Abstract :
Stieltjes polynomials are orthogonal polynomials with respect to the sign changing weight function wPn(·, w), where wPn(·, w) is the nth 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 wλ(x) = (1 − x2)λ − 1/2, 0 ≤ λ ≤ 1, we prove asymptotic representations of the Stieltjes polynomials and of their first derivative, which hold uniformly for x = cos θ, ϵ ≤ π − ϵ, where ϵ ∈ (0, π/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 Gauss-Kronrod quadrature formulae with respect to wλ, 0 ≤ λ ≤ 1, and we prove the precise asymptotical value for the variance of Gauss-Kronrod quadrature formulae in these cases.
