Let $F:\mathbb R ^n\to\mathbb R$ be a function that has a Taylor expansion, then it can be written (expanded at $a$) as $$ F(x)=\sum_{\alpha} \frac{(x_1 - a_1)^{\alpha_1}\dots(x_n - a_n)^{\alpha_n}}{\alpha_1 !\dots \alpha_n !}\frac{\partial^{|\alpha|}F(a)}{\partial_1^{\alpha_1}\dots \partial_n^{\alpha_n}} $$ where multi-index is used, i.e. $\alpha = (\alpha_1,\dots,\alpha_n) \in \mathbb N_0^n$ and $|\alpha| = \alpha_1 + \dots + \alpha_n$.
Now we take $F:\mathbb R^2 \to \mathbb R$ and the quantum mechanical operators $q$ and $p$ which do not commute, $[q,p] = i\hbar$, then (expanding at $a = 0$) there will be an ambiguity in the order (arrangement) of the powers of $q$ and $p$, since in general, for $m,n,j,k \in \mathbb N$, $$ q^m p^n q^j p^k \neq q^{m+j} p^{n+k} \qquad \iff \qquad [q^m p^n q^j p^k,q^{m+j} p^{n+k}] \neq 0 $$ So which one should be in the expansion ? or should they be symmetrised ?
Another question is how do we take partial derivatives of such function of operators. For example of the product $q^m p^n q^j p^k $, \begin{align} \partial_p(q^m p^n q^j p^k) &= n (q^m p^{n-1} q^j p^k) \quad ?\\ &=j (q^m p^n q^{j-1} p^k) \quad ?\\ &= \frac{1}{2}(n (q^m p^{n-1} q^j p^k) + j (q^m p^n q^{j-1} p^k)) \quad ?\\ & = ??? \end{align}
