# differential-difference equations for hypergeometric function

The hypergeometric function^{} satisfies several equations which
relate derivatives with respect to the argument $z$ to
shifting the parameters $a,b,c,d$ by unity (Here, the
prime denotes derivative with respect to $z$.):

$z{F}^{\prime}(a,b;c;z)+aF(a,b;c;z)$ | $=F(a+1,b;c;z)$ | ||

$z{F}^{\prime}(a,b;c;z)+bF(a,b;c;z)$ | $=F(a,b+1;c;z)$ | ||

$z{F}^{\prime}(a,b;c;z)+(c-1)F(a,b;c;z)$ | $=F(a,b;c-1;z)$ | ||

$(1-z)z{F}^{\prime}(a,b;c;z)$ | $=(c-a)F(a-1,b;c;z)+(a-c+bz)F(a,b;c;z)$ | ||

$(1-z)z{F}^{\prime}(a,b;c;z)$ | $=(c-b)F(a,b-1;c;z)+(b-c+az)F(a,b;c;z)$ | ||

$(1-z)z{F}^{\prime}(a,b;c;z)$ | $=z(c-a)(c-b)F(a,b;c+1;z)+zc(a+b-c)F(a,b;c;z)$ |

These equations may readily be verified by differentiating the series
which defines the hypergeometric equation. By eliminating the derivatives
between these equations, one obtains the contiguity relations^{} for the
hypergeometric function. By differentiating them once more and taking
suitable linear combinations^{}, one may obtain the differential equation^{}
of the hypergeometric function.

Title | differential-difference equations for hypergeometric function |
---|---|

Canonical name | DifferentialdifferenceEquationsForHypergeometricFunction |

Date of creation | 2013-03-22 17:36:18 |

Last modified on | 2013-03-22 17:36:18 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 6 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 33C05 |