Heine's method and to transformation formulas.

We apply Heine's method-the key idea Heine used in 1846 to derive his famous transformation formula for series-to multiple basic series over the root system of type . 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 transformations. In addition, we extend previous work of the author regarding a bibasic extension of Andrews' -Lauricella function, and show how to obtain very general transformation formulas of this type. The results obtained include transformations of an -fold sum into an -fold sum.