Let $k$ denote a base field. There's a variety of interesting ways to turn polynomials in $k[x]$ into a "dual operator" that acts maps $k$-linearly from $k[[x]]$ to another $k$-module, such as $k[[x]]$ or $k$ itself. Their kernels tend to be pretty interesting.
Question. I really have a lot of questions about these "dualization notions." Two of particular are interest are:
Are there standard notations for these various "duals"?
Is there any theory surrounding special families of dualization notions?
Example 1. Coefficient extractor notation can be linearized to assign to each polynomial $P \in k[x]$ a corresponding $k$-linear transformation $[P] : k[[x]] \rightarrow k$. This has uses in its connection with generating functions in probability theory, and also elsewhere.
Example 2. Assign to each $x^n$ the differential operator $\frac{\partial^n}{\partial^n x}$ and do this linearly. Then $\cos(x)$ and $\sin(x)$ form a basis for the kernel of the "dual" operator of $x^2 + 1$, perhaps suggesting an interesting connection with the complex numbers, which can be defined as the ring $\mathbb{R}[x]/(x^2+1).$ Indeed, all autonomous ordinary differential equations with constant coefficients arise from polynomials in this way.
Example 3. Assign to each $x^n$ the left-shift operator $\lambda_x$ definition here. In this case, the kernel of $x^2 - x - 1$, has, as a basis, the (generating functions) of the Fibonacci sequence and the Lucas sequence. Indeed, all autonomous recurrences with constant coefficients arise from polynomials in this way.
Remark. In light of examples 2 and 3 above, notions of dualization $\delta$ with the property that $\delta(x^n)$ can be written as $ax^{n-1}$ for some $a \in k$ seem pretty important. I'd be interested to know if there's any general theory surrounding them. There's some (fairly trivial, but also kind of interesting) connections with cool stuff involving Hadamard products, like this, arising from facts like $$\frac{\partial}{\partial x}(P) = \lambda_x((1-\log(1-x)) \star P).$$