A GENERALIZATION OF KUMMER'S IDENTITY

The well-known formula of Kummer evaluates the hypergeometric series $_2{F_1}\left( {\begin{array}{*{20}{c}} {A,B} \\ C \\ \end{array} \left| { - 1} \right.} \right)$ when the relation C — A + B = 1 holds. This paper deals with the 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 Γ-terms multiplied by terminating ₃F₂(1) series. A few such formulas were essentially known to Whipple in the 1920s. Here we give a simpler and more complete overview of this type of evaluation. Additionally, algorithmic aspects of evaluating hypergeometric series are considered. We illustrate Zeilberger's method and discuss its applicability to nonterminating series and present a couple of similar generalizations of other known formulas.