The associated Legendre polynomials are known to be orthogonal in the sense that $$ \int_{-1}^{1}P_{k}^{m}(x)P_{l}^{m}(x)dx=\frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{k,l} $$ This is intricately linked to the orthogonality of spherical harmonics over a sphere. I would like to know the value of $$ \int_{0}^{1}P_{k}^{m}(x)P_{l}^{m}(x)dx $$ because, in my problem, I am only interested in integrating over a hemisphere. I believe a closed-form expression exists because Mathematica will perform the integrals for individual triplets of integers $k$, $l$ and $m$. However, it won't do the general case. I've tried using Rodriguez's formula and integrating by parts, but I'm getting in a mess doing so. I can see there isn't going to be orthogonality, though.
Thanks in advance for any help.