A calculation problem on Conformal Field Theory This problem is on Di Francesco's book I. It's exercise 7.1:
Calculate the norm of the following vector, where $\lvert h\rangle$ is the state of highest weight.
$$L_{-1}^n\lvert h\rangle$$  
I have just tried to use the commutation relations of the $L$ operators and the fact that $L_1$ acts on $\lvert h\rangle$ is $0$. But as the calculation goes on, things began to be troublesome. I just found them too complicated.
Commutation relations:
$$[L_n,L_m]=(n-m)L_{n+m}+\frac{c}{12}\delta_{n+m,0}n(n^2-1) $$
It is in fact asking us to calculate: 
$\langle h|L_1^nL_{-1}^n|h\rangle    $. 
And we have the following relations: 
$$\langle h\rvert L_{-1}=0 \quad\mbox{and}\quad L_1\lvert h\rangle=0.        $$     
 A: I will outline the steps for you and you can fill in the details:
i)Calculate $[L_1, L_{-1}]=\cdots=2L_0$.
ii) Using the fact that $L_1 L_{-1}=[L_1, L_{-1}]+L_{-1} L_{1}$ and that $L_0|h\rangle=h|h\rangle $, try to calculate $\langle h|L_1 L_{-1}|h\rangle =\cdots\stackrel{?}{=}2h$.
iii) Calculate the following quantity that will (probably) be useful later:
$$\langle   h|(L_1 L_{-1})^n|h\rangle   =\cdots\stackrel{?}{=}(2h)^n$$
iv) (The hard part) We have done the case for $n=1$ for the quantity $\langle h |L_1^n L_{-1}^n |h\rangle$ in i), but you will also need to do the case for $n=2$ and $n=3$ by hand, using the formulae in i) and ii) above and from that, try to deduce an inductive formula for the general case. I will have to admit though that deducing the general form (which then can be proved by induction) from a few cases might be quite hard!
Hope this helps!
A: Hint:
$$\langle h|L_1^n L_{-1}^n|h\rangle 
~=~ \sum_{i=0}^{n-1}  \langle h|L_1^{n-1} L_{-1}^{n-1-i}[L_1,L_{-1}]L_{-1}^i|h\rangle
~=~ \ldots $$
$$~=~ 2\sum_{i=0}^{n-1}  \langle h|L_1^{n-1} L_{-1}^{n-1}(L_0-i)|h\rangle 
~=~\ldots $$
$$~=~  n(2h-(n-1)) \langle h|L_1^{n-1} L_{-1}^{n-1}|h\rangle 
~=~\ldots 
~=~n!\prod_{i=0}^{n-1} (2h-i).$$
