Some transformations on the bilateral series \(_2\psi_2\) (Q485893)

The authors study the bilateral series \({}_2\psi_2\) and derive some transformation formulas in terms of unilateral basic hypergeometric series. First of all, they derive a transformation formula for a certain bilateral \({}_2\psi_2\) series as a linear combination of two unilateral \({}_2\phi_1\) series. This formula turns out to be a generalization of some known and new \(q\)-series identities, such as Ramanujan's \({}_1\psi_1\) summation formula and a certain \({}_2\psi_2\) summation formula. Secondly, the authors derive a transformation formula for a certain bilateral \({}_2\psi_2\) series as a linear combination of three unilateral \({}_2\phi_1\) series. This turns out to be a generalization of Bailey's \({}_2\psi_2\) summation formula for instance. Finally, they discuss some transformation formulas between bilateral \({}_2\psi_2\) and unilateral \({}_3\phi_2\) series. One of these generalizes the Bailey-Daum summation formula and a special case of Ramanujan's \({}_1\psi_1\) summation formula for instance. Furthermore, the authors refer to the paper \textit{Z. Zhang} and \textit{C. Zhang} [Util. Math. 92, 365--375 (2013; Zbl 1296.33030)] for another transformation formula between \({}_2\psi_2\) and \({}_3\phi_2\) series.