Rational Functions Certify Combinatorial Identities

This paper presents a general method for proving and discovering combinatorial identities: to prove an identity one can present a certi cate that consists of a pair of functions of two integer variables. To prove the identity, take the two functions that are given, check that condition (1) below is satis ed (a simple mechanical task), and check the equally simple fact that the boundary conditions (F1), (G1), (G2) below are satis ed. The identity is then proved. Alternatively, one can present the identity itself, and a single rational function. To prove the identity the reader would then construct the pair of functions referred to above, and proceed as before (see x3 below). In this paper we present several one-line proofs of hypergeometric identities. All of these one-line proofs were found by using the method presented below, on computers that have strong symbolic manipulation packages. Once the proofs have been found, they can be checked by hand or on small personal computers that would need only minimal symbolic manipulation capability. Not too long ago the world of combinatorial identities consisted of hundreds of individually proved relations (for a valuable collection of these see [10]), mostly involving binomial coe cients. As a result of ideas of H. Bateman (see the introduction to [10]), G. Andrews [1], and others, it is widely recognized that most of these are special cases of relatively few hypergeometric identities, and attention is now being turned to methods of systematizing these higher level relationships. Gosper [9] has shown how to nd inde nite hypergeometric sums, where they exist, by quite a general procedure (see [11]). In this paper we describe a general attack on de nite hypergeometric, and other, sums, continuing the program started in [13{15].