View source for Associated Legendre polynomials

{{Short description|Canonical solutions of the general Legendre equation}}
In [[mathematics]], the '''associated Legendre polynomials''' are the canonical solutions of the '''general Legendre equation'''

<math display="block">\left(1 - x^2\right) \frac{d^2}{d x^2} P_\ell^m(x) - 2 x \frac{d}{d x} P_\ell^m(x) + \left[ \ell (\ell + 1) - \frac{m^2}{1 - x^2} \right] P_\ell^m(x) = 0,</math>

or equivalently

<math display="block">\frac{d}{d x} \left[ \left(1 - x^2\right) \frac{d}{d x} P_\ell^m(x) \right] + \left[ \ell (\ell + 1) - \frac{m^2}{1 - x^2} \right] P_\ell^m(x) = 0,</math>

where the indices ''ℓ'' and ''m'' (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on {{closed-closed|−1, 1}} only if ''ℓ'' and ''m'' are integers with 0 ≤ ''m'' ≤ ''ℓ'', or with trivially equivalent negative values. When in addition ''m'' is even, the function is a [[polynomial]]. When ''m'' is zero and ''ℓ'' integer, these functions are identical to the [[Legendre polynomial]]s.  In general, when ''ℓ'' and ''m'' are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they are not [[polynomial]]s when ''m'' is odd. The fully general class of functions with arbitrary real or complex values of ''ℓ'' and ''m'' are [[Legendre function]]s.  In that case the parameters are usually labelled with Greek letters.

The Legendre [[ordinary differential equation]] is frequently encountered in [[physics]] and other technical fields. In particular, it occurs when solving [[Laplace's equation]] (and related [[partial differential equation]]s) in [[spherical coordinates]].  Associated Legendre polynomials play a vital role in the definition of [[spherical harmonics]].

==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===
The Associated Legendre Polynomial can also be written as:{{Citation needed|reason=This appears correct, at least for positive m, but is non-obvious.|date=March 2024}}
<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]].

==Orthogonality==
The associated Legendre polynomials are not mutually orthogonal in general. For example, <math>P_1^1</math> is not orthogonal to <math>P_2^2</math>. However, some subsets are orthogonal. Assuming 0&nbsp;≤&nbsp;''m''&nbsp;≤&nbsp;''ℓ'', they satisfy the orthogonality condition for fixed ''m'':

<math display="block">\int_{-1}^{1} P_k ^{m} P_\ell ^{m} dx = \frac{2 (\ell+m)!}{(2\ell+1)(\ell-m)!}\ \delta _{k,\ell}</math>

Where {{math|''δ''<sub>''k'',''ℓ''</sub>}} is the [[Kronecker delta]].

Also, they satisfy the orthogonality condition for fixed {{mvar|''ℓ''}}:

<math display="block">\int_{-1}^{1} \frac{P_\ell ^{m} P_\ell ^{n}}{1-x^2}dx = \begin{cases}
0 & \text{if } m\neq n \\
\frac{(\ell+m)!}{m(\ell-m)!} & \text{if } m=n\neq0 \\
\infty & \text{if } m=n=0
\end{cases}</math>

==Negative {{mvar|m}} and/or negative {{mvar|ℓ}}==

The differential equation is clearly invariant under a change in sign of ''m''.

The functions for negative ''m'' were shown above to be proportional to those of positive ''m'':
<math display="block">P_\ell ^{-m} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} P_\ell ^{m}</math>

(This followed from the Rodrigues' formula definition.  This definition also makes the various recurrence formulas work for positive or negative {{mvar|m}}.)
<math display="block">\text{If}\quad |m| > \ell\,\quad\text{then}\quad P_\ell^{m} = 0.\,</math>

The differential equation is also invariant under a change from {{mvar|ℓ}} to {{math|−''ℓ'' − 1}}, and the functions for negative {{mvar|ℓ}} are defined by

<math display="block">P_{-\ell} ^{m} = P_{\ell-1} ^{m},\ (\ell=1,\,2,\, \dots).</math>

==Parity==
From their definition, one can verify that the Associated Legendre functions are either even or odd according to

<math display="block">P_\ell ^{m} (-x) = (-1)^{\ell - m} P_\ell ^{m}(x) </math>

==The first few associated Legendre functions==
[[File:Mplwp legendreP04a0.svg|thumb|300px|Associated Legendre functions for ''m'' = 0]]
[[File:Mplwp legendreP15a1.svg|thumb|300px|Associated Legendre functions for ''m'' = 1]]
[[File:Mplwp legendreP26a2.svg|thumb|300px|Associated Legendre functions for ''m'' = 2]]
The first few associated Legendre functions, including those for negative values of ''m'', are:

<math display="block">P_{0}^{0}(x)=1</math>

<math display="block">\begin{align}
P_{1}^{-1}(x)&=-\tfrac{1}{2}P_{1}^{1}(x) \\
P_{1}^{0}(x)&=x \\
P_{1}^{1}(x)&=-(1-x^2)^{1/2}
\end{align}</math>

<math display="block">\begin{align}
P_{2}^{-2}(x)&=\tfrac{1}{24}P_{2}^{2}(x) \\
P_{2}^{-1}(x)&=-\tfrac{1}{6}P_{2}^{1}(x) \\
P_{2}^{0}(x)&=\tfrac{1}{2}(3x^{2}-1) \\
P_{2}^{1}(x)&=-3x(1-x^2)^{1/2} \\
P_{2}^{2}(x)&=3(1-x^2)
\end{align}</math>

<math display="block">\begin{align}
P_{3}^{-3}(x)&=-\tfrac{1}{720}P_{3}^{3}(x) \\
P_{3}^{-2}(x)&=\tfrac{1}{120}P_{3}^{2}(x) \\
P_{3}^{-1}(x)&=-\tfrac{1}{12}P_{3}^{1}(x) \\
P_{3}^{0}(x)&=\tfrac{1}{2}(5x^3-3x) \\
P_{3}^{1}(x)&=\tfrac{3}{2}(1-5x^{2})(1-x^2)^{1/2} \\
P_{3}^{2}(x)&=15x(1-x^2) \\
P_{3}^{3}(x)&=-15(1-x^2)^{3/2}
\end{align}</math>

<math display="block">\begin{align}
P_{4}^{-4}(x)&=\tfrac{1}{40320}P_{4}^{4}(x) \\
P_{4}^{-3}(x)&=-\tfrac{1}{5040}P_{4}^{3}(x) \\
P_{4}^{-2}(x)&=\tfrac{1}{360}P_{4}^{2}(x) \\
P_{4}^{-1}(x)&=-\tfrac{1}{20}P_{4}^{1}(x) \\
P_{4}^{0}(x)&=\tfrac{1}{8}(35x^{4}-30x^{2}+3) \\
P_{4}^{1}(x)&=-\tfrac{5}{2}(7x^3-3x)(1-x^2)^{1/2} \\
P_{4}^{2}(x)&=\tfrac{15}{2}(7x^2-1)(1-x^2) \\
P_{4}^{3}(x)&= - 105x(1-x^2)^{3/2} \\
P_{4}^{4}(x)&=105(1-x^2)^{2}
\end{align}</math>
<!--
<math display="block">\begin{align}
P_{5}^{-5}(x)&={1\over 3840}\left(\sqrt{1-x^2}\right)^{5} \\
P_{5}^{-4}(x)&={1\over 384}\left(\sqrt{1-x^2}\right)^{4}x \\
P_{5}^{-3}(x)&={1\over 384}\left(\sqrt{1-x^2}\right)^{3}(9x^{2}-1) \\
P_{5}^{-2}(x)&={1\over 16}\left(\sqrt{1-x^2}\right)^{2}(3x^{3}-1x) \\
P_{5}^{-1}(x)&={1\over 16}\left(\sqrt{1-x^2}\right)(21x^{4}-14x^{2}+1) \\
P_{5}^{0}(x)&={1\over 8}(63x^{5}-70x^{3}+15x) \\
P_{5}^{1}(x)&={-15\over 8}\left(\sqrt{1-x^2}\right)(21x^{4}-14x^{2}+1) \\
P_{5}^{2}(x)&={105\over 2}\left(\sqrt{1-x^2}\right)^{2}(3x^{3}-1x) \\
P_{5}^{3}(x)&={-105\over 2}\left(\sqrt{1-x^2}\right)^{3}(9x^{2}-1) \\
P_{5}^{4}(x)&=945\left(\sqrt{1-x^2}\right)^{4}x \\
P_{5}^{5}(x)&=-945\left(\sqrt{1-x^2}\right)^{5}
\end{align}</math>

<math display="block">\begin{align}
P_{6}^{-6}(x)={1\over 46080}\left(\sqrt{1-x^2}\right)^{6} \\
P_{6}^{-5}(x)={1\over 3840}\left(\sqrt{1-x^2}\right)^{5}x \\
P_{6}^{-4}(x)={1\over 3840}\left(\sqrt{1-x^2}\right)^{4}(11x^{2}-1) \\
P_{6}^{-3}(x)={1\over 384}\left(\sqrt{1-x^2}\right)^{3}(11x^{3}-3x) \\
P_{6}^{-2}(x)={1\over 128}\left(\sqrt{1-x^2}\right)^{2}(33x^{4}-18x^{2}+1) \\
P_{6}^{-1}(x)={1\over 16}\left(\sqrt{1-x^2}\right)(33x^{5}-30x^{3}+5x) \\
P_{6}^{0}(x)={1\over 16}(231x^{6}-315x^{4}+105x^{2}-5) \\
P_{6}^{1}(x)={-21\over 8}\left(\sqrt{1-x^2}\right)(33x^{5}-30x^{3}+5x) \\
P_{6}^{2}(x)={105\over 8}\left(\sqrt{1-x^2}\right)^{2}(33x^{4}-18x^{2}+1) \\
P_{6}^{3}(x)={-315\over 2}\left(\sqrt{1-x^2}\right)^{3}(11x^{3}-3x) \\
P_{6}^{4}(x)={945\over 2}\left(\sqrt{1-x^2}\right)^{4}(11x^{2}-1) \\
P_{6}^{5}(x)=-10395\left(\sqrt{1-x^2}\right)^{5}x \\
P_{6}^{6}(x)=10395\left(\sqrt{1-x^2}\right)^{6}
\end{align}</math>

<math display="block">\begin{align}
P_{7}^{-7}(x)&={1\over 645120}\left(\sqrt{1-x^2}\right)^{7} \\
P_{7}^{-6}(x)&={1\over 46080}\left(\sqrt{1-x^2}\right)^{6}x \\
P_{7}^{-5}(x)&={1\over 46080}\left(\sqrt{1-x^2}\right)^{5}(13x^{2}-1) \\
P_{7}^{-4}(x)&={1\over 3840}\left(\sqrt{1-x^2}\right)^{4}(13x^{3}-3x) \\
P_{7}^{-3}(x)&={1\over 3840}\left(\sqrt{1-x^2}\right)^{3}(143x^{4}-66x^{2}+3) \\
P_{7}^{-2}(x)&={1\over 384}\left(\sqrt{1-x^2}\right)^{2}(143x^{5}-110x^{3}+15x) \\
P_{7}^{-1}(x)&={1\over 128}\left(\sqrt{1-x^2}\right)(429x^{6}-495x^{4}+135x^{2}-5) \\
P_{7}^{0}(x)&={1\over 16}(429x^{7}-693x^{5}+315x^{3}-35x) \\
P_{7}^{1}(x)&={-7\over 16}\left(\sqrt{1-x^2}\right)(429x^{6}-495x^{4}+135x^{2}-5) \\
P_{7}^{2}(x)&={63\over 8}\left(\sqrt{1-x^2}\right)^{2}(143x^{5}-110x^{3}+15x) \\
P_{7}^{3}(x)&={-315\over 8}\left(\sqrt{1-x^2}\right)^{3}(143x^{4}-66x^{2}+3) \\
P_{7}^{4}(x)&={3465\over 2}\left(\sqrt{1-x^2}\right)^{4}(13x^{3}-3x) \\
P_{7}^{5}(x)&={-10395\over 2}\left(\sqrt{1-x^2}\right)^{5}(13x^{2}-1) \\
P_{7}^{6}(x)&=135135\left(\sqrt{1-x^2}\right)^{6}x \\
P_{7}^{7}(x)&=-135135\left(\sqrt{1-x^2}\right)^{7}
\end{align}</math>

<math display="block">\begin{align}
P_{8}^{-8}(x)={1\over 10321920}\left(\sqrt{1-x^2}\right)^{8} \\
P_{8}^{-7}(x)&={1\over 645120}\left(\sqrt{1-x^2}\right)^{7}x \\
P_{8}^{-6}(x)&={1\over 645120}\left(\sqrt{1-x^2}\right)^{6}(15x^{2}-1) \\
P_{8}^{-5}(x)&={1\over 15360}\left(\sqrt{1-x^2}\right)^{5}(5x^{3}-1x) \\
P_{8}^{-4}(x)&={1\over 15360}\left(\sqrt{1-x^2}\right)^{4}(65x^{4}-26x^{2}+1) \\
P_{8}^{-3}(x)&={1\over 768}\left(\sqrt{1-x^2}\right)^{3}(39x^{5}-26x^{3}+3x) \\
P_{8}^{-2}(x)&={1\over 256}\left(\sqrt{1-x^2}\right)^{2}(143x^{6}-143x^{4}+33x^{2}-1) \\
P_{8}^{-1}(x)&={1\over 128}\left(\sqrt{1-x^2}\right)(715x^{7}-1001x^{5}+385x^{3}-35x) \\
P_{8}^{0}(x)&={1\over 128}(6435x^{8}-12012x^{6}+6930x^{4}-1260x^{2}+35) \\
P_{8}^{1}(x)&={-9\over 16}\left(\sqrt{1-x^2}\right)(715x^{7}-1001x^{5}+385x^{3}-35x) \\
P_{8}^{2}(x)&={315\over 16}\left(\sqrt{1-x^2}\right)^{2}(143x^{6}-143x^{4}+33x^{2}-1) \\
P_{8}^{3}(x)&={-3465\over 8}\left(\sqrt{1-x^2}\right)^{3}(39x^{5}-26x^{3}+3x) \\
P_{8}^{4}(x)&={10395\over 8}\left(\sqrt{1-x^2}\right)^{4}(65x^{4}-26x^{2}+1) \\
P_{8}^{5}(x)&={-135135\over 2}\left(\sqrt{1-x^2}\right)^{5}(5x^{3}-1x) \\
P_{8}^{6}(x)&={135135\over 2}\left(\sqrt{1-x^2}\right)^{6}(15x^{2}-1) \\
P_{8}^{7}(x)&=-2027025\left(\sqrt{1-x^2}\right)^{7}x \\
P_{8}^{8}(x)&=2027025\left(\sqrt{1-x^2}\right)^{8}
\end{align}</math>

<math display="block">\begin{align}
P_{9}^{-9}(x)&={1\over 185794560}\left(\sqrt{1-x^2}\right)^{9} \\
P_{9}^{-8}(x)&={1\over 10321920}\left(\sqrt{1-x^2}\right)^{8}x \\
P_{9}^{-7}(x)&={1\over 10321920}\left(\sqrt{1-x^2}\right)^{7}(17x^{2}-1) \\
P_{9}^{-6}(x)&={1\over 645120}\left(\sqrt{1-x^2}\right)^{6}(17x^{3}-3x) \\
P_{9}^{-5}(x)&={1\over 215040}\left(\sqrt{1-x^2}\right)^{5}(85x^{4}-30x^{2}+1) \\
P_{9}^{-4}(x)&={1\over 3072}\left(\sqrt{1-x^2}\right)^{4}(17x^{5}-10x^{3}+1x) \\
P_{9}^{-3}(x)&={1\over 3072}\left(\sqrt{1-x^2}\right)^{3}(221x^{6}-195x^{4}+39x^{2}-1) \\
P_{9}^{-2}(x)&={1\over 256}\left(\sqrt{1-x^2}\right)^{2}(221x^{7}-273x^{5}+91x^{3}-7x) \\
P_{9}^{-1}(x)&={1\over 256}\left(\sqrt{1-x^2}\right)(2431x^{8}-4004x^{6}+2002x^{4}-308x^{2}+7) \\
P_{9}^{0}(x)&={1\over 128}(12155x^{9}-25740x^{7}+18018x^{5}-4620x^{3}+315x) \\
P_{9}^{1}(x)&={-45\over 128}\left(\sqrt{1-x^2}\right)(2431x^{8}-4004x^{6}+2002x^{4}-308x^{2}+7) \\
P_{9}^{2}(x)&={495\over 16}\left(\sqrt{1-x^2}\right)^{2}(221x^{7}-273x^{5}+91x^{3}-7x) \\
P_{9}^{3}(x)&={-3465\over 16}\left(\sqrt{1-x^2}\right)^{3}(221x^{6}-195x^{4}+39x^{2}-1) \\
P_{9}^{4}(x)&={135135\over 8}\left(\sqrt{1-x^2}\right)^{4}(17x^{5}-10x^{3}+1x) \\
P_{9}^{5}(x)&={-135135\over 8}\left(\sqrt{1-x^2}\right)^{5}(85x^{4}-30x^{2}+1) \\
P_{9}^{6}(x)&={675675\over 2}\left(\sqrt{1-x^2}\right)^{6}(17x^{3}-3x) \\
P_{9}^{7}(x)&={-2027025\over 2}\left(\sqrt{1-x^2}\right)^{7}(17x^{2}-1) \\
P_{9}^{8}(x)&=34459425\left(\sqrt{1-x^2}\right)^{8}x \\
P_{9}^{9}(x)&=-34459425\left(\sqrt{1-x^2}\right)^{9}
\end{align}</math>

<math display="block">P_{10}^{-10}(x)={1\over 3715891200}\left(\sqrt{1-x^2}\right)^{10}</math>
<math display="block">P_{10}^{-9}(x)={1\over 185794560}\left(\sqrt{1-x^2}\right)^{9}x</math>
<math display="block">P_{10}^{-8}(x)={1\over 185794560}\left(\sqrt{1-x^2}\right)^{8}(19x^{2}-1)</math>
<math display="block">P_{10}^{-7}(x)={1\over 10321920}\left(\sqrt{1-x^2}\right)^{7}(19x^{3}-3x)</math>
<math display="block">P_{10}^{-6}(x)={1\over 10321920}\left(\sqrt{1-x^2}\right)^{6}(323x^{4}-102x^{2}+3)</math>
<math display="block">P_{10}^{-5}(x)={1\over 645120}\left(\sqrt{1-x^2}\right)^{5}(323x^{5}-170x^{3}+15x)</math>
<math display="block">P_{10}^{-4}(x)={1\over 43008}\left(\sqrt{1-x^2}\right)^{4}(323x^{6}-255x^{4}+45x^{2}-1)</math>
<math display="block">P_{10}^{-3}(x)={1\over 3072}\left(\sqrt{1-x^2}\right)^{3}(323x^{7}-357x^{5}+105x^{3}-7x)</math>
<math display="block">P_{10}^{-2}(x)={1\over 3072}\left(\sqrt{1-x^2}\right)^{2}(4199x^{8}-6188x^{6}+2730x^{4}-364x^{2}+7)</math>
<math display="block">P_{10}^{-1}(x)={1\over 256}\left(\sqrt{1-x^2}\right)(4199x^{9}-7956x^{7}+4914x^{5}-1092x^{3}+63x)</math>
<math display="block">P_{10}^{0}(x)={1\over 256}(46189x^{10}-109395x^{8}+90090x^{6}-30030x^{4}+3465x^{2}-63)</math>
<math display="block">P_{10}^{1}(x)={-55\over 128}\left(\sqrt{1-x^2}\right)(4199x^{9}-7956x^{7}+4914x^{5}-1092x^{3}+63x)</math>
<math display="block">P_{10}^{2}(x)={495\over 128}\left(\sqrt{1-x^2}\right)^{2}(4199x^{8}-6188x^{6}+2730x^{4}-364x^{2}+7)</math>
<math display="block">P_{10}^{3}(x)={-6435\over 16}\left(\sqrt{1-x^2}\right)^{3}(323x^{7}-357x^{5}+105x^{3}-7x)</math>
<math display="block">P_{10}^{4}(x)={45045\over 16}\left(\sqrt{1-x^2}\right)^{4}(323x^{6}-255x^{4}+45x^{2}-1)</math>
<math display="block">P_{10}^{5}(x)={-135135\over 8}\left(\sqrt{1-x^2}\right)^{5}(323x^{5}-170x^{3}+15x)</math>
<math display="block">P_{10}^{6}(x)={675675\over 8}\left(\sqrt{1-x^2}\right)^{6}(323x^{4}-102x^{2}+3)</math>
<math display="block">P_{10}^{7}(x)={-11486475\over 2}\left(\sqrt{1-x^2}\right)^{7}(19x^{3}-3x)</math>
<math display="block">P_{10}^{8}(x)={34459425\over 2}\left(\sqrt{1-x^2}\right)^{8}(19x^{2}-1)</math>
<math display="block">P_{10}^{9}(x)=-654729075\left(\sqrt{1-x^2}\right)^{9}x</math>
<math display="block">P_{10}^{10}(x)=654729075\left(\sqrt{1-x^2}\right)^{10}</math>
-->

==Recurrence formula==

These functions have a number of recurrence properties:

<math display="block">(\ell-m-1)(\ell-m)P_{\ell}^{m}(x) = -P_{\ell}^{m+2}(x) + P_{\ell-2}^{m+2}(x) + (\ell+m)(\ell+m-1)P_{\ell-2}^{m}(x)</math>

<math display="block">(\ell-m+1)P_{\ell+1}^{m}(x) = (2\ell+1)xP_{\ell}^{m}(x) - (\ell+m)P_{\ell-1}^{m}(x)</math>

<math display="block">2mxP_{\ell}^{m}(x)=-\sqrt{1-x^2}\left[P_{\ell}^{m+1}(x)+(\ell+m)(\ell-m+1)P_{\ell}^{m-1}(x)\right]</math>

<math display="block">\frac{1}{\sqrt{1-x^2}}P_\ell^m(x) = \frac{-1}{2m} \left[ P_{\ell-1}^{m+1}(x) + (\ell+m-1)(\ell+m)P_{\ell-1}^{m-1}(x) \right]</math>

<math display="block">\frac{1}{\sqrt{1-x^2}}P_\ell^m(x) = \frac{-1}{2m} \left[ P_{\ell+1}^{m+1}(x) + (\ell-m+1)(\ell-m+2)P_{\ell+1}^{m-1}(x) \right]</math>

<math display="block"> \sqrt{1-x^2}P_\ell^m(x) = \frac1{2\ell+1} \left[ (\ell-m+1)(\ell-m+2) P_{\ell+1}^{m-1}(x) - (\ell+m-1)(\ell+m) P_{\ell-1}^{m-1}(x) \right] </math>

<math display="block"> \sqrt{1-x^2}P_\ell^m(x) = \frac{-1}{2\ell+1} \left[ P_{\ell+1}^{m+1}(x) - P_{\ell-1}^{m+1}(x) \right] </math>

<math display="block">\sqrt{1-x^2}P_\ell^{m+1}(x) = (\ell-m)xP_{\ell}^{m}(x) - (\ell+m)P_{\ell-1}^{m}(x)</math>

<math display="block">\sqrt{1-x^2}P_\ell^{m+1}(x) = (\ell-m+1)P_{\ell+1}^m(x) - (\ell+m+1)xP_\ell^m(x)</math>

<math display="block"> \sqrt{1-x^2}\frac{d}{dx}{P_\ell^m}(x) = \frac12 \left[ (\ell+m)(\ell-m+1)P_\ell^{m-1}(x) - P_\ell^{m+1}(x) \right] </math>

<math display="block"> (1-x^2)\frac{d}{dx}{P_\ell^m}(x) = \frac1{2\ell+1} \left[ (\ell+1)(\ell+m)P_{\ell-1}^m(x) - \ell(\ell-m+1)P_{\ell+1}^m(x) \right] </math>

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

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

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

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

Helpful identities (initial values for the first recursion):

<math display="block">P_{\ell +1}^{\ell +1}(x) = - (2\ell+1) \sqrt{1-x^2} P_{\ell}^{\ell}(x)</math>
<math display="block">P_{\ell}^{\ell}(x) = (-1)^\ell (2\ell-1)!!  (1- x^2)^{(\ell/2)}</math>
<math display="block">P_{\ell +1}^{\ell}(x) = x (2\ell+1) P_{\ell}^{\ell}(x)</math>

with {{math|!!}} the [[double factorial]].

==Gaunt's formula==
<!-- This section is linked from [[Gaunt's formula]]. See [[WP:MOS#Section management]] -->

The integral over the product of three associated Legendre polynomials (with orders matching as shown below) is a necessary ingredient when developing products of Legendre polynomials into a series linear in the Legendre polynomials.  For instance, this turns out to be necessary when doing atomic calculations of the [[Hartree–Fock]] variety where matrix elements of the [[Coulomb operator]] are needed.  For this we have Gaunt's formula <ref>From John C. Slater ''Quantum Theory of Atomic Structure'', McGraw-Hill (New York, 1960), Volume I, page 309, which cites the original work of J. A. Gaunt, ''Philosophical Transactions of the Royal Society of London'', A228:151 (1929)</ref>
<math display="block">\begin{align}
\frac{1}{2} \int_{-1}^1 P_l^u(x) P_m^v(x) P_n^w(x) dx
={}&{}(-1)^{s-m-w}\frac{(m+v)!(n+w)!(2s-2n)!s!}{(m-v)!(s-l)!(s-m)!(s-n)!(2s+1)!} \\
   &{}\times \ \sum_{t=p}^q (-1)^t \frac{(l+u+t)!(m+n-u-t)!}{t!(l-u-t)!(m-n+u+t)!(n-w-t)!}
\end{align}</math>
This formula is to be used under the following assumptions: 
# the degrees are non-negative integers <math>l,m,n\ge0</math>
# all three orders are non-negative integers <math>u,v,w\ge 0</math>
# <math>u</math> is the largest of the three orders
# the orders sum up <math>u=v+w</math>
# the degrees obey <math> m\ge n</math>

Other quantities appearing in the formula are defined as
<math display="block"> 2s = l+m+n </math>
<math display="block"> p = \max(0,\,n-m-u) </math>
<math display="block"> q = \min(m+n-u,\,l-u,\,n-w) </math>

The integral is zero unless
# the sum of degrees is even so that <math>s</math> is an integer
# the triangular condition is satisfied <math>m+n\ge l \ge m-n</math>

Dong and Lemus (2002)<ref>Dong S.H., Lemus R., (2002), [http://www.sciencedirect.com/science/article/pii/S0893965902800040 "The overlap integral of three associated Legendre polynomials"], Appl. Math. Lett. 15, 541-546.</ref> generalized the derivation of this formula to integrals over a product of an arbitrary number of associated Legendre polynomials.

==Generalization via hypergeometric functions==
{{main|Legendre function}}
These functions may actually be defined for general complex parameters and argument:

<math display="block">P_{\lambda}^{\mu}(z) = \frac{1}{\Gamma(1-\mu)} \left[\frac{1+z}{1-z}\right]^{\mu/2} \,_2F_1 (-\lambda, \lambda+1; 1-\mu; \frac{1-z}{2})</math>

where <math>\Gamma</math> is the [[gamma function]] and <math> _2F_1</math> is the [[hypergeometric function]]

<math display="block">\,_2F_1 (\alpha, \beta; \gamma; z) = \frac{\Gamma(\gamma)}{\Gamma(\alpha)\Gamma(\beta)} \sum_{n=0}^\infty\frac{\Gamma(n+\alpha)\Gamma(n+\beta)}{\Gamma(n+\gamma)\ n!}z^n,</math>

They are called the '''Legendre functions''' when defined in this more general way. They satisfy the same differential equation as before:

<math display="block">(1-z^2)\,y'' -2zy' + \left(\lambda[\lambda+1] - \frac{\mu^2}{1-z^2}\right)\,y = 0.\,</math>

Since this is a second order differential equation, it has a second solution, <math>Q_\lambda^{\mu}(z)</math>, defined as:

<math display="block">Q_{\lambda}^{\mu}(z) = \frac{\sqrt{\pi}\ \Gamma(\lambda+\mu+1)}{2^{\lambda+1}\Gamma(\lambda+3/2)}\frac{1}{z^{\lambda+\mu+1}}(1-z^2)^{\mu/2} \,_2F_1 \left(\frac{\lambda+\mu+1}{2}, \frac{\lambda+\mu+2}{2}; \lambda+\frac{3}{2}; \frac{1}{z^2}\right)</math>

<math>P_\lambda^{\mu}(z)</math> and <math>Q_\lambda^{\mu}(z)</math> both obey the various recurrence formulas given previously.

==Reparameterization in terms of angles==
These functions are most useful when the argument is reparameterized in terms of angles, letting <math>x = \cos\theta</math>:

<math display="block">P_\ell^{m}(\cos\theta) = (-1)^m (\sin \theta)^m\ \frac{d^m}{d(\cos\theta)^m}\left(P_\ell(\cos\theta)\right)</math>

Using the relation <math>(1 - x^2)^{1 / 2} = \sin\theta</math>, [[#The first few associated Legendre functions|the list given above]] yields the first few polynomials, parameterized this way, as:

<math display="block">\begin{align}
P_0^0(\cos\theta) & = 1 \\[8pt]
P_1^0(\cos\theta) & = \cos\theta \\[8pt]
P_1^1(\cos\theta) & = -\sin\theta \\[8pt]
P_2^0(\cos\theta) & = \tfrac{1}{2} (3\cos^2\theta-1) \\[8pt]
P_2^1(\cos\theta) & = -3\cos\theta\sin\theta \\[8pt]
P_2^2(\cos\theta) & = 3\sin^2\theta \\[8pt]
P_3^0(\cos\theta) & = \tfrac{1}{2} (5\cos^3\theta-3\cos\theta) \\[8pt]
P_3^1(\cos\theta) & = -\tfrac{3}{2} (5\cos^2\theta-1)\sin\theta \\[8pt]
P_3^2(\cos\theta) & = 15\cos\theta\sin^2\theta \\[8pt]
P_3^3(\cos\theta) & = -15\sin^3\theta \\[8pt]
P_4^0(\cos\theta) & = \tfrac{1}{8} (35\cos^4\theta-30\cos^2\theta+3) \\[8pt]
P_4^1(\cos\theta) & = - \tfrac{5}{2} (7\cos^3\theta-3\cos\theta)\sin\theta \\[8pt]
P_4^2(\cos\theta) & =  \tfrac{15}{2} (7\cos^2\theta-1)\sin^2\theta \\[8pt]
P_4^3(\cos\theta) & = -105\cos\theta\sin^3\theta \\[8pt]
P_4^4(\cos\theta) & = 105\sin^4\theta
\end{align}</math>

The orthogonality relations given above become in this formulation:
for fixed ''m'', <math>P_\ell^m(\cos\theta)</math> are orthogonal, parameterized by θ over <math>[0, \pi]</math>, with weight <math>\sin \theta</math>:

<math display="block">\int_0^\pi P_k^{m}(\cos\theta) P_\ell^{m}(\cos\theta)\,\sin\theta\,d\theta = \frac{2 (\ell+m)!}{(2\ell+1)(\ell-m)!}\ \delta _{k,\ell}</math>

Also, for fixed ''ℓ'':

<math display="block">\int_0^\pi P_\ell^{m}(\cos\theta) P_\ell^{n}(\cos\theta) \csc\theta\,d\theta = \begin{cases} 0 & \text{if } m\neq n \\ \frac{(\ell+m)!}{m(\ell-m)!} & \text{if } m=n\neq0 \\ \infty & \text{if } m=n=0\end{cases}</math>

In terms of θ, <math>P_\ell^{m}(\cos \theta)</math> are solutions of

<math display="block">\frac{d^{2}y}{d\theta^2} + \cot \theta \frac{dy}{d\theta} + \left[\lambda - \frac{m^2}{\sin^2\theta}\right]\,y = 0\,</math>

More precisely, given an integer ''m''<math>\ge</math>0, the above equation has
nonsingular solutions only when <math>\lambda = \ell(\ell+1)\,</math> for ''ℓ''
an integer ≥&nbsp;''m'', and those solutions are proportional to
<math>P_\ell^{m}(\cos \theta)</math>.

==Applications in physics: spherical harmonics==
{{main|Spherical harmonics}}
In many occasions in [[physics]], associated Legendre polynomials in terms of angles occur where [[spherical]] [[symmetry]] is involved.  The colatitude angle in [[spherical coordinates]] is
the angle <math>\theta</math> used above.  The longitude angle, <math>\phi</math>, appears in a multiplying factor.  Together, they make a set of functions called [[spherical harmonic]]s. These functions express the symmetry of the [[Riemann sphere|two-sphere]] under the action of the [[Lie group]] SO(3).{{cn|date=July 2022}}

What makes these functions useful is that they are central to the solution of the equation
<math>\nabla^2\psi + \lambda\psi = 0</math> on the surface of a sphere.  In spherical coordinates θ (colatitude) and φ (longitude), the [[Laplacian]] is

<math display="block">\nabla^2\psi = \frac{\partial^2\psi}{\partial\theta^2} + \cot \theta \frac{\partial \psi}{\partial \theta} + \csc^2 \theta\frac{\partial^2\psi}{\partial\phi^2}.</math>

When the [[partial differential equation]]

<math display="block">\frac{\partial^2\psi}{\partial\theta^2} + \cot \theta \frac{\partial \psi}{\partial \theta} + \csc^2 \theta\frac{\partial^2\psi}{\partial\phi^2} + \lambda \psi = 0</math>

is solved by the method of [[separation of variables#pde|separation of variables]], one gets a φ-dependent part <math>\sin(m\phi)</math> or <math>\cos(m\phi)</math> for integer m≥0, and an equation for the θ-dependent part

<math display="block">\frac{d^{2}y}{d\theta^2} + \cot \theta \frac{dy}{d\theta} + \left[\lambda - \frac{m^2}{\sin^2\theta}\right]\,y = 0\,</math>

for which the solutions are <math>P_\ell^{m}(\cos \theta)</math> with <math>\ell{\ge}m</math>
and <math>\lambda = \ell(\ell+1)</math>.

Therefore, the equation

<math display="block">\nabla^2\psi + \lambda\psi = 0</math>

has nonsingular separated solutions only when <math>\lambda = \ell(\ell+1)</math>,
and those solutions are proportional to

<math display="block">P_\ell^{m}(\cos \theta)\ \cos (m\phi)\ \ \ \ 0 \le m \le \ell</math>

and

<math display="block">P_\ell^{m}(\cos \theta)\ \sin (m\phi)\ \ \ \ 0 < m \le \ell.</math>

For each choice of ''ℓ'', there are {{nowrap|2ℓ + 1}} functions
for the various values of ''m'' and choices of sine and cosine.
They are all orthogonal in both ''ℓ'' and ''m'' when integrated over the
surface of the sphere.

The solutions are usually written in terms of [[complex exponential]]s:

<math display="block">Y_{\ell, m}(\theta, \phi) =  \sqrt{\frac{(2\ell+1)(\ell-m)!}{4\pi(\ell+m)!}}\ P_\ell^{m}(\cos \theta)\ e^{im\phi}\qquad -\ell \le m \le \ell.
</math>
The functions <math>Y_{\ell, m}(\theta, \phi)</math> are the [[spherical harmonics]], and the quantity in the square root is a normalizing factor.
Recalling the relation between the associated Legendre functions of positive and negative ''m'', it is easily shown that the spherical harmonics satisfy the identity<ref>This identity can also be shown by relating the spherical harmonics to [[Wigner D-matrix|Wigner D-matrices]] and use of the time-reversal property of the latter. The relation between associated Legendre functions of ±''m'' can then be proved from the complex conjugation identity of the spherical harmonics.</ref>

<math display="block">Y_{\ell, m}^*(\theta, \phi) = (-1)^m Y_{\ell, -m}(\theta, \phi).</math>

The spherical harmonic functions form a complete orthonormal set of functions in the sense of [[Fourier series]]. Workers in the fields of geodesy, geomagnetism and spectral analysis use a different phase and normalization factor than given here (see [[spherical harmonics]]).

When a 3-dimensional spherically symmetric partial differential equation is solved by the method of separation of variables in spherical coordinates, the part that remains after removal of the radial part is typically
of the form

<math display="block">\nabla^2\psi(\theta, \phi) + \lambda\psi(\theta, \phi) = 0,</math>

and hence the solutions are spherical harmonics.

==Generalizations==
The Legendre polynomials are closely related to [[hypergeometric series]]. In the form of spherical harmonics, they express the symmetry of the [[Riemann sphere|two-sphere]] under the action of the [[Lie group]] SO(3). There are many other Lie groups besides SO(3), and analogous generalizations of the  Legendre polynomials exist to express the symmetries of semi-simple Lie groups and [[Riemannian symmetric space]]s. Crudely speaking, one may define a [[Laplacian]] on symmetric spaces; the eigenfunctions of the Laplacian can be thought of as generalizations of the spherical harmonics to other settings.

==See also==
* [[Angular momentum]]
* [[Gaussian quadrature]]
* [[Legendre polynomials]]
* [[Spherical harmonics]]
* [[Whipple's transformation of Legendre functions]]
* [[Laguerre polynomials]]
* [[Hermite polynomials]]

==Notes and references==
{{Reflist}}
* {{citation|last1=Arfken|first1=G.B.|last2=Weber|first2=H.J.|title=Mathematical methods for physicists|year=2001|publisher=Academic Press|isbn=978-0-12-059825-0}}; Section 12.5. (Uses a different sign convention.)
* {{citation|last=Belousov|first=S. L.|year=1962|title=Tables of normalized associated Legendre polynomials|series=Mathematical tables|volume=18|publisher=Pergamon Press}}.
* {{citation|first1=E. U.|last1=Condon|first2=G. H.|last2=Shortley|title=The Theory of Atomic Spectra|year=1970|location=Cambridge, England|publisher=Cambridge University Press|oclc=5388084}}; Chapter 3.
* {{citation|first1=Richard|last1=Courant|authorlink1=Richard Courant|first2=David|last2=Hilbert|authorlink2=David Hilbert|year=1953|title=Methods of Mathematical Physics, Volume 1|publisher=Interscience Publischer, Inc|location=New York}}.
* {{dlmf|first=T. M. |last=Dunster|id=14|title=Legendre and Related Functions}}
* {{citation|first=A.R.|last=Edmonds|title=Angular Momentum in Quantum Mechanics|year=1957|publisher=Princeton University Press|isbn=978-0-691-07912-7|url-access=registration|url=https://archive.org/details/angularmomentumi0000edmo}}; Chapter 2.
* {{citation|first=F. B.|last=Hildebrand|authorlink=Francis B. Hildebrand|title=Advanced Calculus for Applications|year=1976|publisher=Prentice Hall|isbn=978-0-13-011189-0}}.
* {{dlmf|id=18|title=Orthogonal Polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
* Schach, S. R. (1973) ''[http://epubs.siam.org/doi/abs/10.1137/0507007 New Identities for Legendre Associated Functions of Integral Order and Degree ]'', Society for Industrial and Applied Mathematics Journal on Mathematical Analysis, 1976, Vol. 7, No. 1 : pp.&nbsp;59–69

==External links==
* [http://mathworld.wolfram.com/AssociatedLegendrePolynomial.html Associated Legendre polynomials in MathWorld]
* [http://mathworld.wolfram.com/LegendrePolynomial.html Legendre polynomials in MathWorld]
* [https://dlmf.nist.gov/14 Legendre and Related Functions in DLMF]

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[[Category:Atomic physics]]
[[Category:Orthogonal polynomials]]