C-N AND D-N VERY-WELL-POISED (10)PHI(9) TRANSFORMATIONS

In this paper we derive multivariable generalizations of Bailey's clas sical terminating balanced very-well-poised 10 phi 9 transformation. W e work in the setting of multiple basic hypergeometric series very-wel l-poised on the root systems A(n), C-n, and D-n. Following the distill ation of Bailey's ideas by Gasper and Rahman [11], we use a suitable i nterchange of multisums. We obtain C-n, and D-n, 10 phi 9 transformati ons combined with A(n), C-n, and D-n extensions of Jackson's 8 phi 7 s ummation. Milne and Newcomb have previously obtained an analogous form ula for A,, series. Special cases of our 10 phi 9 transformations incl ude several new multivariable generalizations of Watson's transformati on of an 8 phi 7 into a multiple of a 4 phi 3 series. We also deduce m ultidimensional extensions of Sears' 4 phi 3 transformation formula, t he second iterate of Heine's transformation, the q-Gauss summation the orem, and of the q-binomial theorem.