Asymptotic behavior of coefficients

  • Thread starter intervoxel
  • Start date
  • #1
195
1
I have a three term recurrence relation

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

[/itex]

where

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

and

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

with
[itex]
\[
k=\pm\frac{\beta}{\alpha}
\]
[/itex]

I'm interested in the asymptotic behavior of the coefficients

[itex]
\[
a_n^{(1)}\sim ???
\]
[/itex]

and

[itex]
\[
a_n^{(2)}\sim ???
\]
[/itex]

when

[itex]
n\mapsto\infty

[/itex]

Any ideas?
 

Answers and Replies

  • #2
984
174
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
[itex]a_n \sim \lambda^n[/itex]
and solving the resulting equation for [itex]\lambda[/itex]:

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

The solution is our old friend the quadratic formula:
[itex]\lambda = \frac{p_1 \pm \sqrt{p_1^2 + p_2}}{2} = - \frac{2p_2}{p_1 \mp \sqrt{p_1^2 + p_2}}[/itex]
which has these approximate solutions when n is small:
[itex]\lambda \sim p_1 \ll 1[/itex]
and
[itex]\lambda \sim \frac{p_2}{p_1} \sim \frac{\delta^2}{\alpha\gamma}[/itex]
So
[itex]a_n \sim \left(\frac{\delta^2}{\alpha\gamma}\right)^n[/itex]
 
  • #3
195
1
Thank you for the reply.
 

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