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 certificate 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 satisfied (a simple mechanical task), and check the equally simple fact that the boundary conditions (F1), (G1), (G2) below are satisfied. 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 §3 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 coefficients. 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 find indefinite hypergeometric sums, where they exist, by quite a general procedure (see [11]). In this paper we describe a general attack on definite hypergeometric, and other, sums, continuing the program started in [13-15] . • The method can prove, in a unified way, virtually all known hypergeometric sum identities (and therefore legions of binomial coefficient identities too). It does this by means of certificates of proof, each of which consists of a pair of functions (F, G) (a 'WZ-pair') that satisfy certain conditions, described below. As a by-product, each such pair