World Scientific
Skip main navigation

Cookies Notification

We use cookies on this site to enhance your user experience. By continuing to browse the site, you consent to the use of our cookies. Learn More
×

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

Existing users will be able to log into the site and access content. However, E-commerce and registration of new users may not be available for up to 12 hours.
For online purchase, please visit us again. Contact us at customercare@wspc.com for any enquiries.
No Access

Asymptotic expansions for second-order linear difference equations

    This research was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant A7359.

    https://doi.org/10.1142/9789814656054_0026Cited by:0 (Source: Crossref)
    Previous
    Next
    Abstract:

    Wong, R. and H. Li, Asymptotic expansions for second-order linear difference equations, Journal of Computational and Applied Mathematics 41 (1992) 65-94.

    Formal series solutions are obtained for the difference equation

    y(n +2) + a(n)y(n + 1) + b(n)y(n) = 0
    where a(n) and b(n) have asymptotic expansions of the form
    for large values of n, and b0 ≠ 0. These solutions are characterized by the roots of the characteristic equation ρ2 + a0ρ + b0 = 0. Our discussion is divided into three cases, according to whether the roots are distinct, or equal and do not satisfy the auxiliary equation a1ρ + b1 = 0, or equal and do satisfy the auxiliary equation. The last case is further divided into three subcases, according to whether the roots of the indicia) equation α(α -l)ρ2 +(a1α + a2)ρ + b2 = 0 do not differ by a nonnegative integer, or differ by a positive integer, or are equal. In all cases, the formal series solutions will be shown to be asymptotic. Our approach is based on the method of successive approximations.