## Spherical Harmonics decomposition

What is the equivalent of the delay theorem when dealing with Spherical Harmonics ?

Let's define:

$c_f(l,m) = \int_0^\pi \int_0^{2\pi} f(\theta,\phi) Y_l^m(\theta,\phi) \sin(\theta) d\theta d\phi$

I apply a rotation R= that changes $(\theta,\phi) \rightarrow (\theta_R,\phi_R)$ and I define $g(\theta,\phi) = f(\theta_R,\phi_R)$

Is there a way to get:

$c_g(l,m) = \gamma(\theta,\theta_R,\phi,\phi_R) c_f(l,m)$

$P_l(cos(\Theta)) = \sum_{m=-l}^l Y_l^m(\theta,\phi)^{*} Y_l^m(\theta_R,\phi_R)$
with $\Theta$ the angle between $(\theta,\phi)$ and $(\theta_R,\phi_R)$