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We construct the systems of bi-orthogonal polynomials on the unit circlewhere the Toeplitz structure of the moment determinants is replaced by $\det(w_{2j-k})_{0\leq j,k \leq N-1} $ and the corresponding Vandermonde modulussquared is replaced by $ \prod_{1 \le j < k \le N}(\zeta^{2}_k -\zeta^{2}_j)(\zeta^{-1}_k - \zeta^{-1}_j) $. This is the simplest case of ageneral system of $pj-qk$ with $p,q$ co-prime integers. We derive analogues ofthe structures well known in the Toeplitz case: third order recurrencerelations, determinantal and multiple-integral representations, theirreproducing kernel and Christoffel-Darboux sum, and associated(Carath{\'e}odory) functions. We close by giving full explicit details for thesystem defined by the simple weight $ w(\zeta)=e^{\zeta}$, which is aspecialisation of a weight arising from averages of moments of derivatives ofcharacteristic polynomials over $USp(2N)$, $SO(2N)$ and $O^-(2N)$.

Modulated Bi-orthogonal Polynomials on the Unit Circle: The $2j-k$ and $j-2k$ Systems