# Algorithm for evaluation of spin-weighted spherical harmonics

Is there an algorithm to evaluate spin-weighted spherical harmonics (swSH) at arbitrary points on the sphere? In particular I am looking for, e.g. a recursion relation to evaluate the "spin weighted Legendre functions" $${}_sP^m_l$$, which satisfy the follow Sturm-Liouville equation

$$\frac{d}{dx}\left(\left(1-x^2\right)\frac{d{}_sP^m_l(x)}{dx}\right)+\left(s-\frac{(m-sx)^2}{1-x^2}\right){}_sP^m_l(x)=-(l-s)(l+s+1){}_sP^m_l(x)$$

over the interval $$x\in[-1,1]$$. Classic papers on swSH inclde this and this (note I took the above equation from 2.5 of this paper substituting $$x\equiv-\mathrm{cos}\vartheta$$). In particular, I'd like to evaluate integrals such as

$$\int_{-1}^1dx {}_sP^m_l(x)f(x)$$

(where f is some function I do know how to evaluate) using Gaussian quadrature.

For context: there exist algorithms to compute discrete spherical harmonic transforms, which require the evaluation of the associated Legendre functions (note $$P^m_l={}_0P^m_l$$). Is there something similar for swSH?

• welcome to SciComp. It would help if you would provide the exact function in the question body (not only a link to a paper). For the quadrature: are it only these functions you need to integrate or are they part of the integrand (since you speak about an associated integral transform). Again, a bit more information (equations?) in the body of the question would help. Sep 13 '19 at 9:23
• I've edited my question to include definitions of the swSH and the integrals I'd like to compute. Sep 13 '19 at 14:02