doiSerbia

Generalizations of a formula due to Kummer with applications

Kim Yong Sup (Department of Mathematics Education, Wonkwang University, Iksan, Korea), yspkim@wonkwang.ac.kr
Milovanović Gradimir V. (Serbian Academy of Sciences and Arts, Belgrade, Serbia + University of Niš, Faculty of Sciences and Mathematics, Niš, Serbia), gvm@mi.sanu.ac.rs
Wang Xiaoxia (Department of Mathematics, Shanghai University, Shanghai, P. R. China), xiaoxiawang@shu.edu.cn
Rathie Arjun Kumar (Department of Mathematics, Vedant College of Engineering and Technology, Rajasthan Technical University, Rajasthan State, India), arjunkumarrathie@gmail.com

The aim of this research paper is to obtain explicit expressions of 2F1 [a,b 1/2(a+b ± ℓ+1); 1+x/2] in the most general case for any ℓ = 0, 1, 2, ... For ℓ = 0, we have the well known, interesting and useful formula due to Kummer which was proved independently by Ramanujan. The results presented here are obtained with the help of known generalizations of Gauss’s second summation theorem for the series 2F1(1/2), which were given earlier by Rakha and Rathie [Integral Transforms Spec. Func. 22 (11) (2011), 823-840]. The results are further utilized to obtain new hypergeometric identities by using beta integral method developed by Krattenthaler & Rao [J. Comput. Appl. Math. 160 (2003), 159-173]. Several interesting results due to Ramanujan, Choi, et. al. and Krattenthaler & Rao follow special cases of our main findings.

Keywords: generalized hypergeometric function, Kummer’s formula, extension of Gauss second summation theorem, Ramanujan’s identity, Beta integral