Ultraspherical Gauss--Kronrod Quadrature Is Not Possible for $\lambda > 3$

Abstract

With the help of a new representation of the Stieltjes polynomial it is shown by using Bessel functions that the Stieltjes polynomial with respect to the ultraspherical weight function with parameter $\lambda$ has only few real zeros for $\lambda > 3$ and sufficiently large n. Since the nodes of the Gauss--Kronrod quadrature formulae subdivide into the zeros of the Stieltjes polynomial and the Gaussian nodes, it follows immediately that Gauss--Kronrod quadrature is not possible for $\lambda > 3$. On the other hand, for $\lambda = 3$ and sufficiently large n, even partially positive Gauss--Kronrod quadrature is possible.