While studying chapter 1 of Andrews and Berndt’s Lost Notebook, Part II, I stumbled upon a bibasic Heine’s transformation. A special case is Heine’s 1847 transformation. Other special cases include an identity of Ramanujan (c. 1919), and  a 1966 transformation formula of Andrews. Eventually, I realized that it follows from a Fundamental Lemma given by Andrews in 1966. Still, I’m happy to have rediscovered it. Using this formula one can find many identities proximal to Ramanujan’s own $_2\phi_1$ transformations.

And of course, the multiple series extensions (some in this paper, and others appearing in another paper) are all new.

Here is a preprint.

Here is a video of a talk I presented at the Alladi 60 Conference. March 17-21, 2016.

Update (November 10, 2018). The multi-variable version has been accepted for publication in the Ramanujan Journal. This has been made open access. It is now available online, even though the volume and page number has not been decided yet. The title is: Heine’s method and $A_n$ to $A_m$ transformation formulas.

UPDATE (Feb 11, 2016). This has been published. Reference (perhaps to be modified later): A bibasic Heine transformation formula and Ramanujan’s $_2\phi_1$ transformations, in Analytic Number Theory, Modular Forms and q-Hypergeometric Series, In honor of Krishna Alladi’s 60th Birthday, University of Florida, Gainesville, Mar 2016,  G. E. Andrews and F. G. Garvan (eds.), 99-122 (2017)

The book is available here. The front matter from the Springer site.

UPDATE (June 16, 2016).  The paper has been accepted to appear in: Proceedings of the Alladi 60 conference held in Gainesville, FL. (Mar 2016), K. Alladi, G. E. Andrews and F. G. Garvan (eds.)