A hypergeometric series is a series for which and the ratio of consecutive terms is a Rational
Function of the summation index , i.e., one for which
Many sums can be written as Generalized Hypergeometric Functions by inspections of the ratios of consecutive terms in the generating hypergeometric series.
See also Generalized Hypergeometric Function, Geometric Series, Hypergeometric Function, Hypergeometric Identity
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.