Asymptotic Properties of Orthogonal Polynomials with Respect to a Non-discrete Jacobi-Sobolev Inner Product

Let {Qn(α,β)(x)}n=0∞ denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)d\mu_{\alpha,\beta}(x)+\lambda\int_{-1}^{1}f'(x)g'(x)d\nu_{\alpha,\beta}(x)$$\end{document} where λ>0 and dμα,β(x)=(x−a)(1−x)α−1(1+x)β−1dx, dνα,β(x)=(1−x)α(1+x)βdx with a<−1, α,β>0. Their inner strong asymptotics on (−1,1), a Mehler-Heine type formula as well as some estimates of the Sobolev norms of Qn(α,β) are obtained.