Vrftjkry

Let following be the partition function in infinitely many variables $x_i$ the linear coordinates on the vector space $V$.
$$
mathcal{Z}=expBig(sum_{substack{ggeq 0\ngeq 1}}frac{h^{g-1}}{n!}sum_{i_1 ,ldots , i_n}F_{g,n}(i_1 ,ldots , i_n)x_{i_1}cdots x_{i_n} Big)
$$
where $F_{g,n}(i_1 ,ldots , i_n)$ are scalars.

Let $L_{i}in W_{V}^{h}$ are differential equation that annihilates $mathcal{Z}$ that is
$$ L_{i} mathcal{Z}=0.$$
where $W_{V}^{h}=mathbb{C}[hbar]langle (x_{i},partial_i )rangle/langle [partial_i ,x_i]=hbarrangle$

Let
$$
mathcal{K}=expBig(sum_{substack{ggeq 0\}}h^{g-1}sum_{i_1 }F_{g,1}(i_1)x_{i_1} Big)
$$
where $F_{g,1}(i_1)$ are scalars and are the same scalar appearing in $mathcal{Z}$

I was wondering how I can derive $mathcal{K}$ from $mathcal{Z}$ by some operation. The one I had in mind is the following, I choose a parameter $s$ and replace

$$x_irightarrow sx_i.$$
Then $$frac{partial}{partial s}mathcal{Z}|_{s=0}= mathcal{K}$$
(I don't know if I am making any mistake above calculation)

Also, I wonder what can we say about relation of the annihilator of $mathcal{K}$ to $L_i$.