Patrick, this is a great paper. It shows the same care you apply to maintaining gsl. The writing is also very clear, and I love the table of acronyms :-). Do you plan to submit for publication in a numerical analysis journal, or submit to the arXiv?
The only previous quasi-report had been the ongoing design document that I kept going with James and Brian in the early years, but it is a working design doc, not anything of the scope you have shown here. Tiny suggestions: alignment of equation 36 on page 5: "l >= 1" could be moved quite a bit to the left. Maybe an extra & to make it match the start of the l >= 1, m > 0. You might also want to also cite the reference manual (as we ask people to do when they use gsl :-) ). A recent bibtex skeleton on that is the 2019 Network Theory edition. @book{gslteam2019gnuscientificlibrary, title={GNU Scientific Library Reference Manual}, author={Galassi, Mark and Davies, Jim and Theiler, James and Gough, Brian and Jungman, Gerard and Alken, Patrick and Booth, Michael and Rossi, Fabrice and Ulerich, Rhys}, year={2019}, isbn={9780954612078}, publisher={Network Theory Limited} }