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báo cáo hóa học:" Research Article Error Bounds for Asymptotic Solutions of Second-Order Linear Difference Equations II: The First Case"

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 Error Bounds for Asymptotic Solutions of Second-Order Linear Difference Equations II: The First Case | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 594783 19 pages doi 10.1155 2010 594783 Research Article Error Bounds for Asymptotic Solutions of Second-Order Linear Difference Equations II The First Case L. H. Cao1 2 and J. M. Zhang3 1 Department of Mathematics City University of Hong Kong Tat Chee Avenue Kowloon Hong Kong 2 Department of Mathematics Shenzhen University Guangdong 518060 China 3 Department of Mathematics Tsinghua University Beijin 100084 China Correspondence should be addressed to J. M. Zhang jzhang@math.tsinghua.edu.cn Received 13 July 2010 Accepted 27 October 2010 Academic Editor Rigoberto Medina Copyright 2010 L. H. Cao and J. M. Zhang. 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. We discuss in detail the error bounds for asymptotic solutions of second-order linear difference equation y n 2 npa n y n 1 nqb n y n 0 where p and q are integers afn and b n have asymptotic expansions of the form a n ỵf 0 as ns b n f0 bs ns for large values of n a0 0 and b0 0. 1. Introduction Asymptotic expansion of solutions to second-order linear difference equations is an old subject. The earliest work as we know can go back to 1911 when Birkhoff 1 first deal with this problem. More than eighty years later this problem was picked up again by Wong and Li 2 3 . This time two papers on asymptotic solutions to the following difference equations y n 2 a n y n 1 b n y n 0 1.1 y n 2 npafn y n 1 nqb n y n 0 1.2 were published respectively where coefficients a n and b n have asymptotic properties an Ễ Ệ bn ẳ ns L3 1 s 0 for large values of n a0 Ỷ 0 b0 0 and p q e Z. 2 Advances in Difference Equations Unlike the method used by Olver 4 to treat asymptotic solutions of second-order linear differential equations the method used in Wong and Li s papers cannot give us way to .