Any known compact expression for  Is there any known compact expression for the sum
$$S_{k} = \sum_{i=1}^{k} A^{i-1} P Q^{k-i}$$
where $A$, $P$ and $Q$ are respectively  $m \times m$, $m \times n$ and $n \times n$ matrices?. 
You can assume, if needed, that $A$ and $Q$ are invertible.
The trivial relation
$$ AS_{k}-S_{k}Q = A^{k}P - P Q^{k}$$
perhaps provides some clues (fo example it is known that if $A$ and $-Q$ have no common eigenvalues then the last equation has unique solution).
 A: If we define an $(n+m)\times (n+m)$ matrix by
$$C:=\left(\begin{array}{cc}A&P\\\ 0&Q\end{array}\right),$$
then we have
$$C^k=\left(\begin{array}{cc}A^k&\sum_{i=1}^k A^{i-1}PQ^{k-i}\\\ 0&Q^k\end{array}\right)=\left(\begin{array}{cc}A^k&S_k\\\ 0&Q^k\end{array}\right).$$
I don't think that a simpler expression than this is very likely.
A: As HenrikRüping wrote, my comment is false.
Nevertheless, I think the method is interesting (obviously, it isn't mine), although it gives something "explicit", but not "compact". Maybe you could provide us with context? For example, are you interested in the behavior when $k \rightarrow + \infty$ (assuming the field is topological)?
If $XAX^{-1}$ and $YQY^{-1}$ are "nice" (diagonal or Jordan normal form), then make the change of variable (is this English?) $P'=XPY$.
Then viewing $S_k$ as a linear function of $P$, $XS_k(P)Y=\sum_{i=1}^k XAX^{-1} P' YQY^{-1}$, so up to a change of base on $M_{m,n}(K)$, the endomorphism $S_k$ of this vector space is given in a nice form (eigenvalues are known).
But I'm not sure this is really what you're asking for, and your last comment suggests you already know what I just wrote.
