Stieltjes polynomials are orthogonal polynomials with respect to the sign changing weight function (w P_{n}(\cdot, w)), where (P_{n}(\cdot, w)) is the (n)th orthogonal polynomial with respect to w. Zeros of Stieltjes polynomials are nodes of Gauss-Kronrod quadrature formulae, which are basic for the most frequently used quadrature routines with combined practical error estimate. For the ultraspherical weight function (w,(x)=\left(1-x^{2}\right)^{\lambda-12}, 0 \leqslant i \leqslant 1), we prove asymptotic representations of the Stieltjes polynomials and of their first derivative, which hold uniformly for (x=\cos \theta, \varepsilon \leqslant \theta \leqslant \pi-\varepsilon), where (\varepsilon \in(0, \pi / 2)) is fixed. Some conclusions are made with respect to the distribution of the zeros of Stieltjes polynomials, proving an open problem of Monegato [15, p. 235] and Peherstorfer [23, p. 186]. As a further application, we prove an asymptotic representation of the weights of GaussKronrod quadrature formulae with respect to (\psi_{j}, 0 \leqslant \lambda \leqslant 1), and we prove the precise asymptotical value for the variance of Gauss. Kronrod quadrature formulae in these cases. ' 1995 Academic Press. Inc