Can the result of termwise multiplication of power series be found in a closed form?

This is a question that comes out of a combinatorics question that is using generating functions. Let me define what I mean by multiplying power series the wrong way (although you may be able to guess). Let $$f(z)=a_0+a_1z+a_2z^2+\cdots,$$ and $$g(z)=b_0+b_1z+b_2z^2+\cdots.$$ Then the wrong multiplication, which we will be denoted by $*$ will be defined as $$f*g(Z)=a_0b_0+a_1b_1z+a_2b_2z^2+\cdots .$$

Now this is a perfectly nice expression that can be written down. But suppose that $f$ and $g$ are in some compact form such as a rational function, or an exponential or any other commonly occurring functions (closed form expressions for example).

Then can we write the wrong way to multiply these power series in a closed form way (if the original functions are in closed form). If so, what is the formula. If not, what would a counterexample look like.

This is called the Hadamard product of power series. It is quite difficult to compute in closed form in general and has a tendency to "increase the complexity" of the power series you're looking at; consider, for example, the Hadamard product of $e^z$ and itself.

The Hadamard product can sometimes be computed using complex analysis as in this blog post.