Carleman matrices are useful to express the composition of two functions.
Recall that the Carleman matrix of a function $f$ is an infinite matrix whose $i,j$ element is $\frac{1}{j!}\frac{{\rm d}^j}{{\rm d} x^j}(f(x))^i$ evaluated at, e.g., $x = 0$.
In particular $M(f \circ g) = M(f) M(g)$, where $M(\cdot)$ is the Carleman matrix of function $(\cdot)$.
Is there a simple way to express the Carleman matrix of the sum of two functions, $f$ and $g$, as matrix operations of the Carleman matrices $M(f)$ and $M(g)$?
I can show that $M(f+g)_{i,k} = \sum^i_{r=0} \sum^k_{s=0}{i \choose r} M(f)_{r,s} M(g)_{i-r,k-s}$, but this procedure is not expressing $M(f+g)$ as matrix operations of $M(f)$ and $M(g)$. Can this be done?
What if I just want an approximation of $M(f+g)$? Can I do it using only matrix operations (sum, product, inverse, etc.) on $M(f)$ and $M(g)$?