In the last few days I've come across this nice integral involving the associated Legendre functions $$ {\large\int_{-1}^{1}} \frac{P_{\ell}^m\left(u\right) P_{\lambda}^{m}\left(u\right)}{\sqrt{1 - u^2}} du $$ and my research is stuck in it since then. I'm kind expecting the Legendre functions to be orthogonal with respect to this kernel, i.e. the integral should vanish when $\lambda \neq \ell$, but so far neither I was able to prove it myself nor I've found anything related to this. I succeed to prove it when $\ell$ and $\lambda$ have different parities, which is actually quite easy to see by recalling that $P_\ell^m\left(-u\right) = \left(-1\right)^{\ell + m} P_\ell^m\left(u\right)$. Any help with the remaining cases will be very much appreciated.
Just for context, I'm currently studying different representations (set of orthonormal functions) for the electromagnetic field of antennas using spherical harmonics. This particular integral arose when I tried to project $\frac{Y_\ell^m\left(\theta, \phi\right)}{\sin\theta}$ on $Y_\lambda^\mu\left(\theta, \phi\right)$, from which the inner product yields $$ \ \int_{[0, \pi]} \int_{[0, 2\pi]} Y_\ell^m\left(\theta, \phi\right) Y_\lambda^\mu\left(\theta, \phi\right)^* d\phi d\theta $$ and can then be reduced to the original integral.