Asymptotic behaviour of Stieltjes polynomials for ultraspherical weight functions

For the ultraspherical weight functions wλ(x) = (1 − x2)λ − 12, an asymptotic representation of the Stieltjes polynomials is proved for 1<λ⩽2, which holds uniformly in every closed subinterval of (−1, 1). This extends and completes our earlier results (for 0⩽λ⩽1) in the sense that the problem is solved for all ultraspherical weight functions for which Stieltjes polynomials are known to have only real distinct zeros inside (−1, 1) for all n ∈ N. The main result is applied to prove positivity results for Kronrod extensions of Gauss and Lobatto quadrature formulae.