I have created a ticket on Sage trac for this issue: https://trac.sagemath.org/ticket/25034
As I mention in the ticket, I think that this issue raises a question as to whether the Func_assoc_legendre_P class is correctly defined, given that it now seems to cover both the Ferrers and associated Legendre functions. Given that the intervals on which the functions are defined do not overlap, maybe it makes sense to have a single class for both, but to give it amore generic name? On Thursday, 22 March 2018 20:07:08 UTC, Howard Cohl wrote: > > 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.
