  We give a complete characterization of the positive trigonometric polynomials
Q(\theta,\phi) on the bi-circle, which can be factored as
Q(\theta,\phi)=|p(e^{i\theta},e^{i\phi})|^2 where p(z,w) is a polynomial
nonzero for |z|=1 and |w|\leq 1. The conditions are in terms of recurrence
coefficients associated with the polynomials in lexicographical and reverse
lexicographical ordering orthogonal with respect to the weight
1/(4\pi^2Q(\theta,\phi)) on the bi-circle. We use this result to describe how
specific factorizations of weights on the bi-circle can be translated into
identities relating the recurrence coefficients for the corresponding
polynomials and vice versa. In particular, we characterize the Borel measures
on the bi-circle for which the coefficients multiplying the reverse polynomials
associated with the two operators: multiplication by z in lexicographical
ordering and multiplication by w in reverse lexicographical ordering vanish
after a particular point. This can be considered as a spectral type result
analogous to the characterization of the Bernstein-Szeg\H{o} measures on the
unit circle.
