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arXiv · 2607.00231

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

Roozbeh Gharakhloo·Nicholas S. Witte
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Abstract

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.

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Roozbeh Gharakhloo, Nicholas S. Witte. 2026-06-30. The $2j-k$ and $j-2k$ Bi-orthogonal Polynomials on the Unit Circle: Further Properties and Riemann-Hilbert Characterizations. https://arxiv.org/abs/2607.00231

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