Asymptotic behavior of orthogonal polynomials corresponding to a measure with infinite discrete part off an arc (Q5954644)

In this interesting paper, the authors study asymptotics of orthogonal polynomials associated with a positive measure \(\sigma\). It is assumed that \(\sigma= \alpha+\gamma\), where \(\alpha\) is a measure on a complex rectifiable arc \(E\), that is ``smooth'' in the sense that it is \(C^2\), and \(\alpha\) is absolutely continuous on \(E\), satisfying the analogue on \(E\) of the Szegő condition. Moreover, \(\gamma\) is a discrete measure, with point masses at points \(\{z_k\}\) outside \(E\).NEWLINENEWLINENEWLINELet \(\{T_n\}\) denote the monic orthogonal polynomials associated with the measure \(\sigma\). Under certain conditions on the location of the \(\{z_k\}\), the authors establish the asymptotic relation NEWLINE\[NEWLINET_n(z)= (C(E))^n \Phi(z)^n[\psi^*(z)+ o(1)],NEWLINE\]NEWLINE where \(C(E)\) is the logarithmic capacity of \(E\), and \(\Phi\) is the conformal map of the exterior of \(E\) onto the exterior of the unit ball. Here \(\psi^*\) is an explicitly given function. The authors also present \(L_2\) asymptotics on the arc \(E\), and asymptotics for the leading coefficients of the orthonormal polynomials.NEWLINENEWLINENEWLINEThe paper is a useful extension of work of V. Kaliaguine, and will be of interest to orthogonal polynomial enthusiasts.