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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Derivatives of Integrating Functions for Orthonormal Polynomials with Exponential-Type Weights | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 528454 22 pages doi 10.1155 2009 528454 Research Article Derivatives of Integrating Functions for Orthonormal Polynomials with Exponential-Type Weights Hee Sun Jung1 and Ryozi Sakai2 1 Department of Mathematics Education Sungkyunkwan University Seoul 110-745 South Korea 2 Fukami 675 Ichiba-cho Toyota-city Aichi 470-0574 Japan Correspondence should be addressed to Hee Sun Jung hsun90@skku.edu Received 22 December 2008 Revised 26 February 2009 Accepted 22 April 2009 Recommended by Jozsef Szabados Let Wp x x pexp -Q x p -1 2 where Q e C2 -TO to 0 to is an even function. In 2008 we have a relation of the orthonormal polynomial pn wp x with respect to the weight wp x Pn x An x pn-1 x - Bn x pn x - 2pnpn x x where An x and Bn x are some integrating functions for orthonormal polynomials pn wp 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 wp x . Copyright 2009 H. S. Jung and R. 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. 1. Introduction and Results Let R -TO to . Let Q e C2 R R 0 to be an even function and let w x exp -Q x be such that fTOxnw2 x dx TO for all n 0 1 2 . Forp -1 2 we set wp x x pw x x e R. 1.1 Then we can construct the orthonormal polynomials pn p x pn wp x of degree n with respect to wp x . That is -TO Pn p x pmrp x wp x dx ômn Kronecker s delta Pnp x Ynxn Yn Yn p 0. 1.2 2 Journal of Inequalities and Applications A function f R R is said to be quasi-increasing if there exists C 0 such that f x Cf y for 0 x y. For any two sequences bnJX1 and cn O 1 of nonzero real numbers or functions we write bn cn if there exists a constant C 0 independent of n or x such that bn Ccn for n large enough. We write .
