A bibasic Heine transformation formula and Ramanujan's $_2\phi_1$ transformations

TL;DR

This paper explores Ramanujan's transformation formulas, rediscovering a bibasic Heine transformation and extending Andrews' theorem to a multibasic setting, using only the q-binomial theorem.

Contribution

It introduces a bibasic Heine transformation formula and a multibasic generalization of Andrews' theorem, expanding the understanding of q-analogues of classical functions.

Findings

Rediscovery of a bibasic Heine transformation from fundamental lemmas.

Derivation of identities related to Ramanujan's transformation formulas.

Generalization of Andrews' 1972 theorem to multibasic q-analogues.

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'.