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 -