Expectation value of descendant fields I'm trying to calculate the following quantity:
$ \left<(L_{-1}\phi)(w_1)(L_{-1}\phi)(w_2) \ldots (L_{-1}\phi)(w_N) \right>$
where $\phi(w_i)$ is a primary operator and $L_{-1}$ is the Virasoro generator of translations. Guided by equations (6.149), (6.155) and (6.161) in Di Francesco, Mathieu and Senechal, and after calculating a simple contour integral, the result came out to be:
$\sum_{i=1}^{i=N} \partial^2_{i} \left< \phi(w_1) \dots \phi(w_N) \right>$
I was specially guided by the OPE between the EM tensor and $\partial \phi$:
$T(z)\partial \phi(w)= \frac{2h\phi(w)}{(z-w)^3} +\frac{(h+1) \partial\phi(w)}{(z-w)^2} + \frac{\partial^2\phi(w)}{(z-w)}$
from which I was able to calculate the $\left< T(z)(L_{-1}\phi)(w_1)(L_{-1}\phi)(w_2) \ldots (L_{-1}\phi)(w_N)\right>$ and use it to obtain the result mentioned above. I'm not quite confident, however, this is 100% correct.  
The reason I'm not quite confident of the validity of my result is that if I attempt to compute the correlation function as given above, I get mixed derivative terms, specifically $\partial_{w_i}\partial_{w_{i+j}}$ acting on the correlation function of the primary fields. I'm not sure, however, if I should get those terms in the final result. I only want to be left with $\sum_{i=1}^{i=N} \partial^2_{i}$ operator at the end. 
Is there anything basic I'm missing? 
 A: Already by dimensional analysis you can see that your answer is wrong. If $\phi$ has weight $h$, then the correlator without $L_{-1}$ operators has total weight $N\cdot h$, and the one with $L_{-1}$ operators should have total weight $N \cdot (h+1)$. Yet your formula has total weight $N \cdot (h+2)$.  In fact you're way way overcomplicating this. Just look at formula (6.136) of the Yellow Book and you'll get the correct result straight away.
Edit
I think you're confused at a basic level about the notation
$$(L_{-1} \phi)(w).$$
You seem to think that in a correlation function
$$
\langle (L_{-1} \phi)(w) \psi(z) \ldots \rangle
$$
you somehow get many crossterms that involve $L_{-1}$ acting on $\psi$ etc. However, by definition we just have
$$(L_{-1} \phi)(w) = [L_{-1},\phi(w)]$$
and likewise
$$(L_{n} L_{m} \phi)(w) = [L_{n},[L_{m},\phi(w)]]$$
and so forth. If you wish, you can rephrase the above definition in terms of integrating $T(z)$ around a small contour circulating $\phi(w)$. But it's not necessary, if you just use the formula from Di Francesco expressing $[L_{-1},\phi(w)]= \pm \partial_w \phi(w)$ (I forget about the sign).
A: I think I figured it out. The $n$-point correlation function of level-2 null field obeys a 2nd-order differential equation:
$\frac{3}{2(2h+1)} \partial^2_{w_1} \left<\phi(w_1) \dots \phi(w_N) \right> = \left<L_{-2}\phi(w_1) \dots \phi(w_N) \right> = \sum^N_{j=1} \left(\frac{h_j}{(z-z_j)^2} + \frac{\partial_j}{z-z_j} \right) \left<\phi(w_1) \dots \phi(w_N) \right>$.
If you then sum over ALL null fields, you get:
$\sum^N_{i=1}\partial^2_{w_i} \left< \phi(w_1) \dots \phi(w_N) \right> = \frac{2(2h+1)}{3} \sum^N_{i=1} \left( \sum^N_{j\neq i} \left(\frac{h_j}{(z_i-z_j)^2} + \frac{\partial_j}{z_i-z_j} \right)\right) \left< \phi(w_1) \dots \phi(w_N) \right>$
which is the operator I was looking for to start with.
