A MULTIVARIABLE VIEW OF ONE-VARIABLE q-SERIES

1. The q-disease A symptom of the q-disease|that scourge since Euler's times|is that those aaicted feel compelled to retrace steps taken long ago by their masters. For instance, when last counted 4], there were 53 proofs of the Rogers{Ramanujan identities. Other popular identities, such as the q-binomial theorem, Euler's pentagonal number theorem, Jacobi's triple product identity, and Ramanujan's 1 1 sum, have also been the subject of many studies, see 2, 3, 6, 16]. Moreover, these excesses are not limited to a few pretty identities. Indeed, many authors 1, 2, 3, 5, 6, 15, 16, 57] have written their version of how the fundamental formulas for basic hypergeometric series should be organized. The present ooering is yet another survey of these identities. What is diierent here is that the focus is on the theory of multiple basic hypergeometric series, of the kind considered by Gustafson, Milne and their co-workers. Even so, we restrict our attention largely to the one-variable case, and there too merely sketch the ideas involved. More details, and historical references, may be found in Andrews 3] and Gasper and Rahman 16]. In the case of the multivariable theory, a longer introduction appears in 7] which, in turn, is based upon Milne and Lilly's 50] instructive survey. 2. The q-binomial theorem Rogers' 6 5 summation. We begin our organization of the fundamental theorems of basic hypergeometric series. Very often, there are many methods of obtaining these identities, only a few of which generalize to multivariables. To a large extent, Andrews' 3] approach works quite well in this regard. In addition, the interplay between diierent dimensions|unique to the multivariable theory|allows a dramatic simpliication. This simpliication is the subject of this section. Let q be a complex number such that jqj < 1. We deene the q-rising factorial as (2.2) where the equality (2.2) holds when k is a non-negative integer. Classical (one-variable) basic hypergeometric series, or q-hypergeometric series, with r numerator parameters a 1