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})_{0le j,k le N-1} and the corresponding Vandermonde modulus squared is replaced by prod _{1 le j < k le N}(zeta _k - zeta _j)(zeta ^{-2}_k - zeta ^{-2}_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 (Carathéodory) 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 textrm{USp}(2N), textrm{SO}(2N) and textrm{O}^-(2N).