Analogs of the \(m\)-function in the theory of orthogonal polynomials on the unit circle
A key leit-motif which passes through the whole paper is a variety of different functions in the theory of orthogonal polynomials on the unit circle (OPUC) growing from the multitude of applications of the Weyl-Titchmarsh \(m\)-functions associated to the Schrödinger operators on the halfline and infinite Jacobi matrices. For instance, the integral representation \[ m(z,J)=\int \frac{d\mu(t)}{t-z} \] leads to the Carathéodory function \[ F(z)=\int \frac{e^{i\theta}+z}{e^{i\theta-z}}\,d\mu(\theta) \] as a natural ``\(m\)-function'' for OPUC. Yet another reasonable choice for an \(m\)-function for OPUC is the Schur function \(f\) with the Schur parameters \(\{\gamma_0, \gamma_1,\ldots\}\) due to the linear fractional relation between \(f\) and \(f_1\) with the shifted parameters \(\{\gamma_1, \gamma_2,\ldots\}\). The latter is close to the relation between \(m(z,J)\) and \(m(z,J^{(1)})\), where \(J^{(1)}\) is the Jacobi matrix with ``shifted'' entries.
orthogonal polynomials on the unit circle
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