The $2j-k$ and $j-2k$ Bi-orthogonal Polynomials on the Unit Circle: Further Properties and Riemann-Hilbert Characterizations

In previous work \cite{GW}, we developed a theory of modulated $2j-k$ bi-orthogonal polynomial systems $\{P_n(z;r),Q_n(z;r)\}$ and $j-2k$ bi-orthogonal polynomial systems $\{R_n(z;r),S_n(z;r)\}$, which generalize the classical $j-k$ Toeplitz systems. In the present paper, we further develop this theory in several directions. We derive simplified and unified recurrence relations for both families of polynomials, prove a more transparent Christoffel–Darboux formula, and give Riemann–Hilbert characterizations of the $2j-k$ and $j-2k$ systems.