A BC(n) Bailey lemma and generalizations of Rogers-Ramanujan identities

The symmetric rational function W_λ/μ (z; t, q, a, b) in n independent variables z = defined by Gustafson is generalized by introducing an extra parameter r $\in C$ to the ω_λ/μ (z; r, t, q; a, b) function, and a characterization of this extension is given. A BC _n multivariable Jackson sum in terms of ω_λ is proved. Certain fundamental properties of the function ω _λ are established. A BC_n generalization of the classical Bailey Transform and, consequently, a ₁₀ϕ ₉ multiple basic hypergeometric series transformation identity are proved. A BC_n generalization of the Rogers-Selberg identity, and D_n generalizations of the Rogers-Ramanujan identities are given in terms of determinants of theta functions. BC_n extensions of the one and two parameter Bailey Lemmas are proved and D_n generalizations of the extreme cases of the Andrews-Gordon identities are given, again in terms of determinants of theta functions. The Bailey Lemma is interpreted in the setting of a multivariate interpolation problem.