On second kind polynomials associated with rational transformations of linear functionals

In this paper the following construction process of orthogonal polynomials on the unit circle is considered: Let u be a regular and hermitian linear functional and let { Φ n } be the corresponding orthogonal polynomials sequence. We define a new linear functional L by means the following relation with u: λ ( z − β ) L = ( z − α ) u , α , β , λ ∈ C , λ ≠ 0 . In this situation we obtain conditions for the regularity of L , as well as the corresponding orthogonal polynomials sequence. Also, we give one explicit representation for the orthogonal polynomials sequence of the second kind associated to L . For the particular case when α = β , L becomes in the well-known modification of u by addition of a Dirac mass. This case will be studied with special attention.