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.
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