Ratio asymptotics and quadrature formulas
Suppose \(p_n\) \((n=0,1,2,\ldots)\) is a sequence of orthogonal polynomials on the real line, satisfying a three-term recurrence relation \(tp_n(t) = a_{n+1}p_{n+1}(t)+b_np_n(t)+a_np_{n-1}(t)\). The author gives a method for obtaining the asymptotic behaviour of the ratio \(s_n(z)/p_n(z)\) for a comparison sequence \(s_n\) \((n=0,1,2,\ldots)\) of polynomials. Crucial is the investigation of modified moments \(\int r_m(t) d\mu(t)\) where the \(r_m\) are polynomials suitably chosen in terms of the given polynomials \(s_n\) and the orthogonal polynomials \(p_n\) which one wants to study. The method gives ratio asymptotics for a large class of orthogonal polynomials in the class \(M\) of converging recurrence coefficients, but also for unbounded recurrence coefficients (even with exponential growth). In an appendix the author also shows how the technique can be used when the Fourier coefficients of \(s_n(t) = \sum_{k=0}^n c_k p_k(t)\) behave nicely, which in particular gives ratio asymptotics for compact perturbations of orthogonal polynomials.
scientific article; zbMATH DE number 1023015