Heine's method and $A_n$ to $A_m$ transformation formulas

TL;DR

This paper extends Heine's classical transformation method to multiple basic series over root systems of type A, leading to new bibasic and multivariate transformation formulas that generalize Ramanujan's and Andrews' results.

Contribution

It introduces a novel application of Heine's method to multiple series over type A root systems, deriving general transformation formulas including bibasic and multivariate cases.

Findings

Derived bibasic extension of Heine's formula.

Recovered extensions of Ramanujan's $_2\phi_1$ transformations.

Established transformations between n-fold and m-fold sums.

Abstract

We apply Heine's method---the key idea Heine used in 1846 to derive his famous transformation formula for $_2\phi_1$ 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 $_2\phi_1$ 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.