Asymptotic Properties of Derivatives of the Stieltjes Polynomials

Abstract EN

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 ∫−11wλ(x)Pλ,n(x)Eλ,n+1(x)xmdx=0, 0≤m

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.