A pair of Closed Form functions is said to be a Wilf-Zeilberger pair if

(1) |

Wilf-Zeilberger pairs are very useful in proving Hypergeometric Identities of the form

(2) |

(3) |

(4) |

(5) |

(6) |

For example, consider the Binomial Coefficient identity

(7) |

(8) |

(9) |

(10) |

Taking

(11) |

(12) |

(13) |

For any Wilf-Zeilberger pair ,

(14) |

(15) |

(16) |

(17) |

(18) | |||

(19) |

(Amdeberhan and Zeilberger 1997). The latter identity has been used to compute Apéry's Constant to a large number of decimal places (Plouffe).

**References**

Amdeberhan, T. and Zeilberger, D. ``Hypergeometric Series Acceleration via the WZ Method.'' *Electronic J. Combinatorics* **4**, No. 2, R3, 1-3, 1997.
http://www.combinatorics.org/Volume_4/wilftoc.html#R03. Also available at
http://www.math.temple.edu/~zeilberg/mamarim/mamarimhtml/accel.html.

Cipra, B. A. ``How the Grinch Stole Mathematics.'' *Science* **245**, 595, 1989.

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. ``The WZ Phenomenon.'' Ch. 7 in *A=B.*
Wellesley, MA: A. K. Peters, pp. 121-140, 1996.

Wilf, H. S. and Zeilberger, D. ``Rational Functions Certify Combinatorial Identities.'' *J. Amer. Math. Soc.* **3**, 147-158, 1990.

Zeilberger, D. ``The Method of Creative Telescoping.'' *J. Symb. Comput.* **11**, 195-204, 1991.

Zeilberger, D. ``Closed Form (Pun Intended!).'' *Contemporary Math.* **143**, 579-607, 1993.

© 1996-9

1999-05-26