Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle (Q2790189)

The author deals with sets of orthogonal polynomials with respect to the Sobolev inner products \(\langle f, g \rangle_{S^{(b,s,t)}}\) defined on the unit circle as follows, for \(0\leq t <1\), \(s\geq 0\) and \(\mathcal{R}e(b) > - \frac{1}{2}\), NEWLINE\[NEWLINE\langle f, g \rangle_{S^{(b,s,t)}} = \langle f, g \rangle_{\mu^{(b,t)}} +s\langle f^{\prime} , g^{\prime} \rangle_{\mu^{(b+1)}} , \;\; \text{with}NEWLINE\]NEWLINE NEWLINE\[NEWLINE \langle f, g \rangle_{\mu^{(b,t)}} = (1-t) \langle f, g \rangle_{\mu^{(b)}} + t \overline{f(1)} g(1) ,NEWLINE\]NEWLINE NEWLINE\[NEWLINE \langle f, g \rangle_{\mu^{(b)}} = \frac{\tau(b)}{2 \pi} \int_{0}^{2\pi} \overline{f\left( e^{i \theta} \right)} g \left( e^{i \theta}\right) \left( e^{\pi-\theta} \right)^{\mathcal{I}m(b)} \left( \sin^{2}\left( \theta/2 \right) \right)^{\mathcal{R}e(b)}d \theta, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \text{and}\;\; \tau(b) = \frac{2^{b+\overline{b}}\,|\Gamma\left( b+1\right)|^2}{\Gamma\left( b+\overline{b}+1\right)}.NEWLINE\]NEWLINE The sets of monic polynomials orthogonal with regard to the inner products \(\langle f, g \rangle_{S^{(b,s,t)}} \) and \( \langle f, g \rangle_{\mu^{(b,t)}} \) are denoted by \(\left\{ \Psi_{n}^{(b,s,t)} \right\}\) and \(\left\{ \Phi_{n}^{(b,t)} \right\}\), respectively. Recent contributions, like for example [\textit{K. Castillo} et al., J. Approx. Theory 184, 146--162 (2014; Zbl 1291.42021)] and [\textit{M. S. Costa} et al., J. Approx. Theory 173, 14--32 (2013; Zbl 1282.33017)], help to set up the background from which the author establishes recurrence identities connecting \(\left\{ \Psi_{n}^{(b,s,t)} \right\}\) and \(\left\{ \Phi_{n}^{(b,t)} \right\}\). The asymptotic behavior of the corresponding recurrence coefficients is also analysed.