 1975.

17.
J. L. Lavoie, F. Grondin, and A. K. Rathie, Generalizations of Watson's theorem on the sum of $_{3}F_{2}$, Indian J. Math. 34 (1992), no. 1, 23-32.

18.
J. L. Lavoie, F. Grondin, and A. K. Rathie, Generalizations of Whipple's theorem on the sum of $_{3}F_{2}$, J. Comput. Appl. Math. 72 (1996), no. 2, 293-300.

19.
J. L. Lavoie, F. Grondin, A. K. Rathie, and K. Arora, Generalizations of Dixon's theorem on the sum of $_{3}F_{2}$, Math. Comp. 62 (1994), no. 205, 267-276.

20.
S. Lewanowicz, Generalized Watson's summation formula for $_{3}F_{2}$(1), J. Comput. Appl. Math. 86 (1997), no. 2, 375-386.

21.
M. Milgram, On hypergeometric $_{3}F_{2}$(1), Arxiv. Math. CA/0603096 (2006).

22.
R. P. Paris, A Kummer type transformation for a $_{2}F_{2}$ hypergeometric function, J. Comput. Appl. Math. 173 (2005), no. 2, 379-382.

23.
E. D. Rainville, Special Functions, The Macmillan company, New York, 1960.

24.
M. A. Rakha and A. K. Rathie, Generalizations of classical summation theorems for the series $_{2}F_{1}$ and $_{3}F_{2}$ with applications, Integral Transforms and Special Functions (2011), to appear.

25.
A. K. Rathie and J. Choi, Another proof of Kummer's second theorem, Commun. Korean Math. Soc. 13 (1998), no. 4, 933-936.

26.
A. K. Rathie and V. Nargar, On Kummer's second theorem involving product of gener- alized hypergeometric series, Matematiche (Catania) 50 (1995), no. 1, 35-38.

27.
A. K. Rathie and U. Pandey, Another method for a quadratic transformation formula due to Kummer, Proc. National Conference PAMET (2010), 1-5.

28.
A. K. Rathie and R. B. Paris, An extension of the Euler type transformation for the $_{3}F_{2}$ series, Far East J. Math. Sci. 27 (2007), no. 1, 43-48.

29.
A. K. Rathie and T. Pogany, New summation formula for $_-3}F_-2}(\frac-1}-2})$ and a Kummer type II transformation of $_-2}F_-2}$(x), Math. Commun. 13 (2008), no. 1, 63-66.

30.
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.

31.
H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.

32.
H. M. Srivastava and R. Panda, An integral representation for the product of two Jacobi polynomials, J. London Math. Soc. (2) 12 (1975/76), no. 4, 419-425.

33.
R. Videnas, A generalization of Kummer's identity, Rocky Mountain J. Math. 32 (2002), no. 2, 919-936.