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

Abstract

We construct the systems of bi-orthogonal polynomials on the unit circle where the Toeplitz structure of the moment determinants is replaced by $\det (w_{2j-k}{)}_{0\le j,k\le N-1}$ and the corresponding Vandermonde modulus squared is replaced by ${\prod }_{1\le j<k\le N}({\zeta }_k^2-{\zeta }_j^2)({\zeta }_k^{-1}-{\zeta }_j^{-1})$. 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{é}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 $O^{-}(2N)$.

47 pages