Asymptotic behavior of coefficients

[intervoxel]

Given the difference equation

$$a_{n+2}+A_n(\lambda)a_{n+1}+B_n(\lambda)a_n=0$$

where

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

and

$$B_n(\lambda)=\frac{(n+\alpha)(n+\beta)}{2(n+2)(n+1+\gamma)}$$

The asymptotic behavior of the coefficients is given by

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

and

$$a_n^{(2)}\sim n^{-3}\sum_{s=0}^{\infty}\frac{c_s^{(2)}}{n^s}$$


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.

 

[gtoguy97]

But it's not helping me.Unfortunately, there is no single procedure for solving difference equations with variable coefficients. The best approach is to use a combination of analytical and numerical techniques. First, you need to find the general solution of the equation using the standard methods such as undetermined coefficients, variation of parameters, or the z-transform. This will provide the form of the solution as a sum of terms with unknown coefficients (which are usually functions of the parameter).Next, you can use numerical methods such as power series expansion or the WKB method to approximate the unknown coefficients of the solution.Finally, you can use analytical methods such as asymptotic analysis to determine the asymptotic behavior of the solution. This involves finding the dominant terms in the solution as n -> infinity.