Partial derivatives of spherical harmonics Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic?
 A: 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 $(-1)^m$. 
