# incomplete gamma function recurrence formula

The incomplete gamma function^{} satisfies the following recurrence formula:

$$\gamma (a+1,x)=a\gamma (a,x)-{x}^{a}{e}^{-x}$$ |

This can be derived by integration by parts:

${\int}_{0}^{x}}{e}^{-t}{t}^{a}\mathit{d}t$ | $=$ | $-{\displaystyle {\int}_{0}^{x}}{t}^{a}\mathit{d}{e}^{-t}$ | ||

$=$ | $a{\displaystyle {\int}_{0}^{x}}{t}^{a-1}\mathit{d}{e}^{-t}-{x}^{a}{e}^{-x}$ |

In terms of other variants of the incomplete gamma function, the recursion relation appears as follows:

$P(a+1,x)$ | $=$ | $P(a,x)-{\displaystyle \frac{{x}^{a}{e}^{-x}}{\mathrm{\Gamma}(a+1)}}$ | ||

${\gamma}^{*}(a-1,x)$ | $=$ | $x{\gamma}^{*}(a,x)+{\displaystyle \frac{{e}^{-x}}{\mathrm{\Gamma}(a)}}$ |

Title | incomplete gamma function recurrence formula |
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Canonical name | IncompleteGammaFunctionRecurrenceFormula |

Date of creation | 2013-03-22 15:36:50 |

Last modified on | 2013-03-22 15:36:50 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 6 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 30D30 |

Classification | msc 33B15 |