Transformations and summations for bilateral basic hypergeometric series

Abstract: We derive transformation and summation formulas for bilateral basic hypergeometric series. As a starting point, we use two transformations of bilateral basic very-well-poised ${}_8\Psi_8$. The first transformation is given as a sum of two nonterminating ${}_8W_7$'s and the second is given in terms of a sum of a ${}_4\psi_4$ and two balanced ${}_4\phi_3$'s. From these transformations we derive limiting transformations with vanishing denominator elements which shed light on the transformation properties of these bilateral basic hypergeometric series. We also study tuple product identities, namely triple, quintuple, sextuple, septuple, octuple, nonuple and undecuple, which are given in terms of sums of bilateral basic hypergeometric series.