limiting point of a difference equation with coefficients related to a characteristic polynomial

Question

I have a difference equation of the form \begin{equation} \mathbf{x(k+D+1)}= -(\alpha_D x(k+D)+\cdots+\alpha_1 x(k-1)+\alpha_0 x(k)) - c \end{equation} $c$ is a constant, $\alpha_D,\alpha_{D-1},\ldots,\alpha_0$ is the coefficients of characteristic polynomials of a matrix, say $\mathbf{A}$, whose all eigenvalues have magnitude less than 1. Additionally, I know that the sequence $x(0),x(1),\ldots$ converges to a limit say $x_\mathrm{limit}$. Final value theorem can be used to solve this if $\mathbf{A}$ had $1$ as a single eigenvalue. Other than this is there any general solution for $x_\mathrm{limit}$ (i.e. even if $1$ is not an eigenvalue of $\mathbf{A}$)

Answer 1

I think I got the answer also, please correct me if I am wrong. Since the sequence is converging, I could safely assume as $k \to \infty, x(k-n) = x_\mathrm{limit}$. Hence, I get $(1 + \sum{\alpha_i}) x_\mathrm{limit} = -c$, giving $x_\mathrm{limit} = \frac{-c}{(1 + \sum{\alpha_i})}$.