Advances in Difference Equations
Volume 2010 (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

1Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
2Department of Mathematics, Shenzhen University, Guangdong 518060, China
3Department of Mathematics, Tsinghua University, Beijin 100084, China

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.

Abstract

We discuss in detail the error bounds for asymptotic solutions of second-order linear difference equation 𝑦 ( 𝑛 + 2 ) + 𝑛 𝑝 𝑎 ( 𝑛 ) 𝑦 ( 𝑛 + 1 ) + 𝑛 𝑞 𝑏 ( 𝑛 ) 𝑦 ( 𝑛 ) = 0 , where 𝑝 and 𝑞 are integers, 𝑎 ( 𝑛 ) and 𝑏 ( 𝑛 ) have asymptotic expansions of the form ∑ 𝑎 ( 𝑛 ) ∼ ∞ 𝑠 = 0 ( 𝑎 𝑠 / 𝑛 𝑠 ) , ∑ 𝑏 ( 𝑛 ) ∼ ∞ 𝑠 = 0 ( 𝑏 𝑠 / 𝑛 𝑠 ) , for large values of 𝑛 , 𝑎 0 ≠ 0 , and 𝑏 0 ≠ 0 .