- #1
intervoxel
- 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?
[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?