Elementary Derivations of Identities for Bilateral Basic Hypergeometric Series

We give elementary derivations of several classical and some new summation and transformation formulae for bilateral basic hypergeometric series. For purpose of motivation, we review our previous simple proof (“A simple proof of Bailey’s very-well-poised 6ψ6 summation”, Proc. Amer. Math. Soc., to appear) of Bailey’s very-well-poised 6ψ6 summation. Using a similar but different method, we now give elementary derivations of some transformations for bilateral basic hypergeometric series. In particular, these include M. Jackson’s very-well-poised 8ψ8 transformation, a very-well-poised 10ψ10 transformation, by induction, Slater’s general transformation for very-well-poised 2rψ2r series, and Slater’s transformation for general rψr series. Finally, we derive some new transformations for bilateral basic hypergeometric series of Chu–Gasper– Karlsson–Minton-type.