\((1,1)\)-coherent pairs on the unit circle
From the abstract: ``A pair \((\mathcal{U}, \mathcal{V})\) of Hermitian regular linear functionals on the unit circle is said to be a \((1,1)\)-coherent pair if their corresponding sequences of orthogonal polynomials, \(\{\phi_n(x)\}_{n \geq 0}\) and \(\{\psi_n(x)\}_{n \geq 0}\) satisfy \(\phi_n^{[1]}(z)+a_n \phi_{n-1}^{[1]}(z)=\psi_n(z)+b_n\psi_{n-1}(z)\), \(a_n \geq 0\), \(n \geq 1\), where \(\phi_n(z)^{[1]}=\phi_{n+1}'(z)/(n+1)\).'' The authors consider \((1,1)\) coherent pairs where \(\mathcal{U}\) is the linear functional associated with the Lebesgue measure \(d\theta/2\pi\), and when \(\mathcal{U}\) is the Bernstein-Szegő linear functional associated with the measure \(((1-|C|^2)/|1+Ce^{i\theta}|^2)(d\theta/2\pi)\). In both cases, the authors derive formulae for \(\psi_n\), \(n \geq 1\), as well as the sequence of moments for the functional \(\mathcal{V}\). The authors also consider the measures corresponding to the linear functional \(\mathcal{V}\) in each case.
scientific article; zbMATH DE number 6306641
scientific article