Editing Associated Legendre polynomials (section)

===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.