Asymptotic behaviour of Stieltjes polynomials for ultraspherical weight functions (Q1917935)

This paper is concerned with the Stieltjes polynomials \(E^{(\lambda)}_{n+1}(x)\) defined by \[ \int^1_{-1}w_\lambda(x)P^{(\lambda)}_n(x)E^{(\lambda)}_{n+1}(x)x^kdx=0,\quad k=0,1,2,\dots,n, \] where \(w_\lambda(x)=(1-x^2)^{\lambda-1/2}\), \(\lambda>-1/2\), is the ultraspherical weight function and \(P^{(\lambda)}_n(x)\) are the ultraspherical polynomials. The author gives an asymptotic representation of \(E^{(\lambda)}_{n+1}(\cos\vartheta)\), as \(n\to\infty\), in the case \(1<\lambda\leq 2\). This representation, which holds uniformly for \(\varepsilon\leq\vartheta\leq\pi-\varepsilon\), \(\varepsilon>0\), completes the study of the ultraspherical weight functions for which Stieltjes polynomials are known to have only real distinct zeros inside \((-1,1)\) for all \(n\). The asymptotic behaviour obtained is successfully applied in proving interesting positivity results for Kronrod extensions of Gauss and Lobatto quadrature rules.