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.

Abstract

Let wρ(x):=|x|ρexp(−Q(x)), ρ>−1/2, where Q∈C2:(−∞,∞)→[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)pn−1(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).