Gaurav Bhatnagar and Michael J. Schlosser

Cn and Dn very-well-poised 10φ9 transformations

(37 pages)

Abstract. In this paper, we derive multivariable generalizations of Bailey's classical terminating balanced very-well-poised 10φ9 transformation. We work in the setting of multiple basic hypergeometric series very-well-poised on the root systems An, Cn, and Dn. Following the distillation of Bailey's ideas by Gasper and Rahman, we use a suitable interchange of multisums. We obtain Cn and Dn 10φ9 transformations from an interchange of multisums, combined with An, Cn, and Dn extensions of Jackson's 8φ7 summation. Milne and Newcomb have previously obtained an analogous formula for An series. Special cases of our 10φ9 transformations include several new multivariable generalizations of Watson's transformation of an 8φ7 into a multiple of a 4φ3 series. We also deduce multidimensional extensions of Sears' 4φ3 transformation formula, the second iterate of Heine's transformation, the q-Gauss summation theorem, and of the q-binomial theorem.

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