Orthogonal basic hypergeometric Laurent polynomials

The Askey-Wilson polynomials are orthogonal polynomials in x = cos θ, which are given as a terminating 4φ3 basic hypergeometric series. The non-symmetric Askey- Wilson polynomials are Laurent polynomials in z = e^iθ, which are given as a sum of two terminating 4φ3's. They satisfy a biorthogonality relation. In this paper new orthogonality relations for single 4φ3's which are Laurent polynomials in z are given, which imply the non-symmetric Askey-Wilson biorthogonality. These results include discrete orthogonality relations. They can be considered as a classical analytic study of the results for non-symmetric Askey-Wilson polynomials which were previously obtained by affine Hecke algebra techniques.