Is there any closed form formula (or some procedure) to find all $n$th partial derivatives of a spherical harmonic?
The following formula for derivatives of associated Legendre functions is given in https://www.sciencedirect.com/science/article/pii/S0377042709004385 (New formulae for higher order derivatives and applications, by R.M. Slevinsky and H. Safouhi): $$\frac{d^k}{dx^k}P_l^m(x)=\frac{(1)^m}{2^ll!}\sum\limits_{n=0}^k\binom{k}{n}\left\{\left[\sum\limits_{i=[\frac{n+1}{2}]}^n\hat A_n^ix^{2in}(2)^i(1x^2)^{\frac{m}{2}i}\prod_{j=0}^{i1}\left(\frac{m}{2}j\right)\right]\,\left[\sum\limits_{i=[\frac{l+m+kn+1}{2}]}^{l+m+kn}\hat A_{l+m+kn}^ix^{2ilmk+n}\,2^i(x^21)^{li}\prod_{j=0}^{i1}\left(lj\right)\right ]\right\},$$ where $$\hat A_k^i=\sum_{j=0}^i\frac{(1)^{ij}(2jk+1)_k}{2^ij!(ij)!},$$ $(x)_k$ being the Pochhammer symbol.
See also https://www.sciencedirect.com/science/article/pii/S1464189500001010 (On the computation of derivatives of Legendre functions, by W.Bosch) for numerically stable recursive calculation of derivatives. Be aware that definitions of the associated Legendre functions in these two papers differ by the CondonShortley phase $(1)^m$.

$\begingroup$ Thank you very much for the formulas and papers. I have a quick question: How this formula would work if $k=1$? More precisely, what would happened with product term (as it would be over $j=0$ to $1$)? Thank you. $\endgroup$ – user124297 Jun 4 '18 at 1:00

$\begingroup$ If $k=1$, $i$ in the first product will be either 0 or 1. As these product terms are related to the identities (37) from the paper, it follows from this identities that they must be supplemented by the convention (when $i=0$) $$\prod_{j=0}^{1}\left(\frac{m}{2}j\right)=1$$ and $$\prod_{j=0}^{1}\left(lj\right)=1.$$ If $i=1$, then $$\prod_{j=0}^{0}\left(\frac{m}{2}j\right)=\frac{m}{2}$$ and $$\prod_{j=0}^{0}\left(lj\right)=l.$$ $\endgroup$ – Zurab Silagadze Jun 4 '18 at 6:10

$\begingroup$ Spherical harmonics are a two variable functions. I don't see any partial derivatives in the above. $\endgroup$ – user64494 Jun 4 '18 at 7:22

$\begingroup$ Spherical harmonics are functions of $\phi$ and $x=\cos{\theta}$ of the form $$Y_l^m(\theta,\phi)=\sqrt{\frac{(2l+1)(lm)!}{4\pi(l+m)!}}P_l^m(\cos{\theta})e^{im\phi}.$$ Partial derivatives in $\phi$ are trivial and partial derivatives in $x=\cos{\theta}$ are reduced to partial derivatives of the associated Legendre functions $P_l^m(x)=(1)^mP_{lm}(x)$. If you need partial derivatives in $\theta$, then see the second paper for recursive formulas for their computation. $\endgroup$ – Zurab Silagadze Jun 5 '18 at 4:23