<s>[INST] Integration by parts with Legendre Functions I need help deriving 
$\int_{-l}^l [P_l^m(x)]^2 = \frac{2}{2l+1} \frac{(l+m)!}{(l-m)!}$
for the associated Legendre functions
I am supposed to use 
$P_l^m(x) = (-1)^{-m}\int_{-l}^l \frac{(1-x^2)^{\frac{m}{2}}}{2^ll!}\frac{d^{l+m}}{dx^{l+m}}(x^2-1)^l$
and
$P_l^m(x) = \frac{1}{2^ll!}^{-m}\int_{-l}^l(1-x^2)^{\frac{m}{2}}\frac{d^{l-m}}{dx^{l-m}}(x^2-1)^l$
I tried to multiply them together, ignoring the terms without x and got
$\int_{-l}^l\frac{d^{l+m}}{dx^{l+m}}(x^2-1)^l\frac{d^{l-m}}{dx^{l-m}}(x^2-1)^l$  
I'm supposed to integrate by parts m-times to get both derivates to be $\frac{d^{l}}{dx^{l}}$
This is where I think I'm making my mistake, after doing the integration by parts and getting the same number of derivatives for each term, $\frac{d^{l}}{dx^{l}}$
I get
$(-1)^m\int_{-l}^l(\frac{d^{l}}{dx^{l}}(x^2-1)^l)^2$
I guess the (-1)^m will cancel with part of the original equations but I don't see how to get to the final conclusion 
$\frac{2}{2l+1} \frac{(l+m)!}{(l-m)!}$
Any guidance or help would be greatly appreciated, Thanks a lot!
 [/INST] The Rodrigues' formulae you supplied have gone somewhat astray; they should be
$P_n^m(x)=\frac{1}{2^n n!} (x^2-1)^{m⁄2} [(x^2-1)^n ]^{(n+m)}$
and
$P_n^{-m}(x)=\frac{1}{2^n n!} (x^2-1)^{-m⁄2} [(x^2-1)^n ]^{(n-m)}$
Let us derive the value of $‖P_n^m‖^2$ for positive m:
$\newcommand{\partial}[1]{\left[#1\right]}$
$\newcommand{\bracket}[1]{\left(#1\right)}$
\begin{equation}
\begin{split}
‖P_n^m‖^2&=\int_{-1}^{+1} [P_n^m]^2 dx
\\ &=\frac{1}{2^{2n} n!^2} \int_{-1}^{+1}(x^2-1)^m\cdot [(x^2-1)^n ]^{(n+m)}\cdot [(x^2-1)^n ]^{(n+m)}dx
\end{split}
\end{equation}
Making use of
\begin{equation}
\begin{split}
[(x^2-1)^n]^{(n-m)}=\frac{(n-m)!}{(n+m)!} (-1)^m (1-x^2)^m [(x^2-1)^n]^{(n+m)}
\end{split}
\end{equation}
we get
\begin{equation}
\begin{split}
‖P_n^m‖^2&=\frac{1}{2^{2n} n!^2} \int_{-1}^{+1}(-1)^m\frac{(n+m)!}{(n-m)!}[(x^2-1)^n ]^{(n+m)}\cdot [(x^2-1)^n]^{(n-m)}dx
\end{split}
\end{equation}
Partial integration $n-m$ times then gives
\begin{equation}
\begin{split}
‖P_n^m‖^2&=\frac{1}{2^{2n} n!^2}(-1)^m\frac{(n+m)!}{(n-m)!} \cdot (-1)^{n-m}\int_{-1}^{+1}[(x^2-1)^n ]^{(2n)}\cdot [(x^2-1)^n]^{(0)}dx
\\ &=\frac{1}{2^{2n} n!^2}(-1)^n\frac{(n+m)!}{(n-m)!} \int_{-1}^{+1}(2n)!(x^2-1)^ndx
\\ &=\frac{1}{2^{2n} n!^2}(-1)^n\frac{(n+m)!}{(n-m)!}(2n)!(-1)^n\frac{n!^2}{(2n)!}\cdot \frac{2^{2n+1}}{2n+1}
\\ &=\frac{(n+m)!}{(n-m)!} \frac{2}{2n+1}
\end{split}
\end{equation}
And it is easy (using $P_n^{-m}(x) = (-1)^m \frac{(n-m)!}{(n+m)!} P_n^m(x)$) to derive that
\begin{equation}
\begin{split}
‖P_n^{-m}‖^2&=\frac{(n-m)!^2}{(n+m)!^2} ‖P_n^m‖^2
\\ &=\frac{(n-m)!}{(n+m)!} \frac{2}{2n+1}
\end{split}
\end{equation}
Whence
\begin{equation}
\begin{split}
‖P_n^{m}‖^2&=\frac{(n+|m|)!}{(n-|m|)!}\frac{2}{2n+1}
\end{split}
\end{equation}
For negative and positive $m$, $-n \le m \le n$.
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