Multiplication-like binary operator based on Maclaurin series of functions

Question

Let us define the binary function operator $\maltese$ based on the Maclaurin series of real-valued functions $f(x)$ and $g(x)$: $$\left(\sum_{n=0}^{\infty}f^{(n)}(0)\cdot\frac{x^n}{n!}\right)\maltese\left(\sum_{n=0}^{\infty}g^{(n)}(0)\cdot\frac{x^n}{n!}\right)=f \operatorname{\maltese} g=\sum_{n=0}^{\infty}g^{(n)}(0)\cdot f^{(n)}(0)\cdot\frac{x^n}{n!}$$ Some obvious properties of the $\maltese$ operator: $$f\operatorname{\maltese}g=g\operatorname{\maltese}f$$ $$(f\operatorname{\maltese}g)\operatorname{\maltese}h=f\operatorname{\maltese}(g\operatorname{\maltese}h)$$ $$(f+g)\operatorname{\maltese}h=f\operatorname{\maltese}h+g\operatorname{\maltese}h$$ $$(C\cdot f)\operatorname{\maltese}g=C\cdot(f\operatorname{\maltese}g)\quad(C=\mathrm{const})$$ $$f\operatorname{\maltese}0=0$$ $$f\operatorname{\maltese}\exp=f$$ $$\sin\operatorname{\maltese}\cos=0$$ Has such operator been described or studied anywhere? Do we know any non-trivial properties of the $\maltese$ operator?

Answer 1

This is often called the Hadamard product of a power series and certainly has been studied. It is closely related to the Cartesian product operation on combinatorial species. There are many other applications, of course.