Transformation formulas for multivariable basic hypergeometric series

Abstract

We study multivariable (bilateral) basic hypergeometric series associated with (type A) Macdonald polynomials. We derive several transformation and summation properties for such series, including analogues of Heine's 2^1 transformation, the g-Pfaff-Kummer and Euler transformations, the g-Saalschiitz summation formula, and Sear's transformation for terminating, balanced 4(^3 series. For bilateral series, we rederive Kaneko's analogue of the i^i summation formula, and give multivariable extensions of Bailey's 2^2 transformations.