Asymptotic Properties of Derivatives of the Stieltjes Polynomials

Let wλ x : 1 − x2 λ−1/2 and Pλ,n x be the ultraspherical polynomials with respect to wλ x . Then, we denote the Stieltjes polynomials with respect to wλ x by Eλ,n 1 x satisfying ∫1 −1 wλ x Pλ,n x Eλ,n 1 x x dx 0, 0 ≤ m < n 1, ∫1 −1 wλ x Pλ,n x Eλ,n 1 x xdx / 0, m n 1. In this paper, we investigate asymptotic properties of derivatives of the Stieltjes polynomials Eλ,n 1 x and the product Eλ,n 1 x Pλ,n x . Especially, we estimate the even-order derivative values of Eλ,n 1 x and Eλ,n 1 x Pλ,n x at the zeros of Eλ,n 1 x and the product Eλ,n 1 x Pλ,n x , respectively. Moreover, we estimate asymptotic representations for the odd derivatives values of Eλ,n 1 x and Eλ,n 1 x Pλ,n x at the zeros of Eλ,n 1 x and Eλ,n 1 x Pλ,n x on a closed subset of −1, 1 , respectively. These estimates will play important roles in investigating convergence and divergence of the higher-order Hermite-Fejér interpolation polynomials.