Distributional equation for Laguerre–Hahn functionals on the unit circle

Let u be a Hermitian linear functional defined in the linear space of Laurent polynomials and F its corresponding Carathéodory function. We establish the equivalence between a Riccati differential equation with polynomial coefficients for F , z A F ′ = B F 2 + C F + D , and a distributional equation for u , D ( A u ) = B ̃ u 2 + C ̃ u + H ̃ L , where L is the Lebesgue functional, and the polynomials B ̃ , C ̃ , H ̃ are defined in terms of the polynomials A , B , C , D .