Lemma: Taylor reduction of coefficients at a delta derivative

Let $c$ be smooth in $(q,\xi )$ near $q=0$ and $j\ge 0$. Then, as distributions in $q$ depending smoothly on $\xi$,

In particular $q{\delta }^{(j)}(q)=-j{\delta }^{(j-1)}(q)$ and $q\delta (q)=0$.

Proof. $⟨c{\delta }^{(j)},\varphi ⟩=(-1{)}^j{{\partial }_q}^j(c\varphi )(0)={\sum }_i(\genfrac{}{}{0ex}{}{j}{i})(-1{)}^{j-i}{{\partial }_q}^{j-i}c(0)(-1{)}^i{{\partial }_q}^i\varphi (0)$ by Leibniz, and $(-1{)}^i{{\partial }_q}^i\varphi (0)=⟨{\delta }^{(i)},\varphi ⟩$.