ON AN EXTENSION OF KUMMER-TYPE II TRANSFORMATION

Year 2014, Volume: 4 Issue: 1, 80 - 85, 01.06.2014

Medhat A. Rakha Arjun K. Rathie

Abstract

In the theory of hypergeometric and generalized hypergeometric series, Kummer’s type I and II transformations play an important role. In this short research paper, we aim to establish the explicit expression of e − x 2 2F2   a, d + n; x 2a + n, d;   for n = 3. For n = 0, we have the well known Kummer’s second transformation. For n = 1, the result was established by Rathie and Pogany [12] and later on by Choi and Rathie [2]. For n = 2, the result was recently established by Rakha, et al. [10]. The result is derived with the help of Kummer’s second transformation and its contiguous results recently obtained by Kim, et. al.[4]. The result established in this short research paper is simple, interesting, easily established and may be potentially useful.

Keywords

Generalized Hypergeometric Series, Kummer’s type I and II transformations

References

• Bailey, W. N., Products of generalized hypergeometric series, Proc. London Math. Soc., 28, 242 - 250 (1928).
• Choi, J. and Rathie, A. K., Another proof of Kummer’s second theorem, Commun. Korean Math. Soc., 13, 933 - 936 (1998).
• Kim, Y. S., Rakha M. A., and Rathei, A. K., Extensions of classical summation theorems for the series2F1,3F2and4F3with applications in Ramanujan’s summations, Int. J. Math. & Math. Sci., ID , 26 pages, (2010).
• Kim, Y. S., Rakha, M. A., and Rathei, A. K., Generalizations of Kummer’s second theorem with applications, Comput. Math. & Math. Phys., 50 (3), 387 - 402 (2010).
• Kim, Y. S., Choi, J. and Rathie, A. K., Two results for the terminating3F2(2) with applications, Bull. Korean Math. Soc., 49 (3),621 – 633 (2012).
• Kummer, E. E., ¨Uber die hypergeometridche Reihe . . . , J. Reine Angew. Math., 15, 39 - 83 (1836).
• Paris, R. B., A Kummer type transformation for a hypergeometric function, J. Comput. Appl. Math., 173, 379 - 382 (2005).
• Rainville, E. D., Special Functions, The Macmillan Company, New York (1960).
• Rakha, M. A. and Rathie, A. K., Generalizations of classical summation theorems for the series2F1 and3F2with applications, Integral Transform and Special Functions, 22 (11), 823 - 840 (2011).
• Rakha, M. A., Awad, M. M., and Rathie, A. K., On an extension of Kummer’s second theorem, Abstract and Applied Analysis, Volume 2013, Article ID 128458, 6 pages.
• Rakha, M. A. A note on Kummer-type II transformation for the generalized hypergeometric function F2, Mathematical Notes, 19 (1), 154 - 156 (2012).
• Rathie, A. K. and Pogany, K. New summation formula for3F21 (1) tion, Math. Communic, 13, 63 - 66 (2008).