ON AN EXTENSION OF KUMMER-TYPE II TRANSFORMATION

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,

PDF

___

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).