On the unit sphere, each square-integrable function can be expanded as a linear combination of spherical harmonics :
$$ f(\theta,\phi) = \Sigma_{l=0}^\infty \Sigma_{m=-l}^{+l} f_{lm} Y_{lm} (\theta,\phi)$$
With, reciprocally : $$ f_{lm} = \int_0^{2\pi} d\phi \int_0^{\pi} d\theta sin(\theta) Y_{lm}^* (\theta,\phi) f(\theta,\phi)$$
This means that $$f(\theta,\phi)=\sum_{m=0}^{\infty}\frac{1}{2\pi A_m}Y_{m+1,m}(\theta,\phi)=\sum_{m=0}^{\infty}(-1)^m2^{\frac{m}{2}+1}\sqrt{\frac{\pi\cdot m!}{(2m+3)!!}}\;Y_{m+1,m}(\theta,\phi).$$
How to find $n$-th derivative of function $f(\theta, \phi)$?