### Abstract

The hypergeometric function _{2}F_{1} [a_{1}, a_{2} ; a_{3} ; z] plays an important role in mathematical analysis and its application. Gauss defined two hypergeometric functions to be contiguous if they have the same power-series variable, if two of the parameters are pairwise equal, and if the third pair differs by ±1. He showed that a hypergeometric function and any two other contiguous to it are linearly related. In this paper, we present an interesting formula as a linear relation of three shifted Gauss polynomials in the three parameters a_{1}, a_{2} and a_{3}. More precisely, we obtained a recurrence relation including _{2}F_{1} [a_{1} + α_{1}, a_{2} ; a_{3} ; z], _{2}F_{1} [a_{1}, a_{2} + α_{2} ; a_{3} ; z] and _{2}F_{1} [a_{1}, a_{2} ; a_{3} + α_{3} ; z] for any arbitrary integers α_{1}, α_{2} and α_{3}.

Original language | English |
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Pages (from-to) | 1918-1926 |

Number of pages | 9 |

Journal | Computers and Mathematics with Applications |

Volume | 56 |

Issue number | 8 |

DOIs | |

Publication status | Published - Oct 2008 |

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### Keywords

- F hypergeometric function
- Computer algebra
- Contiguous function relation
- Gauss hypergeometric function
- Linear recurrence relation

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Modelling and Simulation

### Cite this

*Computers and Mathematics with Applications*,

*56*(8), 1918-1926. https://doi.org/10.1016/j.camwa.2008.04.018