Bilateral inversions and terminating basic hypergeometric series identities

The qq-analogue of Legendre inversions is established and generalized to bilateral sequences. They are employed to investigate the dual relations of three basic formulae due to Jackson and Bailey, on balanced 3ϕ23ϕ2-series, well-poised 8ϕ78ϕ7-series and bilateral 6ψ66ψ6-series. Several terminating well-poised series identities are consequently derived, including the qq-Dixon formulae on terminating 3ψ33ψ3-series and two terminating well-poised 5ψ55ψ5-series identities due to [F.H. Jackson, Certain qq-identities, Quart. J. Math. (Oxford) 12 (1941) 167–172; W.N. Bailey, On the analogue of Dixon’s theorem for bilateral basic hypergeometric series, Quart. J. Math. (Oxford) 1 (1950) 318–320].