International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 309503, 26 pages
doi:10.1155/2010/309503
Research Article

Extensions of Certain Classical Summation Theorems for the Series 2 𝐹 1 , 3 𝐹 2 , and 4 𝐹 3 with Applications in Ramanujan's Summations

Yong Sup Kim,1 Medhat A. Rakha,2,3 and Arjun K. Rathie4

1Department of Mathematics Education, Wonkwang University, Iksan 570-749, Republic of Korea
2Mathematics Department, College of Science, Suez Canal University, Ismailia 41522, Egypt
3Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, Muscat, Alkhod 123, Oman
4Vedant College of Engineering and Technology, Village-Tulsi, Post-Jakhmund, Bundi, Rajasthan State 323021, India

Received 20 May 2010; Revised 7 September 2010; Accepted 23 September 2010

Academic Editor: Teodor Bulboacă

Copyright © 2010 Yong Sup Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Motivated by the extension of classical Gauss's summation theorem for the series 2 𝐹 1 given in the literature, the authors aim at presenting the extensions of various other classical summation theorems such as those of Kummer, Gauss's second, and Bailey for the series 2 𝐹 1 , Watson, Dixon and Whipple for the series 3 𝐹 2 , and a few other hypergeometric identities for the series 3 𝐹 2 and 4 𝐹 3 . As applications, certain very interesting summations due to Ramanujan have been generalized. The results derived in this paper are simple, interesting, easily established, and may be useful.