Infinite series for recurrence Question 1
If I define $A(z) = \sum_{n \ge 0} a_n \frac{z^n}{n!} \tag 1$
(where $a_n$ are $3\times 3$ constant matrices indexed with n),
then can we re-write  $\sum_{n \ge 1} a_{n-1} \frac{z^n}{n!} \tag 2 $ and  $\sum_{n \ge 2} a_{n-2} \frac{z^n}{n!} \tag 3 $   in terms of A(z)? 
Question 2
Can we write $\psi(z)=\sum_{n \ge 0} na_{n}  z^n  \tag 4 $ 
in terms of any any existing infinite series such that we can also re write 
$\sum_{n \ge 1} a_{n-1}  z^n    \tag 5 $ and  $\sum_{n \ge 2} a_{n-2}  z^n    \tag 6 $   in terms of $\psi(z)$?
NB :: Thease all are part of my attempts to solve $na_n=a_{n-1}+a_{n-2}$ without using ODE. Means by using power series or any other infinte series

*

*The nine entries in your matrix are independent, so it is slightly simpler to think of nine separate equations.

*Differentiate the first equation (2).  Then let $m=n-1$ and relabel the result.  Unfortunately, this turns it into a differential equation.

*Let $\xi(z)$ be equation (3).  Then $\xi^{\prime}(z)$ can be rearranged in terms of $\psi(z)$.
$B(z)=\sum_{n\geq1}a_{n-1}{z^n\over n!}\\B\,^{\prime}(z)=\sum_{n\geq1}a_{n-1}{z^{n-1}\over (n-1)!}=\sum_{m\geq0}a_m{z^m\over m!}=B(z)$
