Bijective proofs of basic hypergeometric series identities

Abstract

Bijections are given which prove the following theorems: the ^-binomial theorem, Heine's 2 transformation, the g-analogues of Gauss', Rummer's, and Saalschtz's theorems, the very well poised 4 3 and 6 5 evaluations, and Watson's transformation of an 8 7 to a 4 3 . The proofs hold for all values of the parameters. Bijective proofs of the terminating cases follow from the general case. A bijective version of limiting cases of these series is also given. The technique is to mimic the classical proofs, based upon a bijective proof of the ^-binomial theorem and sign-reversing involutions which cancel infinite products.