Consider the algebra of operators acting on wavefunctions ($L^2(\mathbb{R})$) generated by $X$ and $P = -i\hbar (\partial/\partial x)$. For some operator $A$ in this algebra, or possibly in a restricted class of nice operators, is there a canonical Taylor expansion in terms of $X$ and $P$? In other words, is there a canonical way to write $A$ as an infinite (scaler-weighted) sum of finite products of $X$ and $P$,
$$A = a_0 I + a_{\mathrm{x}}X + a_{\mathrm{p}}P + a_{\mathrm{xx}}X^2 + a_{\mathrm{xp}}XP + a_{\mathrm{px}}PX + a_{\mathrm{pp}}P^2 + \cdots$$
where the scalar weights $a_i$ can be computed with derivative-like commutators, e.g., $$a_{\mathrm{x}} = \frac{\partial A }{\partial X} := -i[A,P]$$
or something like that?
One possibility is to consider the Wigner transform $W[A]$ of $A$, take the normal Taylor expansion of $W[A]$ in terms of phase-space scalars $x$ and $p$, and then do the Weyl transform (i.e, inverse Wigner transform) on each term in the series. That should probably converge (in which norm?) for at least some restricted set of well-behaved operators $A$. But the Wigner transform is well-known to be just one choice of phase-space transform associated with a particular operator ordering; presumably this series would then change if a phase-space transform associated with a different operator ordering were used. Would the series still sum to $A$ regardless?
Are Bopp operators relevant here?
And is there a nice way to do this without the Wigner transform? In particular, is there something like Taylor's theorem that can be expressed purely in terms of commutators and a series?
EDIT: An $n$-th degree polynomial in operators $X$ and $P$ will correspond to an $n$-th degree polynomial in classical variables $x$ and $p$ for all choices of phase-space transform because the degree of a polynomial in $X$ and $P$ is not changed by re-ordering (i.e., commuting the operators around). (Both sides of $XP = PX + i\hbar$ are second degree.) Therefore, if the $n$-th degree Taylor approximation to an operator is defined with a phase-space transform, it is independent of the particular phase-space transform. (This point was emphasized to me by Dan Ranard.)
