Transformation formulae 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 ${}_2\phi_1$ transformation, the $q$-Pfaff-Kummer and Euler transformations, the $q$-Saalsch\"utz summation formula and Sear's transformation for terminating, balanced ${}_4\phi_3$ series. For bilateral series, we rederive Kaneko's analogue of the ${}_1\psi_1$ summation formula and give multivariable extensions of Bailey's ${}_2\psi_2$ transformations.