Distributional equation for Laguerre–Hahn functionals on the unit circle

摘要

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, zAF′=BF2+CF+D, and a distributional equation for u, D(Au)=B̃u2+C̃u+H̃L, where L is the Lebesgue functional, and the polynomials B̃,C̃,H̃ are defined in terms of the polynomials A,B,C,D.