TY - JOUR
T1 - Generalization of Kummer's second theorem with applications
AU - Kim, Yong Sup
AU - Rakha, M. A.
AU - Rathie, A. K.
N1 - Funding Information:
ACKNOWLEDGMENTS The first author was supported by Wonkwang University Research Fund, 2010. This paper was pre sented at the 16th International Conference on Finite or Infinite Dimensional Analysis and Applications (ICFID 16) held at Dongguk University, Gyeongju, South Korea during July 28–August 1, 2008 by the third author.
PY - 2010
Y1 - 2010
N2 - The aim of this research paper is to obtain single series expression for i = 0, ±1, ±2, ±3, ±4, ±5, where 1F1(·) is the function of Kummer. For i = 0, we have the well known Kummer second theorem. The results are derived with the help of generalized Gauss second summation theorem obtained earlier by Lavoie et al. In addition to this, explicit expressions of each for i = 0, ±1, ±2, ±3, ±4, ±5 are also given. For i = 0, we get two interesting and known results recorded in the literature. As an applications of our results, explicit expressions for i, j = 0, ±1, ±2, ±3, ±4, ±5 and for j = 0, ±1, ±2, ±3, ±4, ±5 are given. For i = j = 0 and j = 0, we respectively get the well known Preece identity and a well known quadratic transformation formula due to Kummer. The results derived in this paper are simple, interesting, easily established and may be useful in the applicable sciences.
AB - The aim of this research paper is to obtain single series expression for i = 0, ±1, ±2, ±3, ±4, ±5, where 1F1(·) is the function of Kummer. For i = 0, we have the well known Kummer second theorem. The results are derived with the help of generalized Gauss second summation theorem obtained earlier by Lavoie et al. In addition to this, explicit expressions of each for i = 0, ±1, ±2, ±3, ±4, ±5 are also given. For i = 0, we get two interesting and known results recorded in the literature. As an applications of our results, explicit expressions for i, j = 0, ±1, ±2, ±3, ±4, ±5 and for j = 0, ±1, ±2, ±3, ±4, ±5 are given. For i = j = 0 and j = 0, we respectively get the well known Preece identity and a well known quadratic transformation formula due to Kummer. The results derived in this paper are simple, interesting, easily established and may be useful in the applicable sciences.
KW - Dixon theorem
KW - Function of Kummer
KW - Generalization of Kummer theorem
KW - Generalized gipergeometric function
KW - Hypergeometric Gauss summation theorem
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U2 - 10.1134/S0965542510030024
DO - 10.1134/S0965542510030024
M3 - Article
AN - SCOPUS:77951791118
SN - 0965-5425
VL - 50
SP - 387
EP - 402
JO - Computational Mathematics and Mathematical Physics
JF - Computational Mathematics and Mathematical Physics
IS - 3
ER -