Asymptotic behavior of coefficients

  1. Given the difference equation

    [tex]
    a_{n+2}+A_n(\lambda)a_{n+1}+B_n(\lambda)a_n=0
    [/tex]

    where

    [tex]
    A_n(\lambda)=-\frac{(n+1)(2\delta+\epsilon+3(n+\gamma))+Q}{s(n+2)(n+1+\gamma)}
    [/tex]

    and

    [tex]
    B_n(\lambda)=\frac{(n+\alpha)(n+\beta)}{2(n+2)(n+1+\gamma)}
    [/tex]

    The asymptotic behavior of the coefficients is given by

    [tex]
    a_n^{(1)}\sim 2^{-n}n^{-1-\lambda/2}\sum_{s=0}^{\infty}\frac{c_s^{(1)}}{n^s}
    [/tex]

    and

    [tex]
    a_n^{(2)}\sim n^{-3}\sum_{s=0}^{\infty}\frac{c_s^{(2)}}{n^s}
    [/tex]


    I have to do the a similar calculation in my research project but I couldn't find out the procedure used. Please, someone can show me the steps to such a solution?

    I tried to helplessly follow the text by Saber Elaydi, An Introduction to Difference Equations.
     
  2. jcsd
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