Summary: Consider a positive Borel measure \(\mu\), supported on an arc \(\Gamma_\alpha= \{e^{i\theta}: 0\leq \alpha\leq \theta\leq 2\pi- \alpha\}\), absolutely continuous with respect to the linear Lebesgue measure, such that \[ \int_\alpha^{2\pi- \alpha} {{\mu' (\theta) d\theta} \over {\sqrt {(\theta- \alpha) (2\pi- \alpha- \theta)}}}< \infty \] and \(\mu'>0\) a.e. on \(\Gamma_\alpha\). Then, if \(\varphi_n (\mu, z)= \kappa_n (\mu) z^n+ \dots\) is the corresponding \(n\)-th orthonormal polynomial with \(\kappa_n (\mu) >0\), we have \(\lim_{n\to \infty} {{\kappa_n (\mu)} \over {\kappa_{n+1} (\mu)}}= \cos (\alpha/2)\).
For the entire collection see [Zbl 0836.00031].