Algorithms for m-Fold Hypergeometric Summation

Abstract Zeilberger's algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms F ( n, k ) is extended to certain nonhypergeometric terms. An expression F ( n, k ) is called hypergeometric term if both F ( n +1, k )/ F ( n, k ) and F ( n , k +1)/ F ( n, k ) are rational functions. Typical examples are ratios of products of exponentials, factorials, Γ function terms, binomial coefficients, and Pochhammer symbols that are integer-linear with respect to n and k in their arguments. We consider the more general case of such ratios that are rational-linear with respect to n and k in their arguments, and present an extended version of Zeilberger's algorithm for this case, using an extended version of Gosper's algorithm for indefinite summation. In a similar way the Wilf-Zeilberger method of rational function certification of integer-linear hypergeometric identities is extended to rational-linear hypergeometric identities. The given algorithms on definite summation apply to many cases in the literature to which neither the Zeilberger approach nor the Wilf-Zeilberger method is applicable. Examples of this type are given by theorems of Watson and Whipple, and a large list of identities ("Strange evaluations of hypergeometric series") that were studied by Gessel and Stanton. Finally we show how the algorithms can be used to generate new identities.