# Symmetry Groups Underlying Bailey's Transformations for ${}_{10}\phi_9$-series

Author: Stijn Lievens (Ghent University)

### Abstract

The concept of symmetry groups associated with two term transformations for basic hypergeometric series is well known, and most of them have been studied and identified (J. Math. Phys. 1999:6692+ and references therein). One two term identity for which the invariance group, to our knowledge, was not written down explicitly is Bailey's four term transformation for non-terminating ${}_{10}\phi_9$-series considered as a two term transformation between a linear combination of such series which we call $\Phi$. It is shown that the invariance group of this transformation is the Weyl group of type $E_6$.

We demonstrate that the group associated with a three term transformation between $\Phi$-series, each admitting Bailey's two term transformation, is the Weyl group of type $E_7$. We do this by giving a description of the root system of type $E_7$ that allows to find a transformation between equivalent three term identities in an easy way. We also show how one can find a prototype of each of the five essentially different three term identities between $\Phi$-series.