Symmetry groups of Bailey's transformations for 10φ9-series

Although most of the symmetry groups or “invariance groups” associated with two term transformations between (basic) hypergeometric series have been studied and identified, this is not the case for the most general transformation formulae in the theory of basic hypergeometric series, namely Bailey's transformations for φ910-series. First, we show that the invariance group for both Bailey's two term transformations for terminating φ910-series and Bailey's four term transformations for non-terminating φ910-series (rewritten as a two term transformation of a so-called Φ-series) is isomorphic to the Weyl group of type E6. We continue our recent research concerning the group structure underlying three term transformations [S. Lievens, J. Van der Jeugt, Invariance groups of three term transformations for basic hypergeometric series, J. Comput. Appl. Math. 197 (2006) 1-14] and demonstrate that the group associated with a three term transformation between these Φ-series, each admitting Bailey's two term transformation, is the Weyl group of type E7. We do this by giving a description of the root system of type E7 that allows to find a transformation between equivalent three term identities in an easy way. A computation shows that there are five, essentially different, three term transformations between these Φ-series; we give an explicit form of each of these five transformations in an elegant way. To our knowledge only one of these transformations has appeared in the literature.