Orthogonal basic hypergeometric Laurent polynomials

The Askey-Wilson polynomials are orthogonal polynomials in x = \cos \theta, which are given as a terminating 4\phi_3 basic hypergeometric series. The non-symmetric Askey-Wilson polynomials are Laurent polynomials in z=e^{i\theta}, which are given as a sum of two terminating _4\phi_3's. They satisfy a biorthogonality relation. In this paper we give new orthogonality relations for single 4\phi_3's which are Laurent polynomials in z, which imply the non-symmetric Askey-Wilson biorthogonality. These results can be considered as a classical analytic study of the results for non-symmetric Askey-Wilson polynomials which are previously obtained by affine Hecke algebra techniques.