Invariance groups of three term transformations for basic hypergeometric series

The study of invariance groups associated with two term transformations between (basic) hypergeometric series has received its fair share of attention, and indeed, for most two term transformations between (basic) hypergeometric series, the underlying invariance group is explicitly known. In this article, we study the group structure underlying some three term transformation formulae, thereby giving an explicit and simple realization that is helpful in determining whether two of these transformation formulae are equivalent or not.