A bibasic Heine transformation formula and Ramanujan's ${}_2{\varphi }_1$ transformations

Abstract

We study Andrews and Berndt's organization of Ramanujan's transformation formulas in Chapter 1 of their book Ramanujan's Lost Notebook, Part II. In the process, we rediscover a bibasic Heine's transformation, which follows from a Fundamental Lemma given by Andrews in 1966, and obtain identities proximal to Ramanujan's entries. We also provide a multibasic generalization of Andrews' 1972 theorem concerning a $q$-analogue of the Lauricella function. Our results only require the $q$-binomial theorem, and are an application of what Andrews and Berndt call 'Heine's Method'.

Accepted for the proceedings of the Alladi 60th birthday conference. In this version, changes based on referee's comments, and some minor grammatical corrections