On a peculiar operation on polynomials Consider the polynomials with natural number coefficients. Take two polynomials $P, Q$. If $Q(x) = \sum_{i} c_i x^i$, then we define
$$(Q*P)(x) = \prod_{i} [P(x+i)]^{c_i}.$$
Is this operation studied anywhere in the literature? What is the name for it? I do not know what to search for, so I didn't find much. This operation satisfies some interesting properties, for instance
$$P*(Q*R) = (P \times Q)*R; \quad (P*R)\times(Q*R) = (P+Q)*R$$
where $P\times Q$ is the usual multiplication of polynomials.
This makes the operation behave very much like exponentiation. I think this can even generalize to the multivariable case. Also, this seems to work for infinite (formal) power series, so perhaps it is relevant to generating functions?
 A: The reason the operation behaves like exponentiation is because you defined it using exponentiation, it doesn't seem to be deeper than that I think. There are many operations on polynomials that behave the way you describe, and unless your definition is inspired by a concrete problem, I don't think it has been studied.
To illustrate what I mean: denote by $\mathbb{N}[x]$ the set of polynomials with natural coefficients (here $0\in\mathbb{N}$). Now let $f:\mathbb{N}[x]\to\mathbb{N}[x]$ be any multiplicative function, and define
$$
Q*_f P:=\prod_i [f^{\circ i}(P)]^{q_i},
$$
where $f^{\circ i}$ means that we apply $f$ exactly $i$ times (i.e. $f^{\circ 3}=f\circ f\circ f$, and by convention $f^{\circ 0}=\operatorname{Id}$). Then $*_f$ satisfies the same properties as your operation. If we chose $f$ to be the function $f(P(x))=P(x+1)$ (which is multiplicative), then $*_f$ is exactly the operation you describe. But we can chose any multiplicative $f$ however wild we would like it to be. To see that such multiplicative functions may be very wild, consider the fact that we can freely choose the value at any irreducible polynomial, and extend the function to all of $\mathbb{N}[x]$. So unless your operation $∗$ has some more interesting features, there are many others like it.
