The generalized hypergeometric function is given by a Hypergeometric Series, i.e., a series for which the ratio of
successive terms can be written

(1) |

(2) | |||

(3) |

where is the Pochhammer Symbol or Rising Factorial

(4) |

(5) |

is ``the'' Hypergeometric Function, and is the Confluent Hypergeometric Function. A function of the form is called a Confluent Hypergeometric Limit Function.

The generalized hypergeometric function

(6) |

(7) |

(8) |

(9) |

A generalized hypergeometric equation is termed ``well posed'' if

(10) |

(11) |

(12) |

(13) |

Gosper (1978) discovered a slew of unusual hypergeometric function identities, many of which were subsequently proven by Gessel and Stanton (1982). An important generalization of Gosper's technique, called Zeilberger's Algorithm, in turn led to the powerful machinery of the Wilf-Zeilberger Pair (Zeilberger 1990).

Special hypergeometric identities include Gauss's Hypergeometric Theorem

(14) |

(15) |

(16) |

(17) |

(18) |

(19) |

Gessel (1994) found a slew of new identities using Wilf-Zeilberger Pairs, including the following:

(20) |

(21) |

(22) |

(23) |

**References**

Bailey, W. N. *Generalised Hypergeometric Series.* Cambridge, England: Cambridge University Press, 1935.

Dwork, B. *Generalized Hypergeometric Functions.* Oxford, England: Clarendon Press, 1990.

Exton, H. *Multiple Hypergeometric Functions and Applications.* New York: Wiley, 1976.

Gessel, I. ``Finding Identities with the WZ Method.'' *Theoret. Comput. Sci.* To appear.

Gessel, I. and Stanton, D. ``Strange Evaluations of Hypergeometric Series.'' *SIAM J. Math. Anal.* **13**, 295-308, 1982.

Gosper, R. W. ``Decision Procedures for Indefinite Hypergeometric Summation.'' *Proc. Nat. Acad. Sci. USA* **75**, 40-42, 1978.

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. *A=B.* Wellesley, MA: A. K. Peters, 1996.

Saxena, R. K. and Mathai, A. M. *Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences.*
New York: Springer-Verlag, 1973.

Slater, L. J. *Generalized Hypergeometric Functions.* Cambridge, England: Cambridge University Press, 1966.

Zeilberger, D. ``A Fast Algorithm for Proving Terminating Hypergeometric Series Identities.'' *Discrete Math.* **80**, 207-211, 1990.

© 1996-9

1999-05-25