A New Transformation Formula for 2ψ2

Many bilateral basic hypergeometric series can be derived from unilateral basic hypergeometric series, in this paper, we study the nonterminating unilateral basic hypergeometric series shifting into bilateral basic hypergeometric series. Firstly,using the bilateral extension method, we derive a new transformation formula between bilateral basic hypergeometric series 2ψ2 and unilateral basic hypergeometric series 21 through the nonterminating form of the q-Saalschüz formula. Secondly, we also deduce another transformation formula between bilateral basic hypergeometric series 2ψ2 and unilateral basic hypergeometric series 21 by using the Heine’s 21 transformation formula to the 21 series in theory 1. Finally, we give a new proof of the nonterminating extension of the q-Saalschüz formula by taking special value f=q in theory 1.