Recursion relations of Associated Legendre Polynomials with Schmidt Semi-normalization

Can someone please provides me with the recursion relations of Associated Legendre Polynomials when using Schmidt quasi-normalization? I need that in the context of Geomagnetism to obtain the Spherical Harmonics coefficients g and h.

FYI, the usual reference relied on for the equations used in geomagnetism is:

• Langel, R. A. "Chapter four: Main field." Geomagnetism, edited by JA Jacobs (1987).

If it wasn't for lockdown I'd have copied from the book locked in my office to make sure I got it right...but until someone corrects me...

The Schmidt quasi/semi-normalisations for the associated Legendre polynomials are given by these recursions, in the form you'd actually use for calculations typically:

$S_{0,0}=1$

$S_{n,0}=S_{n-1,0}\left&space;(&space;\frac{2n-1}{n}&space;\right&space;)$

$S_{n,m}=S_{n,m-1}&space;\sqrt{&space;\frac{(n-m+1)(\delta_m^1+1)}{n+m}&space;}$

Note that if you use Schmidt normalisation of Legendre polynomials from a given software language to do calculations, you do not want to include the Condon-Shortley phase factor of $(-1)^{m}$. So check if it is included in the Legendre polynomial and/or normalisation formula.

Edit: Some links to open source geomagnetism compatible software for calculating Schmidt normalised associated Legendre polynomials.

Python: ChaosMagPy

Python/Fortran-95: SHTools

• That exactly what I was looking for, except I couldn't find the mentioned reference anywhere on the web, so thanks for the hint. However, I really need a confirmation about the formula correctness including the derivatives too. Aug 24 '20 at 16:06
• I don't think there is a digital version of Langel (1987) anywhere online, but I just found I could view enough of the following reference through Google Books to confirm the equations: Wertz, J. R. "Spacecraft Attitude Determination and Control", (1978). Use the "Search inside" option and search for "Appendix H", you can see the first few pages about Schmidt normalisation.
– WJB
Aug 24 '20 at 16:27
• That was useful, thanks again! So, the factors S are used with Gauss functions to obtain Schmidt functions, where: P_Schmidt = S * P_Gauss. But, it's still a long way to go with. I would love to compute directly the P_Schmidt functions since we know that there's recursion relations for them out there. Aug 24 '20 at 17:33
• Are you actually after the final recursive formula for the Schmidt P_n^m(cos(theta)) functions themselves, or are you ultimately trying to write code to calculate these values for geomag problems? I can add links to open source code in a few languages much more easily than I can work back and type up the formula working from those codes right now!
– WJB
Aug 24 '20 at 19:15
• Exactly I'm trying to implement a code to calculate those values, any open source code is welcomed! but I can't just take the code without the formula since I need to document that. Aug 24 '20 at 21:22