The Bailey transform is

*G. E. Andrews*’ [Proc. Symp. Pure Math. Am. Math. Soc., Columbus, Ohio 1978, Proc. Symp. Pure Math. 34, 1-24 (1979;

Zbl 0403.33002)] codification in terms of matrix inversions of a technique of W. N. Bailey for generating new Rogers-Ramanujan type identities. It can also be used to exhibit the equivalence of certain pairs of identities such as the summation formulæ for the terminating balanced

${}_{3}{\phi}_{2}$ and the very-well-poised

${}_{6}{\phi}_{5}$. In this paper, the author moves all of this machinery up to the context of multiple basic hypergeometric series very-well-poised on

$U(n+1)$. It is then used to recreate much of the classical theory of basic hypergeometric series in the more general context of multiple series over

$U(n+1)$.