Ultraspherical Stieltjes polynomials and Gauss-Kronrod quadrature behave nicely for lambda < 0

摘要

We show that the zeros of the Stieltjes polynomials associated with the ultraspherical weight function with parameter lambda < 0 are real and simple and, except for two of them, belong to (-1, 1). We also prove that they strictly interlace with the zeros of the corresponding ultraspherical polynomials. Consequently, a Gauss-Kronrod quadrature formula with two nodes outside the interval [-1, 1] is obtained. We prove that the coefficients of such quadrature rules are positive. Finally, an asymptotic representation of Stieltjes polynomials which converges uniformly on the whole interval [-1, 1] is provided.