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

Abstract

AbstractWe construct the systems of bi-orthogonal polynomials on the unit circle where the Toeplitz structure of the moment determinants is replaced by $$\det (w_{2j-k})_{0\le j,k \le N-1} $$ det ( w 2 j - k ) 0 ≤ j , k ≤ N - 1 and the corresponding Vandermonde modulus squared is replaced by $$\prod _{1 \le j < k \le N}(\zeta _k - \zeta _j)(\zeta ^{-2}_k - \zeta ^{-2}_j) $$ ∏ 1 ≤ j < k ≤ N ( ζ k - ζ j ) ( ζ k - 2 - ζ j - 2 ) . This is the simplest case of a general system of $$pj-qk$$ p j - q k 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 $$ w(\zeta )=e^{\zeta }$$ w ( ζ ) = e ζ , which is a specialisation of a weight arising from averages of moments of derivatives of characteristic polynomials over $$\textrm{USp}(2N)$$ USp ( 2 N ) , $$\textrm{SO}(2N)$$ SO ( 2 N ) and $$\textrm{O}^-(2N)$$ O - ( 2 N ) .