### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 025, 44 pages      arXiv:1704.00020      https://doi.org/10.3842/SIGMA.2018.025
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

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

Gaurav Bhatnagar and Michael J. Schlosser
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

Received September 01, 2017, in final form March 13, 2018; Published online March 22, 2018

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.

Key words: $A_n$ elliptic and basic hypergeometric series; elliptic and basic hypergeometric series on root systems; well-poised Bailey transform and lemma.

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