### 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 |
---|---|

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 |

### Fingerprint

### 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

**Contiguous relations and their computations for 2F1 hypergeometric series.** / Ibrahim, Adel K.; Rakha, Medhat A.

Research output: Contribution to journal › Article

*Computers and Mathematics with Applications*, vol. 56, no. 8, pp. 1918-1926. https://doi.org/10.1016/j.camwa.2008.04.018

}

TY - JOUR

T1 - Contiguous relations and their computations for 2F1 hypergeometric series

AU - Ibrahim, Adel K.

AU - Rakha, Medhat A.

PY - 2008/10

Y1 - 2008/10

N2 - The hypergeometric function 2F1 [a1, a2 ; a3 ; 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 a1, a2 and a3. More precisely, we obtained a recurrence relation including 2F1 [a1 + α1, a2 ; a3 ; z], 2F1 [a1, a2 + α2 ; a3 ; z] and 2F1 [a1, a2 ; a3 + α3 ; z] for any arbitrary integers α1, α2 and α3.

AB - The hypergeometric function 2F1 [a1, a2 ; a3 ; 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 a1, a2 and a3. More precisely, we obtained a recurrence relation including 2F1 [a1 + α1, a2 ; a3 ; z], 2F1 [a1, a2 + α2 ; a3 ; z] and 2F1 [a1, a2 ; a3 + α3 ; z] for any arbitrary integers α1, α2 and α3.

KW - F hypergeometric function

KW - Computer algebra

KW - Contiguous function relation

KW - Gauss hypergeometric function

KW - Linear recurrence relation

UR - http://www.scopus.com/inward/record.url?scp=51049122983&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51049122983&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2008.04.018

DO - 10.1016/j.camwa.2008.04.018

M3 - Article

VL - 56

SP - 1918

EP - 1926

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 8

ER -