Say I have two square matrices $A$ and $P$. My series is as follows:
$$\begin{aligned} X_1 &= P\\ X_2 &= PA+A^TP\\ &\vdots\\ X_R &= X_{R-1}A+A^T X_{R-1}\end{aligned}$$
What function is this? It is fairly similar to the binomial expansion, since the coefficients of the expanded terms lead to the Pascal triangle, yet it is somewhat different.
The matrix equation $$X_{k+1} = X_kA + A^TX_k$$ can be converted (by column stacking) into a vector equation $$\eqalign{ x_{k+1} &= \left(A^T\otimes I+I\otimes A^T\right) x_k \\ &\doteq (B+C)x_k \\ &= (B+C)^kp \\ }$$ So it is a binomial expansion of $(B+C)$
In general matrix multiplication is non-commutative, however these particular matrices do commute, so the expansion can be treated exactly like the scalar case.
And should $(B+C)^k$ converge to a limit $$\eqalign{ L &= \lim_{k\to\infty}(B+C)^k \\ x &= \lim_{k\to\infty}x_{k+1} = Lp \\ }$$