Zero asymptotics of Laurent orthogonal polynomials (Q1915583)

The two-point Padé approximants \(P_n(z)/ h_n(z)\) for a Stieltjes function \(\widehat{\rho}(z)= \int^\infty_0 (t-z)^{-1} d\rho(t)\) are investigated, with \(\lambda_n\) interpolation conditions at the origin and \(2n-\lambda_n\) interpolation conditions at \(-\infty\). For the integers \(\lambda_n\) one assumes that \(\lim_{n\to\infty} \lambda_n/2n= \theta\in [0,1]\). The denominator polynomials are known to be orthogonal with varying weight \(x^{-\lambda_n} d\rho(x)\) on \([0,\infty)\) and this information is used to describe their asymptotic zero distribution. This is done in two different ways. First, contracted zero distributions are investigated, where all zeros are scaled with a scaling factor \(c_n\) that tends to infinity, and each scaled zero gets the same weight \(1/n\). Secondly, one also investigates the weighted zero distribution by giving each zero a weight, with large zeros receiving little weight. In both cases the asymptotic zero distribution is obtained and depends on the parameter \(\theta\).