Distributional equation for Laguerre-Hahn functionals on the unit circle

Abstract

Let u be a hermitian linear functional defined in the linear space of Laurent polynomials and F its corresponding Carath´eodory 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) = B1u2+ C1u+H1L, where L is the Lebesgue functional, and the polynomials B1,C1,D1 are defined in terms of the polynomials A,B,C,D