Asymptotic behavior of coefficients

[intervoxel]

I have a three term recurrence relation

$\[<br /> a_0=1,<br /> \]<br /> \[<br /> a_1=p_1(1)a_0,<br /> \]<br /> \begin{equation}\label{recurr}<br /> \begin{array}{ccc}<br /> a_{n}=p_1(n) a_{n-1}+p_2(n) a_{n-2}, && n\ge2.\\<br /> \end{array}<br /> \end{equation}<br />$

where

$\[<br /> p_1(n)=\frac{\delta^2/\alpha\gamma}{(n+k)^2-\beta^2/\alpha^2}<br /> \]$

and

$\[<br /> p_2(n)=-\frac{1}{(n+k)^2-\beta^2/\alpha^2},<br /> \]$

with
$\[<br /> k=\pm\frac{\beta}{\alpha}<br /> \]$

I'm interested in the asymptotic behavior of the coefficients

$\[<br /> a_n^{(1)}\sim ?<br /> \]$

and

$\[<br /> a_n^{(2)}\sim ?<br /> \]$

when

$n\mapsto\infty<br />$

Any ideas?

 

[lpetrich]

Here's a trick that's often used in such problems. Take n large, then treat p₀(n) and p₁(n) as constants. You get a recurrence relation with constant coefficients, and it's possible to solve it by taking
$a_n \sim \lambda^n$
and solving the resulting equation for $\lambda$:

$\lambda^2 = p_1 \lambda + p_0$
where
$p_1 \sim \frac{\delta^2/\alpha\gamma}{n^2}$
$p_2 \sim - \frac{1}{n^2}$

The solution is our old friend the quadratic formula:
$\lambda = \frac{p_1 \pm \sqrt{p_1^2 + p_2}}{2} = - \frac{2p_2}{p_1 \mp \sqrt{p_1^2 + p_2}}$
which has these approximate solutions when n is small:
$\lambda \sim p_1 \ll 1$
and
$\lambda \sim \frac{p_2}{p_1} \sim \frac{\delta^2}{\alpha\gamma}$
So
$a_n \sim \left(\frac{\delta^2}{\alpha\gamma}\right)^n$

 

[intervoxel]

Thank you for the reply.