# an algebraic identity leading to Wilson’s theorem

For any positive integer $n$ and any real or complex number^{} $x$,

$$\sum _{k=0}^{n}{(-1)}^{k}\left(\genfrac{}{}{0pt}{}{n}{k}\right){(x-k)}^{n}=n!$$ |

Furthermore, if $n>m$ then

$$\sum _{k=0}^{n}{(-1)}^{k}\left(\genfrac{}{}{0pt}{}{n}{k}\right){(x-k)}^{m}=0$$ |

## References

- 1 S. M Ruiz. An algebraic identity leading to Wilson’s theorem. The Mathemtical Gazette, 80(489):579–582, November 1996. http://arxiv.org/abs/math.GM/0406086math.GM/0406086.

Title | an algebraic identity leading to Wilson’s theorem |
---|---|

Canonical name | AnAlgebraicIdentityLeadingToWilsonsTheorem |

Date of creation | 2013-03-22 14:31:38 |

Last modified on | 2013-03-22 14:31:38 |

Owner | GeraW (6138) |

Last modified by | GeraW (6138) |

Numerical id | 16 |

Author | GeraW (6138) |

Entry type | Result |

Classification | msc 11B65 |

Classification | msc 05A10 |

Related topic | Factorial |

Related topic | WilsonsTheorem |