Editing Associated Legendre polynomials (section)

==Definition for non-negative integer parameters {{mvar|ℓ}} and {{mvar|m}}==
These functions are denoted <math>P_\ell^{m}(x)</math>, where the superscript indicates the order and not a power of ''P''.  Their most straightforward definition is in terms
of derivatives of ordinary [[Legendre polynomials]] (''m'' ≥ 0)

<math display="block"> P_\ell^{m}(x) = (-1)^m (1-x^2)^{m/2} \frac{d^m}{dx^m} \left( P_\ell(x) \right), </math>

The {{math|(−1)<sup>''m''</sup>}} factor in this formula is known as the [[Spherical harmonics#Condon–Shortley phase|Condon–Shortley phase]].  Some authors omit it.  That the functions described by this equation satisfy the general Legendre differential equation with the indicated values of the parameters ''ℓ'' and ''m'' follows by differentiating ''m'' times the Legendre equation for {{math|''P''<sub>''ℓ''</sub>}}:<ref>{{harvnb|Courant|Hilbert|1953|loc=V, §10}}.</ref>
<math display="block">\left(1-x^2\right) \frac{d^2}{dx^2}P_\ell(x) -2x\frac{d}{dx}P_\ell(x)+ \ell(\ell+1)P_\ell(x) = 0.</math>

Moreover, since by [[Rodrigues' formula]],
<math display="block">P_\ell(x) = \frac{1}{2^\ell\,\ell!} \  \frac{d^\ell}{dx^\ell}\left[(x^2-1)^\ell\right],</math>
the ''P''{{su|b=''ℓ''|p=''m''}} can be expressed in the form
<math display="block">P_\ell^{m}(x) = \frac{(-1)^m}{2^\ell \ell!} (1-x^2)^{m/2}\  \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^\ell.</math>

This equation allows extension of the range of ''m'' to: {{math|−''ℓ'' ≤ ''m'' ≤ ''ℓ''}}. The definitions of {{math|''P''<sub>''ℓ''</sub><sup>±''m''</sup>}}, resulting from this expression by substitution of {{math|±''m''}}, are proportional. Indeed, equate the coefficients of equal powers on the left and right hand side of
<math display="block">\frac{d^{\ell-m}}{dx^{\ell-m}} (x^2-1)^{\ell} = c_{lm} (1-x^2)^m  \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^{\ell},</math>
then it follows that the proportionality constant is
<math display="block">c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} ,</math>
so that
<math display="block">P^{-m}_\ell(x) = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} P^{m}_\ell(x).</math>

===Alternative notations===
The following alternative notations are also used in literature:<ref>{{Abramowitz_Stegun_ref|8|332}}</ref>
<math display="block">P_{\ell m}(x) = (-1)^m P_\ell^{m}(x) </math>

===Closed Form===

Starting from the explicit form provided in the article of [[Legendre polynomials|Legendre Polynomials]]

<math>
P_l(x)=2^l\sum_{k=0}^l x^k\binom{l}{k}\binom{(l+k-1)/2}{l}
</math>

one obtains with the standard rules for <math>m</math>-fold derivatives for powers

<math display="block"> P_l^m(x)=(-1)^{m} \cdot 2^{l} \cdot (1-x^2)^{m/2} \cdot  \sum_{k=m}^l \frac{k!}{(k-m)!}\cdot x^{k-m} \cdot \binom{l}{k} \binom{\frac{l+k-1}{2}}{l} </math>

with simple monomials and the [[Binomial coefficient#Generalization and connection to the binomial series|generalized form of the binomial coefficient]]. The sum effectively extends only over terms where <math>l-k</math> is even, because for odd <math>l-k</math> the binomial factor <math>\binom{(l+k-1)/2}{l}</math> is zero.

Summarizing results of Doha
<ref>{{Cite journal |last=Doha |first=E. H. |year=1991|title=The coefficients of differentiated expansions and derivatives of ultraspherical polynomials |journal=Computers & Mathematics with Applications |volume=21 |issue=2 |pages=115–122 |doi=10.1016/0898-1221(91)90089-M |issn=0898-1221}}</ref>
the expansion of derivatives into Legendre Polynomials defines coefficients <math>\tau</math>

<math>
\frac{d^m}{dx^m}P_l(x)
= \sum_{t=0}^{\lfloor (l-m)/2\rfloor} \tau_{l,m,t} P_{l-m-2t}(x)
,
</math>

where

<math>
\tau_{l,m,t}
=
\epsilon_{l-t}
\frac{l-m-2t+1/2}{2l-2t+1}\frac{(2m)!}{2^mm!}
\binom{2l-2t+1}{2m}
\frac{m}{m+t}\binom{m+t}{t}
\frac{1}{\binom{l-t}{m}}
,
</math>

and where

<math>
\epsilon_q\equiv \begin{cases}
1, & q=0;\\
2, & q\ge 1
\end{cases}
</math>

is the Neumann factor.