Distributional equation for Laguerre–Hahn functionals on the unit circle

Let uu be a Hermitian linear functional defined in the linear space of Laurent polynomials and FF its corresponding Carathéodory function. We establish the equivalence between a Riccati differential equation with polynomial coefficients for FF, zAF′=BF2+CF+DzAF′=BF2+CF+D, and a distributional equation for uu, D(Au)=B̃u2+C̃u+H̃L, where LL is the Lebesgue functional, and the polynomials B̃,C̃,H̃ are defined in terms of the polynomials A,B,C,DA,B,C,D.