A note on the bilateral series \(_{2}\psi_{2}\) (Q2870973)

The first author and \textit{Q. Hu} have proved in [Util. Math. 77, 277--285 (2008; Zbl 1160.05004)] several tansformation formulas between the basic hypergeometric series \({_3\phi_2}\) and \({_2\phi_1}\). In the present paper, the authors give a new relation between the bilateral hypergeometric series \({_2\psi_2}\) and the hypergeometric series \({_3\phi_2}\). The method of proof is based on an extension of \textit{M. Jackson's} method from [J. Lond. Math. Soc. 25, 189--196 (1950; Zbl 0036.32601) and Q. J. Math., Oxf. II. Ser. 1, 63--68 (1950; Zbl 0035.16703)].