## Journal of Applied Mathematics

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

#### Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 482935, 25 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.jam/1355495242

Digital Object Identifier
doi:10.1155/2012/482935

Mathematical Reviews number (MathSciNet)
MR2959981

Zentralblatt MATH identifier
1252.41004

#### Citation

Jung, Hee Sun; Sakai, Ryozi. Asymptotic Properties of Derivatives of the Stieltjes Polynomials. J. Appl. Math. 2012 (2012), Article ID 482935, 25 pages. doi:10.1155/2012/482935. https://projecteuclid.org/euclid.jam/1355495242