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

TL;DR

This paper introduces and analyzes bi-orthogonal polynomial systems on the unit circle based on $2j-k$ and $j-2k$ structures, extending classical Toeplitz theory to new algebraic configurations with applications in random matrix theory.

Contribution

It constructs the first explicit bi-orthogonal polynomial systems for the $pj-qk$ class, deriving their fundamental properties and connecting them to random matrix averages.

Findings

Derived third order recurrence relations for the new systems.

Established determinantal and integral representations.

Connected the systems to averages of characteristic polynomial derivatives.

Abstract

We construct the systems of bi-orthogonal polynomials on the unit circle where 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 modulus squared 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 a general system of $pj-qk$ with $p,q$ co-prime integers. We derive analogues of the structures well known in the Toeplitz case: third order recurrence relations, determinantal and multiple-integral representations, their reproducing kernel and Christoffel-Darboux sum, and associated (Carath{\'e}odory) functions. We close by giving full explicit details for the system defined by the simple weight $ w(\zeta)=e^{\zeta}$, which is a specialisation of a weight arising from averages of moments of derivatives of characteristic polynomials over $USp(2N)$, $SO(2N)$ and $O^-(2N)$.