Heine’s method and [Formula: see text] to [Formula: see text] transformation formulas

We apply Heine’s method—the key idea Heine used in 1846 to derive his famous transformation formula for [Formula: see text] series—to multiple basic series over the root system of type A. In the classical case, this leads to a bibasic extension of Heine’s formula, which was implicit in a paper of Andrews which he wrote in 1966. As special cases, we recover extensions of many of Ramanujan’s [Formula: see text] transformations. In addition, we extend previous work of the author regarding a bibasic extension of Andrews’ q-Lauricella function, and show how to obtain very general transformation formulas of this type. The results obtained include transformations of an n-fold sum into an m-fold sum.