Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic?



Partial derivatives of spherical harmonics
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):
$$\frac{d^k}{dx^k}P\_l^m(x)=\frac{(-1)^m}{2^ll!}\sum\limits\_{n=0}^k\binom{k}{n}\left\{\left[\sum\limits\_{i=[\frac{n+1}{2}]}^n\hat A\_n^ix^{2i-n}(-2)^i(1-x^2)^{\frac{m}{2}-i}\prod\_{j=0}^{i-1}\left(\frac{m}{2}-j\right)\right]\,\left[\sum\limits\_{i=[\frac{l+m+k-n+1}{2}]}^{l+m+k-n}\hat A\_{l+m+k-n}^ix^{2i-l-m-k+n}\,2^i(x^2-1)^{l-i}\prod\_{j=0}^{i-1}\left(l-j\right)\right ]\right\},$$
where $$\hat A\_k^i=\sum\_{j=0}^i\frac{(-1)^{i-j}(2j-k+1)\_k}{2^ij!(i-j)!},$$ $(x)\_k$ 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 Condon-Shortley phase $(-1)^m$. 


