Bilateral inversions and terminating basic hypergeometric series identities

The q -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 3 ϕ 2 -series, well-poised 8 ϕ 7 -series and bilateral 6 ψ 6 -series. Several terminating well-poised series identities are consequently derived, including the q -Dixon formulae on terminating 3 ψ 3 -series and two terminating well-poised 5 ψ 5 -series identities due to [F.H. Jackson, Certain q -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].