Invariance groups of transformations of basic hypergeometric series

We show that certain two-term transformation formulas between basic hyperge ometric series can easily be described by means of invariance groups. For t he transformations of nonterminating (3)phi(2) series, and those of termina ting balanced (4)phi(3) series, these invariance groups are symmetric group s. For transformations of (2)phi(1) series the invariance group is the dihe dral group of order 12. Transformations of terminating (3)phi(2) series are described by means of some subgroup of S-6, and finally the invariance gro up of transformations of very-well-poised nonterminating (8)phi(7) series i s shown to be isomorphic to the Weyl group of a root system of type D-5. (C ) 1999 American Institute of Physics. [S0022- 2488(99)00512-5].