Two-term recurrence relation

Question

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 $a_n =b_n=o(n)$

Answer 1

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.