Transformations and Summations for Bilateral Basic Hypergeometric Series

We review and derive transformation and summation formulas for bilateral basic hypergeometric series. Our study focuses on consequences of certain bilateral extensions of two important results by Bailey, namely a transformation for very-well-poised $$_8\!W_7$$ 8 W 7 series in terms of two balanced $$_4\phi _3$$ 4 ϕ 3 series, and a transformation connecting three $$_8\!W_7$$ 8 W 7 series. Two rather recently discovered transformations of bilateral basic very-well-poised $${}_8\!\Psi _8$$ 8 Ψ 8 , one by Zhang and Zhang, the other by Wei and Yu, serve as the starting point of our investigations. From these transformations we work out interesting special cases that were not considered before, including explicit bilateral quadratic and cubic summations. We further explicitly record noteworthy lower-level transformations derived by taking suitable limits and deduce more transformations by exploiting the symmetry of the parameters in the series.