Integrate product of two spherical harmonics over half-sphere

Question

When calculating quantum-mechanical probability that a particle is located in half-space where $z \ge 0$, I need to evaluate the integral $$ I_{\ell \ell'}^{mm'} = \int\limits_0^{2\pi} \int\limits_0^{\color{red} {\pi/2}} Y_{\ell m}(\vartheta, \varphi) Y_{\ell'm'}^*(\vartheta, \varphi) \sin \vartheta \, \mathrm{d}\vartheta \, \mathrm{d}\varphi \,. $$ I am able to deal with the integration over $\varphi$, leading to $$ I_{\ell \ell'}^{mm'} \propto \delta_{mm'} \int\limits_{\color{red} 0}^{1} P_{\ell}^{m}(x) P_{\ell'}^m(x) \mathrm{d}x \,, $$ but then I get stuck when integrating the remaining pair of associated Legendre polynomials. I could integrate them term-by-term using their closed form, but this results in annoying long formula with summations, binomial coefficients and lots of gamma functions. Is there a more direct approach to get $I_{\ell\ell'}^{mm'}$?

Answer 1

did you find any solution to this question? I have seen results for integrals for only Legendre Polynomials but not for associated Legendre Polynomials. I have a hunch the half-space integrals will be zero for certain combinations of the ALP and perhaps 1/2 of the result for integral over the whole domain [-1,1]. But I am not sure which one exactly.