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.