  We give multidimensional generalizations of several transformation formulae
for basic hypergeometric series of a specific type. Most of the upper
parameters of the series differ multiplicatively from corresponding lower
parameters by a nonnegative integer power of the base q. In one dimension,
formulae for such series have been found, in the q -> 1 case, by B. M. Minton
and P. W. Karlsson, and in the basic case by G. Gasper, by W. C. Chu, and more
recently by the author. Our identities involve multilateral basic
hypergeometric series associated to the root system A_r (or equivalently, the
unitary group U(r+1)).
