Recall that, for fixed $j \in Z_+\cup \{0\}$, a spherical harmonic of degree $j$ is the restriction to $S^{n-1}$ of a harmonic polynomial on $R^n$ that is homogeneous of degree $j$. Let $Y_{j,1}\, ,Y_{j,2}\,\ldots, Y_{j,d_{n-1}(j)}$ be an orthonormal basis of the subspace in $L^{2}(S^{n-1})$ of degree $j$ spherical harmonics; here, $\displaystyle{d_{n-1}(j):={n+j-1\choose j}}$.
Consider for $j$ even the following expansion: $$ F=Y_{0,1}+\sum_{j,k}(-1)^{j/2}\frac{\Gamma\left(\frac{n+j}{2}\right)}{\Gamma\left(\frac j2\right)}f_{j,k} Y_{j,k}, $$
How to find all $n$-th derivatives of $F$?