Sums with unity Numerator and Factorials in the Denominator which can be expressed analytically include

(1) | |||

(2) | |||

(3) | |||

(4) |

where is a Hypergeometric Function and is a Gamma Function.

Sums with in the Numerator having analytic solutions include

(5) | |||

(6) | |||

(7) | |||

(8) |

A sum with in the Numerator is

(9) |

Sums of factorial Powers include

(10) |

(11) |

(12) | |||

(13) | |||

(14) |

(Beeler

**References**

Beeler, M.; Gosper, R. W.; and Schroeppel, R. *HAKMEM.* Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

© 1996-9

1999-05-26