Suppose we have a sequence $\{a_n\}_{n=0}^{\infty}$ which is generated by \begin{align*} a_{n+1} - \left(q+ \frac{A} {n+1} \right) a_n - \frac B n a_{n-1} = C, \end{align*} for $n \ge 1$, where $q, A, B, C$ are fixed positive constants and $0 < q < 1$.
Suppose the initial condition is $a_{0} = a_{1} = 1$. It is easy to see the sequence $\{a_n\}_{n=2}^{\infty}$ generated is positive and lower bounded by $C$. I want to argue the limit of the sequence exits and come up with the limit.
If we know there is a limit for the sequence, then by taking limits, it seems like $\lim_{n \to \infty} a_n = C/(1-q)$ and also this limit is not a fixed point of the recurrence. I guess the sequence will approximate this limit. But I cannot see how to argue the limit is guaranteed to exist. Any comments will be helpful. Thanks.
Update: I guess one way is to write the relation recursively, i.e., \begin{align*} a_{n+1} &= C + \left( q+\frac{A}{n+1} \right) a_n + \frac{B}{n} a_{n-1} \\ &= C + \left( q+\frac{A}{n+1} \right) \left( C + \left( q+\frac{A}{n} \right) a_{n-1} + \frac{B}{n-1} a_{n-2} \right) + \frac{B} {n} a_{n-1}, \end{align*} Since we want the limiting behavior, any term with $n$ in the denominator will vanish. Then if we keep expanding, we get \begin{align*} a_{n+1} = C + Cq + Cq^2 + \dots + C q^{n+1} + o(n). \end{align*} It seems right except it seems a bit too messy. If anyone has a neat proof, please point me out. Another question is whether we can estimate the convergence rate of $\{a_k\}$ to $C/(1-q)$. From the expansion the slowest mode seems to $AC/(n+1)$ and thus the rate is determined by this term. Is this correct? Thanks.