Differential properties for a class of Sobolev orthogonal polynomials

We study the orthogonal polynomials with respect to a Sobolev inner product of the following type:〈f,g〉s=∫02πf(eiθ)g(eiθ)|Bh(eiθ)|2 dθ2π+1λ∫02πf′(eiθ)g′(eiθ) dθ2π, z=eiθ,where Bh(z) is a complex polynomial of degree h, dθ/2π is the normalized Lebesgue measure and λ is a positive real number. ymptotic behavior in the complex plane, as well as the differential equations satisfied by the orthogonal polynomials are obtained. As an application, two differential problems are solved, one of them is like a Dirichlet boundary value problem.