Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems

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 ${}_{10}{\varphi }_9$ transformation formulas when the nome $p$ is $0$, and two multiple series extensions of Frenkel and Turaev’s sum.