### 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 |
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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 |

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### 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