Journal of Applied Mathematics
Volume 2012 (2012), Article ID 482935, 25 pages
http://dx.doi.org/10.1155/2012/482935
Research Article

Asymptotic Properties of Derivatives of the Stieltjes Polynomials

Hee Sun Jung1 and Ryozi Sakai2

1Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, Republic of Korea
2Department of Mathematics, Meijo University, Nagoya 468-8502, Japan

Received 16 March 2012; Accepted 24 May 2012

Academic Editor: Jin L. Kuang

Copyright © 2012 Hee Sun Jung and Ryozi Sakai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝑤 𝜆 ( 𝑥 ) ∶ = ( 1 − 𝑥 2 ) 𝜆 − 1 / 2 and 𝑃 𝜆 , 𝑛 ( 𝑥 ) be the ultraspherical polynomials with respect to 𝑤 𝜆 ( 𝑥 ) . Then, we denote the Stieltjes polynomials with respect to 𝑤 𝜆 ( 𝑥 ) by 𝐸 𝜆 , 𝑛 + 1 ( 𝑥 ) satisfying ∫ 1 − 1 𝑤 𝜆 ( 𝑥 ) 𝑃 𝜆 , 𝑛 ( 𝑥 ) 𝐸 𝜆 , 𝑛 + 1 ( 𝑥 ) 𝑥 𝑚 𝑑 𝑥 = 0 , 0 ≤ 𝑚 < 𝑛 + 1 , ∫ 1 − 1 𝑤 𝜆 ( 𝑥 ) 𝑃 𝜆 , 𝑛 ( 𝑥 ) 𝐸 𝜆 , 𝑛 + 1 ( 𝑥 ) 𝑥 𝑚 𝑑 𝑥 ≠ 0 , 𝑚 = 𝑛 + 1 . In this paper, we investigate asymptotic properties of derivatives of the Stieltjes polynomials 𝐸 𝜆 , 𝑛 + 1 ( 𝑥 ) and the product 𝐸 𝜆 , 𝑛 + 1 ( 𝑥 ) 𝑃 𝜆 , 𝑛 ( 𝑥 ) . Especially, we estimate the even-order derivative values of 𝐸 𝜆 , 𝑛 + 1 ( 𝑥 ) and 𝐸 𝜆 , 𝑛 + 1 ( 𝑥 ) 𝑃 𝜆 , 𝑛 ( 𝑥 ) at the zeros of 𝐸 𝜆 , 𝑛 + 1 ( 𝑥 ) and the product 𝐸 𝜆 , 𝑛 + 1 ( 𝑥 ) 𝑃 𝜆 , 𝑛 ( 𝑥 ) , respectively. Moreover, we estimate asymptotic representations for the odd derivatives values of 𝐸 𝜆 , 𝑛 + 1 ( 𝑥 ) and 𝐸 𝜆 , 𝑛 + 1 ( 𝑥 ) 𝑃 𝜆 , 𝑛 ( 𝑥 ) at the zeros of 𝐸 𝜆 , 𝑛 + 1 ( 𝑥 ) and 𝐸 𝜆 , 𝑛 + 1 ( 𝑥 ) 𝑃 𝜆 , 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.