Expansion formulas for elliptic hypergeometric series
Gaurav Bhatnagar, Archna Kumari
TL;DR
The article develops a matrix-analytic framework for elliptic hypergeometric series by modifying the well-poised Bailey lemma through inverse matrix pairs ${\mathbf F}$ and ${\mathbf G}$ and a central matrix ${\mathbf M}$ with entries $M_{km}(a,b,q,p)$. Leveraging a finite $q$-Lagrange inversion lemma, this approach produces four expansion formulas parameterized by an arbitrary sequence ${\alpha_j}$, which, when combined with established summation theorems, yield 24 transformation formulas—eight of which are new in the elliptic setting. In the limit $p\to 0$, the method also yields new basic hypergeometric (bibasic) transformations, enriching known Bailey-type identities. Overall, the paper provides a productive pipeline for generating elliptic hypergeometric identities via matrix inversions and summations, with potential extensions to root systems and related algebraic structures.
Abstract
We provide an alternate approach to obtaining expansion formulas on the lines of the well-poised Bailey lemma. We recover results due to Spiridonov and Warnaar and one new formula of this type. These formulas contain an arbitrary sequence as an argument, and are thus flexible in the number of parameters they contain. As a result, we are able to derive $19$ new transformation formulas for elliptic hypergeometric series. These transformation formulas appear to be new even in the basic hypergeometric case, when $p=0$.