A generalization of Kummer's identity

Details A generalization of Kummer's identity A generalization of Kummer's identity

2000-05-10

arxiv.org/abs/math/0005095v3

Mathematics

Classical Analysis and ODEs

13 pages; classical proofs simplified, possible transformations reviewed; in the algoritmic part similar evaluations of other

Scientific paper

The well-known Kummer's formula evaluates the hypergeometric series 2F1(A,B;C;-1) when the relation B-A+C=1 holds. This paper deals with evaluation of 2F1(-1) series in the case when C-A+B is an integer. Such a series is expressed as a sum of two \Gamma-terms multiplied by terminating 3F2(1) series. A few such formulas were essentially known to Whipple in 1920's. Here we give a simpler and more complete overview of this type of evaluations. Additionally, algorithmic aspects of evaluating hypergeometric series are considered. We illustrate Zeilberger's method and discuss its applicability to non-terminating series, and present a couple of similar generalizations of other known formulas.

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