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.