TITLE: Two-term recurrence relation
QUESTION [9 upvotes]: We consider the following system of recurrence relations for $n \in \mathbb Z$ and $\vert \lambda \vert=1$ with $\lambda \in \mathbb{C}$
$$a_{n+1} = \lambda a_{n-1}+ \lambda^* a_n + \lambda^* n b_n $$
$$b_{n+1} = \lambda^* b_{n-1}+ \lambda b_n -  \lambda n a_n. $$
Here, $\lambda^*$ is the complex conjugate of $\lambda.$
I am interested in non-zero initial conditions under which $a_n,b_n$ tend to zero for $n \rightarrow \pm \infty.$
Observation: If there is a limit $a_n \rightarrow a$ and $b_n \rightarrow b$, then indeed by the last term in each row, it has to be zero and $a_n =b_n=o(n)$

REPLY [2 votes]: I assume that the initial conditions $a_0,a_1,b_0,b_1$ and that $n\to +\infty$.
Let $A(x):=\sum_{n\geq 0} a_n x^n$ and $B(x):=\sum_{n\geq 0} a_n x^n$. Then the recurrence relations become:
$$\begin{cases}
A(x) - a_1x - a_0 = \lambda x^2 A(x) + \lambda^* x (A(x)-a_0) + \lambda^* x^2 B'(x), \\
B(x) - b_1x - b_0 = \lambda^* x^2 B(x) + \lambda x (B(x)-b_0) - \lambda x^2 A'(x).
\end{cases}$$
That is, we have a system of 2 linear first-order ODE:
$$
\begin{bmatrix} A'(x)\\ B'(x)\end{bmatrix}
=
\begin{bmatrix} 0 & -\lambda^* x^{-2}+x^{-1}+\lambda^{*2}\\ 
\lambda x^{-2} - x^{-1} - \lambda^2 & 0\end{bmatrix}
\cdot
\begin{bmatrix} A(x)\\ B(x)\end{bmatrix}
+
\begin{bmatrix} \lambda^*(b_0x^{-2} + (b_1-\lambda b_0)x^{-1})\\
-\lambda(a_0x^{-2} + (a_1-\lambda^*a_0)x^{-1})\end{bmatrix},
$$
which may be analyzed with the standard methods.