Asymptotic behavior of coefficients

I have a three term recurrence relation

$$a_0=1,$ $a_1=p_1(1)a_0,$ \label{recurr} \begin{array}{ccc} a_{n}=p_1(n) a_{n-1}+p_2(n) a_{n-2}, && n\ge2.\\ \end{array}$

where

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

and

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

with
$$k=\pm\frac{\beta}{\alpha}$$

I'm interested in the asymptotic behavior of the coefficients

$$a_n^{(1)}\sim ???$$

and

$$a_n^{(2)}\sim ???$$

when

$n\mapsto\infty$

Any ideas?

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
$a_n \sim \left(\frac{\delta^2}{\alpha\gamma}\right)^n$