Studying the book "Physics of Atoms and Molecules" by B.H Brandsden and C.J. Joachain I stumbled upon this given result (without any proof):
$$ Y_{m_1}^{\ell_1}Y_{m_2}^{\ell_2}=\sum_{\ell=|\ell_1-\ell_2|}^{\ell_1+\ell_2}\sum_{m=-\ell}^{\ell}\sqrt{\dfrac{(2\ell_1+1)(2\ell_2+1)}{4\pi(2\ell+1)}}\langle\ell_1,\ell_2, 0,0|\ell, 0\rangle\langle\ell_1,\ell_2, m_1, m_2|\ell, m\rangle Y_{m}^{\ell} $$
- $\ell$ is the quantum number associated with the operator $L^2$,
- $m$ is the quantum number associated with the operator $L_z$,
- $Y_{m}^{\ell}$ is the spherical harmonic associated with the quantum numbers $\ell$ and $m$,
- $\langle \ell',\ell'',m_{\ell'},m_{\ell''}|\mathcal{L},\mathcal{M}\rangle$ is shorthand notation for the Clebsh-Gordan coefficient $\langle \ell',\ell'',m_{\ell'},m_{\ell''}|\ell',\ell'',\mathcal{L},\mathcal{M}\rangle$.
I tried to get this expression a proof but with no success. Does anybody know how to prove it or some literature where to find the proof?
It is (3.8.72) and background sections (3.6, 3.8) of Modern Quantum Mechanics, by Sakurai & Napolitano (2010 Addison Wesley), ISBN-13: 978-0805382914 .
The point is $$ \langle \theta, \phi| l,m\rangle= Y^l_m(\theta,\phi) \tag{3.6.23} $$ and you apply these to the Clebsch series for rotation matrix representations, directly defined through spherical harmonics, as demonstrated in (3.8.72) of that text, well-worth reviewing.