# Asymptotic behavior of coefficients

by intervoxel
Tags: asymptotic, behavior, coefficients
 P: 134 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.

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