Recursion relation for orthogonal polynomials on the complex plane

Abstract

The article deals with orthogonal polynomials on compact infinite subsets of the complex plane. Orthogonal polynomials are treated as coordinates of generalized eigenvector of a normal operator $A$. It is shown that there exists a recursion that gives the possibility to reconstruct these polynomials. This recursion arises from generalized eigenvalue problem and, actually, this means that every gene alized eigenvector of $A$ is also a generalized eigenvector of $A^*$ with the complex conjugated eigenvalue. If the subset is actually the unit circle, it is shown that the presented algorithm is a generalization of the well-known Szego recursion from OPUC theory.