On the order of reflection coefficients for generalized Jacobi weights (Q1945245)

Denote by \(\varphi_n(W,z)\), shortly \(\varphi_n(z)\), the monic orthogonal polynomials associated with the weight function \(W(\theta)\) which, under some conditions (known in the literature) allows that there exists a unique system of orthogonal polynomials on the unit circle with respect to \(w\). In the present paper the authors consider trigonometric weight functions and investigate the reflection coefficients \[ \varphi_n(0) : = \varphi_n(W,0). \] The trigonometric weight \(W(\theta)\) is a generalized trigonometric Jacobi weight (\(W \in\) GTJ). The authors develop results of their study on the proposed theme, the reflection coefficients \[ \varphi_n(0) : = \varphi_n(W,0), \] corresponding to the generalized trigonometric Jacobi weights, \(W\in \) GTJ. After recalling the definition of \(W(\theta) \in \) GTJ (given, e.g., by \textit{G. Mastroianni} and \textit{P. Vértesi} [Acta Math. Hung. 77, No.~4, 323--357 (1997; Zbl 0913.42018)]), the authors are able to state the main result of their paper: \[ \limsup_{n\to +\infty} n| \varphi_n(W,0)| < +\infty. \]