Oh, by the way, Wolfram Mathworld is just completely wrong on this page you referenced. There is a huge difference between the two functions. Also, there is no such thing as an associated Legendre polynomial. There is a Legendre polynomials, but if you take the degrees and orders to be integers for associated Legendre functions, they don't always end up being polynomials, as you can see from the formulas which I showed earlier in this thread.
On Thursday, March 22, 2018 at 8:49:16 AM UTC-7, James Womack wrote: > > Thanks. If that is the case, then presumably this *is* a bug in Sage Math > and Func_assoc_legendre_P should distinguish the special cases for n == m > when x > 1 or x < 1 when evaluating associated Legendre polynomials. > > Would you be able to clarify the distinction between Ferrers functions of > the first kind and associated Legendre functions for a non-expert? Wolfram > Mathworld seems to suggest that they are the same: > http://mathworld.wolfram.com/FerrersFunction.html > > On Thursday, 22 March 2018 15:23:03 UTC, Howard Cohl wrote: >> >> >> >> On Thursday, March 22, 2018 at 3:25:06 AM UTC-7, Samuel Lelievre wrote: >>> >>> Ralf wrote: >>> > Thanks, >>> > P.S. Still someone should contact DLMF with the right arguments. >>> >>> I just emailed them with cc to sage-devel. >>> >> >> There's nothing wrong with the formula. The Legendre function in the DLMF >> is for arguments greater than 1, and is not valid for arguments less than >> one. For arguments less than one the correct formula is >> >> P_m^m(x)=(-1)^m (2m)!/(2^m m!) (1-x^2)^(m/2). >> >> Both of these are easy to derive using the well-known formulae for >> P_\nu^{-\nu} and {\sf P}_\nu^{-\nu} and the connection formulas which >> relate P_{\nu}^{-m} to P_{\nu}^m, and for Ferrers functions. See >> http://dlmf.nist.gov/14.5.iv <https://dlmf.nist.gov/14.5.iv> and >> https://dlmf.nist.gov/14.9. >> Where P is the associated Legendre function of the first kind, and {\sf >> P} is the Ferrers function of the first kind. >> > -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
