With the help of a new representation of the Stieltjes polynomial it is sho
wn 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 poly
nomial and the Gaussian nodes, it follows immediately that Gauss-Kronrod qu
adrature 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.