On differential equations for orthogonal polynomials on the unit circle
In the paper under review, the authors study orthogonal polynomials on the unit circle whose Carathéodory function \(F\) satisfies the differential equation \(zAF'=BF^2+CF+D\) with polynomial coefficients and \(A\neq 0\). The set of such Carathéodory functions is well-known also as the Laguerre-Hahn class on the unit circle. In the Theorems 1 and 2 the authors give a characterization of the above mentioned class. The formulation is rather long and they split the proofs in some lemmas and corollaries. In the last section, they investigate the semi-classical class on the unit circle and its characterization in terms of second order linear differential equations. More precisely, they consider Carathéodory functions associated to a measure \(\mu\) of the type \(d\mu=wd\theta+\sum_{k=1}^N\lambda_k\delta_{z_k}\), \((\lambda_k\geq 0\), \(N\in\mathbb{N})\), where \(w\) is the absolutely continuous part with respect to the Lebesgue measure \(d\theta\), and \(\delta_{z_k}\) is the Dirac measure at \(z_k\) with \(z_k\) on the unit circle for \(k=1,\dots,N\). The authors give the result of this section in Theorem 3, and they use some known lemmas to prove it.
orthogonal polynomials on the unit circle
scientific article; zbMATH DE number 5563865
scientific article