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Asymptotic behavior of coefficients 
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#1
Feb311, 07:06 AM

P: 135

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. 


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