### Abstract

The aim of this research paper is to obtain single series expression for i = 0, ±1, ±2, ±3, ±4, ±5, where _{1}F_{1}(·) 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.

Original language | English |
---|---|

Pages (from-to) | 387-402 |

Number of pages | 16 |

Journal | Computational Mathematics and Mathematical Physics |

Volume | 50 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 |

### Fingerprint

### Keywords

- Dixon theorem
- Function of Kummer
- Generalization of Kummer theorem
- Generalized gipergeometric function
- Hypergeometric Gauss summation theorem

### ASJC Scopus subject areas

- Computational Mathematics

### Cite this

*Computational Mathematics and Mathematical Physics*,

*50*(3), 387-402. https://doi.org/10.1134/S0965542510030024

**Generalization of Kummer's second theorem with applications.** / Kim, Yong Sup; Rakha, M. A.; Rathie, A. K.

Research output: Contribution to journal › Article

*Computational Mathematics and Mathematical Physics*, vol. 50, no. 3, pp. 387-402. https://doi.org/10.1134/S0965542510030024

}

TY - JOUR

T1 - Generalization of Kummer's second theorem with applications

AU - Kim, Yong Sup

AU - Rakha, M. A.

AU - Rathie, A. K.

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

UR - http://www.scopus.com/inward/record.url?scp=77951791118&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951791118&partnerID=8YFLogxK

U2 - 10.1134/S0965542510030024

DO - 10.1134/S0965542510030024

M3 - Article

AN - SCOPUS:77951791118

VL - 50

SP - 387

EP - 402

JO - Computational Mathematics and Mathematical Physics

JF - Computational Mathematics and Mathematical Physics

SN - 0965-5425

IS - 3

ER -