Asymptotic Properties of Derivatives of the Stieltjes Polynomials

Abstract

Let $w_{\lambda }(x):={(1-x^2)}^{\lambda -1/2}$ and $P_{\lambda ,n}(x)$ be the ultraspherical polynomials with respect to $w_{\lambda }(x)$. Then, we denote the Stieltjes polynomials with respect to $w_{\lambda }(x)$ by $E_{\lambda ,n+1}(x)$ satisfying ${\displaystyle {\int }_{-1}^1{w}_{\lambda }(x)P_{\lambda ,n}(x)E_{\lambda ,n+1}(x)x^{m}dx}=0$, , ${\displaystyle {\int }_{-1}^1{w}_{\lambda }(x)P_{\lambda ,n}(x)E_{\lambda ,n+1}(x)x^{m}dx}\ne 0$, $m=n+1$. In this paper, we investigate asymptotic properties of derivatives of the Stieltjes polynomials $E_{\lambda ,n+1}(x)$ and the product $E_{\lambda ,n+1}(x)P_{\lambda ,n}(x)$. Especially, we estimate the even-order derivative values of $E_{\lambda ,n+1}(x)$ and $E_{\lambda ,n+1}(x)P_{\lambda ,n}(x)$ at the zeros of $E_{\lambda ,n+1}(x)$ and the product $E_{\lambda ,n+1}(x)P_{\lambda ,n}(x)$, respectively. Moreover, we estimate asymptotic representations for the odd derivatives values of $E_{\lambda ,n+1}(x)$ and $E_{\lambda ,n+1}(x)P_{\lambda ,n}(x)$ at the zeros of $E_{\lambda ,n+1}(x)$ and $E_{\lambda ,n+1}(x)P_{\lambda ,n}(x)$ 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.