Hypergeometric Series
A hypergeometric series 
is a series for which 
and the ratio of consecutive terms is a RationalFunction of the summation index 
, i.e., one for which
 | (1) |
and 
Polynomials. The functions generated by hypergeometric series are calledHypergeometric Functions or, more generally, Generalized HypergeometricFunctions. If the polynomials are completely factored, the ratio of successiveterms can be written | (2) |
in the Denominator is present for historical reasons of notation, and the resultingGeneralized Hypergeometric Function is written | (3) |
and 
, the function becomes a traditional Hypergeometric Function 
.Many sums can be written as Generalized Hypergeometric Functions byinspections of the ratios of consecutive terms in the generating hypergeometric series.
See also Generalized Hypergeometric Function, Geometric Series, Hypergeometric Function,Hypergeometric Identity
References
Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. ``Hypergeometric Series,'' ``How to Identify a Series as Hypergeometric,'' and ``Software That Identifies Hypergeometric Series.'' §3.2-3.4 in A=B. Wellesley, MA: A. K. Peters, pp. 34-42, 1996.
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