Has the following operation $*$ on formal power series $f,g$ been studied before?

$$[X^n](f*g) = n! \cdot [X^n]f \cdot [X^n]g$$

where $n$ is a nonnegative integer? This is the typical Hadamard product multiplied by $n!$.

Clearly the identity is $\exp$, our product is associative and commutative — this means infinitely differentiable functions without pole at $x=0$ form a monoid under our product.

Other straightforward identities (forgive my abuse of notation):

• $e^{ax} * e^{bx} = e^{abx}$;
• $(cf)*g = f*(cg) = c(f*g)$;
• $\cosh x * e^{ix} = \cos x$; and
• If $c_n = \frac{f^{(n+1)}(0)}{f^{(n)}(0)}$, then the exponential generating function $\mathrm{EG}(c_n; - ) * f = f'$.