Symmetry groups of Baileyʹs transformations for 109-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 Baileyʹs transformations for φ 9 10 -series. First, we show that the invariance group for both Baileyʹs two term transformations for terminating φ 9 10 -series and Baileyʹs four term transformations for non-terminating φ 9 10 -series (rewritten as a two term transformation of a so-called Φ -series) is isomorphic to the Weyl group of type E 6 . 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 Baileyʹs 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. 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.