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2012 Asymptotic Properties of Derivatives of the Stieltjes Polynomials
Hee Sun Jung, Ryozi Sakai
J. Appl. Math. 2012: 1-25 (2012). DOI: 10.1155/2012/482935

## 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 ${\int }_{-1}^{1}{w}_{\lambda }(x){P}_{\lambda ,n}(x){E}_{\lambda ,n+1}(x){x}^{m}dx=0$, $0\le m, ${\int }_{-1}^{1}{w}_{\lambda }(x){P}_{\lambda ,n}(x){E}_{\lambda ,n+1}(x){x}^{m}dx\ne 0$, $m=n+1$. 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.

## Citation

Hee Sun Jung. Ryozi Sakai. "Asymptotic Properties of Derivatives of the Stieltjes Polynomials." J. Appl. Math. 2012 1 - 25, 2012. https://doi.org/10.1155/2012/482935

## Information

Published: 2012
First available in Project Euclid: 14 December 2012

zbMATH: 1252.41004
MathSciNet: MR2959981
Digital Object Identifier: 10.1155/2012/482935