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?

  • $\begingroup$ 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. $\endgroup$
    – GertVdE
    Sep 13, 2019 at 9:23
  • $\begingroup$ I've edited my question to include definitions of the swSH and the integrals I'd like to compute. $\endgroup$ Sep 13, 2019 at 14:02

1 Answer 1


If I'm not mistaken, these spin-weighted spherical harmonic functions are equivalent to the Generalized Associated Legendre functions. In the reference work Virchenko & Fedotova, you can find recurrence relations in chapter 5, p32. Chapter 15, p. 96, of the same reference discusses integral transforms with the Generalized Associated Legendre functions.


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