A NONTERMINATING 8φ7 SUMMATION FOR THE ROOT System Cr

where aq = bcdef (cf. [9, Eq. (2.11.7)]), is one of the deepest results in the classical theory of basic hypergeometric series. It contains many important identities as special cases (such as the nonterminating 3φ2 summation, the terminating 8φ7 summation, and all their specializations including the q-binomial theorem). One way to derive (1.1) is to start with a particular rational function identity, namely Bailey’s [5] very-well-poised 10φ9 transformation, and apply a nontrivial limit procedure, see the exposition in Gasper and Rahman [9, Secs. 2.10 and 2.11]. Basic hypergeometric series (and, more generally, q-series) have various applications in combinatorics, number theory, representation theory, statistics, and physics, see Andrews [1], [2]. For a general account of the importance of basic hypergeometric series in the theory of special functions see Andrews, Askey, and Roy [4]. There are different types of multivariable series. The one we are concerned with are so-called multiple basic hypergeometric series associated to root systems (or, equivalently, to Lie algebras). This is mainly just a classification of certain multiple series according to the type of specific factors (such as a Vandermonde determinant)