The Ferrers functions are defined on the real segment (-1,1). The associated Legendre functions are in general defined on the Complex plane except for the ray (-\infty,1].
Typically Ferrers functions are written with argument x=\cos\theta, |x|<1 and associated Legendre functions are written with argument z=\cosh\eta, z>1, or something to that effect. One can obtain the Ferrers functions through a limiting process from the associated Legendre functions by taking an appropriately weighted average of x+i0, and x-i0, ,where x\in(-1,1). If you read the DLMF chapter on associated Legendre functions, then all this is explained quite clearly. The DLMF uses different symbols for the associated Legendre functions vs. the Ferrers functions. An italic P and Q for the associated Legendre functions and a sans serif P and Q for the Ferrers functions. Their definitions and properties are all given in that DLMF chapter. In Abramowitz and Stegun, the same symbol is used for both functions, but you can tell they are using which by the argument z (for associated Legendre) vs. x (for Ferrers functions). If you have any other questions, please free to email me hcohl -at- nist dot gov. 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.
