# Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems

## TL;DR

This paper extends elliptic Bailey transforms and lemmas to root systems $A_n$, $C_n$, and $D_n$, introduces new transformation formulas, and generalizes key elliptic hypergeometric summation identities.

## Contribution

It provides the first $A_n$, $C_n$, and $D_n$ extensions of elliptic Bailey transforms and introduces new transformation formulas and series extensions.

## Key findings

- Discovered two new $A_n$ ${}_{12}V_{11}$ transformation formulas.
- Derived two new $A_n$ extensions of Bailey's $_{10}\phi_9$ transformation.
- Extended Frenkel and Turaev's summation formula to multiple series.

## Abstract

We list $A_n$, $C_n$ and $D_n$ extensions of the elliptic WP Bailey transform and lemma, given for $n=1$ by Andrews and Spiridonov. Our work requires multiple series extensions of Frenkel and Turaev's terminating, balanced and very-well-poised ${}_{10}V_9$ elliptic hypergeometric summation formula due to Rosengren, and Rosengren and Schlosser. In our study, we discover two new $A_n$ ${}_{12}V_{11}$ transformation formulas, that reduce to two new $A_n$ extensions of Bailey's $_{10}\phi_9$ transformation formulas when the nome $p$ is $0$, and two multiple series extensions of Frenkel and Turaev's sum.