Some \(q\)-series identities related to the \(q\)-triplicate inverse (Q635286)

For the open question presented by \textit{W. Chu} [Rocky Mt. J. Math. 32, No. 2, 561--587 (2002; Zbl 1038.33002)], \textit{Y. Zhang} and \textit{W. Chen} [Rocky Mt. J. Math. 35, 1407--1427 (2005; Zbl 1087.33008)] established and proved the \(q\)-triplicate inversion formula with the help of \(q\)-finite difference method. In the present paper, by applying the \((f, g)\)-inversion series relations due to \textit{X. Ma} [Adv. Appl. Math. 38, No. 2, 227--257 (2007; Zbl 1130.33008)], the authors gives a simple proof of the \(q\)-triplicate inversion formula. Then by applying the \(q\)-triplicate inversion formula to Watson's transformation formula on \(_{8}\Phi_{7}\) series, the authors also derives several \(q\)-series identities. The paper is a good contribution to the further development of \(q\)-series identities and may find applications in number theory and modular equations.