Error Bounds for Asymptotic Solutions of Second-Order Linear Difference Equations II: The First Case
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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 Department of Mathematics, Shenzhen University, Guangdong 518060, China 3 Department of Mathematics, Tsinghua University, Beijin 100084, China 2
Correspondence should be addressed to J. M. Zhang, [email protected] Received 13 July 2010; Accepted 27 October 2010 Academic Editor: Rigoberto Medina Copyright q 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 yn 2 np anyn 1 nq bnyn p and q are integers, an and bn have 0, where ∞ s s asymptotic expansions of the form an ∼ ∞ s0 as /n , bn ∼ s0 bs /n , for large values of n, 0, and b0 / 0. a0 /
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: yn 2 anyn 1 bnyn 0 p
q
yn 2 n anyn 1 n bnyn 0
1.1 1.2
were published, respectively, where coefficients an and bn have asymptotic properties an ∼
∞ as s0
, ns
for large values of n, a0 / 0, b0 / 0, and p, q ∈ Z.
bn ∼
∞ bs , s n s0
1.3
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 obtain error bounds of these asymptotic solutions. Only order estimations were given in their papers. The estimations of error bounds for these asymptotic solutions to 1.1 were given in 5 by Zhang et al. But the problem of obtaining error bounds for these asymptotic solutions to 1.2 is still open. The purpose of this and the next paper Error bounds for asymptotic solutions of second-order linear difference equations II: the second case is to estimate error bounds for solutions to 1.2. The idea used in this paper is similar to that of Olver to obtain error bounds to the Liouville-Green WKB asymptotic expansion of solutions to second-order differential equations. It should be pointed out that similar method appeared in some early papers, such as Spigler and Vianello’s papers 6–9. In Wong and Li’s second paper 3, two different cases were given according to different values of parameters. The first case is devoted to the situation when k > 0, and in the second case as k < 0 where k 2p − q. The whole proof of
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