Elementary derivations of identities for bilateral basic hypergeometric series

Abstract.

We give elementary derivations of several classical and some new summation and transformation formulae for bilateral basic hypergeometric series. For motivation, we review our previous simple proof (Proc. Amer. Math. Soc. 130 (2002), 1103-1111) of Bailey's very-well-poised $_6\psi_6$ 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 $_8\psi_8$ transformation, a very-well-poised $_{10}\psi_{10}$ transformation, by induction, Slater's general transformation for very-well-poised $_{2r}\psi_{2r}$ series, and Slater's transformation for general $_{r}\psi_{r}$ series. Finally, we derive some new transformations for bilateral basic hypergeometric series of a specific type.