Legendre Polynomial Identity

Question

I was looking for a method to do the following integral: $\int^1_{-1}(1-x^2)\frac{dP_m(x)}{dx}P_n(x)~dx$

I know there should be an explicit representation for the result but I am struggling to work it out.

Answer 1

By Rodrigues' formula and the Legendre DE, you essentially want to find $$ \int_{-1}^1 [(x^2-1)^m]^{(m-1)}[(x^2-1)^n]^{(n)}\,\mathrm{d}x $$ If $n<m$, integrate by parts $n$ times gives $$ \pm\int_{-1}^1 [(x^2-1)^m]^{(m-n-1)}\underbrace{[(x^2-1)^n]^{(2n)}}_{=(2n)!}\,\mathrm{d}x $$ which is $0$ unless $m=n+1$, in which case you want to calculate $$ \int_{-1}^1(x^2-1)^m\,\mathrm{d}x $$ which with $x=\sin\theta$ and $\int_0^{\pi/2}\cos^{2m+1}\theta\,\mathrm{d}\theta=\frac{(2m)!!}{(2m+1)!!}$ nails it. Similar for the case $n\geq m$.