C n and D n Very-Well-Poised 10 φ 9 Transformations

Abstract

In this paper we derive multivariable generalizations of Bailey's classical terminating balanced very-well-poised 10 \(\phi\) 9 transformation. We work in the setting of multiple basic hypergeometric series very-well-poised on the root systems A n , C n , and D n . Following the distillation of Bailey's ideas by Gasper and Rahman [11], we use a suitable interchange of multisums. We obtain C n and D n 10 \(\phi\) 9 transformations combined with A n , C n , and D n extensions of Jackson's 8 \(\phi\) 7 summation. Milne and Newcomb have previously obtained an analogous formula for A n series. Special cases of our 10 \(\phi\) 9 transformations include several new multivariable generalizations of Watson's transformation of an 8 \(\phi\) 7 into a multiple of a 4 \(\phi\) 3 series. We also deduce multidimensional extensions of Sears' 4 \(\phi\) 3 transformation formula, the second iterate of Heine's transformation, the q -Gauss summation theorem, and of the q -binomial theorem.