Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 528454, 22 pages
Research Article

Derivatives of Integrating Functions for Orthonormal Polynomials with Exponential-Type Weights

1Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, South Korea
2Fukami 675, Ichiba-cho, Toyota-city, Aichi 470-0574, Japan

Received 22 December 2008; Revised 26 February 2009; Accepted 22 April 2009

Academic Editor: Jozsef Szabados

Copyright © 2009 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.


Let wρ(x):=|x|ρexp(Q(x)), ρ>1/2, where QC2:(,)[0,) is an even function. In 2008 we have a relation of the orthonormal polynomial pn(wρ2;x) with respect to the weight wρ2(x); pn(x)=An(x)pn1(x)Bn(x)pn(x)2ρnpn(x)/x, where An(x) and Bn(x) are some integrating functions for orthonormal polynomials pn(wρ2;x). In this paper, we get estimates of the higher derivatives of An(x) and Bn(x), which are important for estimates of the higher derivatives of pn(wρ2;x).