Add polynomials for spin-weighted harmonics? about associatedlegendrepolynomials.jl HOT 3 OPEN

Great! This looks like a great package and it would be fantastic to have it support spin-weighted transforms.

Yes the Legendre polynomials are just a special/particular case of Jacobi polynomials, and all the identities are formulas that people use for them generally apply to Jacobi polynomials more broadly, just with modified coefficients. In general the "spin weight" $s$ in the spin weighted spherical harmonics tells you how different the Jacobi $(a,b)$ parameters are from those for the regular associated Legendre functions. In theory the spin weight can be arbitrarily large, in which case the polynomials behave quite differently. But in practice, $s$ is typically just 0 or 1 or 2 (corresponding to the rank of the tensor field you're interested in), so the recursions and envelopes, etc., behave pretty similarly to the regular associated Legendre functions.

There are many different recurrence formulas that can be used, but the easiest place to start is probably to just generalize the current recurrences to work for general spin weights. I'll need to take a closer look to understand how over/underflow in the recursions is being handled here, but the typical strategies should also work for moderate spin weights.

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@jmert do you have an opinion on this ☝🏼 ? It shouldn't be much work on your side, but at the moment it sounds easier to make this part of AssociatedLegendrePolynomials than to branch this idea off somehow.

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Yes, I'm generically interested in supporting enough computation that one could calculate the spin-weighted spherical harmonics. I've never actually done that, though, because the reference I've seen before always resort to calculate the spin-weighted spherical harmonic in terms of the Wigner d-matrices, which end up being a completely different calculation.

I'll put reviewing the paper you referenced on my TODO list to review more closely. On a quick glance, it looks like the function P used in the equations you referenced are for the Jacobi polynomials rather than the Legendre polynomials, but wikipedia says they're related.

If there's a way to "boost" by ±1 spin with some kind of simple relation, I think that could be accommodated with relative ease. The Jacobi recurrence relations have a mixture of coefficients, so maybe there's a not-too-complicated way to mix the results of the spin-0 terms to iteratively generate different spins. (Unless we're very lucky, I'd expect that route will be neither the most performant nor the most numerically accurate, but the proof-of-concept would be a big step on its own worth doing.)

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