Asymptotic properties of derivatives of the Stieltjes polynomials (Q1760702)

Summary: 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 \(\int^1_{-1} w_\lambda(x)P_{\lambda, n}(x)E_{\lambda, n+1}(x)x^m dx = 0, 0 \leq m < n + 1, \int^1_{-1} w_\lambda(x)P_{\lambda, n}(x)E_{\lambda, n+1}(x)x^m dx \neq 0, m = n + 1\). 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.