Relative asymptotics for general orthogonal polynomials (Q2396621)

The paper under review concerns some asymptotic properties of orthogonal polynomials with respect to probability measures in the complex plane. Given such a measure \(\sigma\) with compact and infinite support, the author starts out with the sequences of orthonormal and monic orthogonal polynomials \(p_n(z,\sigma)=\kappa_n(\sigma)z^n+\ldots\), \(P_n(z,\sigma)=\kappa_n^{-1}(\sigma)p_n(z,\sigma)=z^n+\ldots\), respectively, and defines a number \(R_\sigma=\max\{|z|:z\in \text{supp}\,\sigma\}\). Two such measures, \(\mu\) and \(\nu\), are said to exhibit relative ratio asymptotics if there is a nonnegative integer \(q\) so that \[ \lim_{n\to\infty}\left(\frac{P_{n-1}(z,\mu)}{P_{n}(z,\mu)}-\frac{P_{n-q-1}(z,\nu)}{P_{n-q}(z,\nu)}\right)=0, \quad |z|>\max\{R_\mu, R_\nu\}. \] To characterize this property the author invokes the Bergman shift matrix \(M_\sigma=\|(M_\sigma)_{ij}\|\), which is a matrix representation of the multiplication operator with respect to the basis \(\{p_n(\cdot,\sigma)\}_{n\geq0}\), on the linear span of polynomials in \(L^2_\sigma\). Theorem. Two measures \(\mu\) and \(\nu\) exhibit relative ratio asymptotics if and only if for each nonnegative integer \(j\) \[ \lim_{n\to\infty}\left(\frac{\kappa_{n-1-j}(\mu)}{\kappa_{n-1}(\mu)}\,(M_\mu)_{n-j,n} - \frac{\kappa_{n-q-1-j}(\nu)}{\kappa_{n-q-1}(\nu)}\,(M_\nu)_{n-q-j,n-q}\right)=0. \]