In the expression
$\begin{bmatrix}A & C \\ 0 & B\end{bmatrix}^n = \begin{bmatrix}A^n & * \\ 0 & B^n\end{bmatrix},$ I wonder whether the term denoted by * can be expressed in a simple form when we assume the following: (1) $A$ has its eigenvalues on or inside the unit circle. Those on the unit circle are simple; (2) $B$ has its eigenvalues strictly inside the unit circle; (3) $A$ and $B$ may have different dimensions.
In fact, I am interested for the value of * as $n \rightarrow \infty$. It would be $C(I-B)^{-1}$ when $A=I$ but in a general case, $A^n$, though bounded, may not converge as $n \rightarrow \infty$. So, I wonder whether * can have a simple expression.