Differential properties for Sobolev orthogonality on the unit circle

The aim of this paper is to study differential properties of the sequence of monic orthogonal polynomials with respect to the following Sobolev inner product:〈f,g〉s=∫02πf(eiθ)g(eiθ) dμ(θ)+1λ∫02πf′(eiθ)g′(eiθ) dθ2π,where μ is a finite positive Borel measure on [0,2π] verifying the following conditions: the Carathéodory function associated with μ has an analytic extension outside the unit disk and the induced norm is equivalent to the Lebesgue norm in the space L2. Here dθ/2π is the normalized Lebesgue measure and λ is a positive real number. The nonhomogeneous second-order differential equations satisfied by the sequence of monic Sobolev orthogonal polynomials are obtained. Moreover, as an application, a sample of Dirichlet boundary value problem is solved.