Editing Legendre polynomials (section)

== Definition and representation ==

=== Definition by construction as an orthogonal system ===
In this approach, the polynomials are defined as an orthogonal system with respect to the weight function <math>w(x) = 1</math> over the interval <math> [-1,1]</math>. That is, <math>P_n(x)</math> is a polynomial of degree <math>n</math>, such that
<math display="block">\int_{-1}^1 P_m(x) P_n(x) \,dx = 0 \quad \text{if } n \ne m.</math>

With the additional standardization condition <math>P_n(1) = 1</math>, all the polynomials can be uniquely determined. We then start the construction process: <math>P_0(x) = 1</math> is the only correctly standardized polynomial of degree 0. <math>P_1(x)</math> must be orthogonal to <math>P_0</math>, leading to <math>P_1(x) = x</math>, and <math>P_2(x)</math> is determined by demanding orthogonality to <math>P_0</math> and <math>P_1</math>, and so on. <math>P_n</math> is fixed by demanding orthogonality to all <math>P_m</math> with <math> m < n </math>. This gives <math> n </math> conditions, which, along with the standardization <math> P_n(1) = 1</math> fixes all <math> n+1</math> coefficients in <math> P_n(x)</math>. With work, all the coefficients of every polynomial can be systematically determined, leading to the explicit representation in powers of <math>x</math> given below.

This definition of the <math>P_n</math>'s is the simplest one. It does not appeal to the theory of differential equations. Second, the completeness of the polynomials follows immediately from the completeness of the powers 1, <math> x, x^2, x^3, \ldots</math>. Finally, by defining them via orthogonality with respect to the [[Lebesgue measure]] on <math> [-1, 1] </math>, it sets up the Legendre polynomials as one of the three [[classical orthogonal polynomials|classical orthogonal polynomial systems]]. The other two are the [[Laguerre polynomials]], which are orthogonal over the half line <math>[0,\infty)</math> with the weight <math> e^{-x} </math>, and the [[Hermite polynomials]], orthogonal over the full line <math>(-\infty,\infty)</math> with weight <math> e^{-x^2} </math>.

=== Definition via generating function ===
The Legendre polynomials can also be defined as the coefficients in a formal expansion in powers of <math>t</math> of the [[generating function]]<ref>{{harvnb|Arfken|Weber|2005|loc=p.743}}</ref>
{{NumBlk||<math display="block">\frac{1}{\sqrt{1-2xt+t^2}} = \sum_{n=0}^\infty P_n(x) t^n \,.</math>|{{EquationRef|2}}}}

The coefficient of <math>t^n</math> is a polynomial in <math> x </math> of degree <math>n</math> with <math>|x| \leq 1</math>. Expanding up to <math>t^1</math> gives
<math display="block">P_0(x) = 1 \,,\quad P_1(x) = x.</math>
Expansion to higher orders gets increasingly cumbersome, but is possible to do systematically, and again leads to one of the explicit forms given below.

It is possible to obtain the higher <math>P_n</math>'s without resorting to direct expansion of the [[Taylor series]], however. Equation&nbsp;{{EquationNote|2}} is differentiated with respect to {{mvar|t}} on both sides and rearranged to obtain
<math display="block">\frac{x-t}{\sqrt{1-2xt+t^2}} = \left(1-2xt+t^2\right) \sum_{n=1}^\infty n P_n(x) t^{n-1} \,.</math>
Replacing the quotient of the square root with its definition in Eq.&nbsp;{{EquationNote|2}}, and [[equating the coefficients]] of powers of {{math|''t''}} in the resulting expansion gives ''Bonnet’s recursion formula''
<math display="block"> (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x)\,.</math>
This relation, along with the first two polynomials {{math|''P''<sub>0</sub>}} and {{math|''P''<sub>1</sub>}}, allows all the rest to be generated recursively.

The generating function approach is directly connected to the [[multipole expansion]] in electrostatics, as explained below, and is how the polynomials were first defined by Legendre in 1782.

=== Definition via differential equation ===
A third definition is in terms of solutions to '''Legendre's differential equation''':
{{NumBlk||<math display="block">(1 - x^2) P_n''(x) - 2 x P_n'(x) + n (n + 1) P_n(x) = 0.</math>|{{EquationRef|1}}}}

This [[differential equation]] has [[regular singular point]]s at {{math|1=''x'' = ±1}} so if a solution is sought using the standard [[Frobenius method|Frobenius]] or [[power series]] method, a series about the origin will only converge for {{math|{{abs|''x''}} < 1}} in general. When {{math|''n''}} is an integer, the solution {{math|''P<sub>n</sub>''(''x'')}} that is regular at {{math|1=''x'' = 1}} is also regular at {{math|1=''x'' = −1}}, and the series for this solution terminates (i.e. it is a polynomial). The orthogonality and completeness of these solutions is best seen from the viewpoint of [[Sturm–Liouville theory]]. We rewrite the differential equation as an eigenvalue problem,
<math display="block">\frac{d}{dx} \left( \left(1-x^2\right) \frac{d}{dx} \right) P(x) = -\lambda P(x) \,,</math>
with the eigenvalue <math>\lambda</math> in lieu of <math> n(n+1)</math>. If we demand that the solution be regular at
<math>x = \pm 1</math>, the [[differential operator]] on the left is [[Hermitian]]. The eigenvalues are found to be of the form
{{math|''n''(''n'' + 1)}}, with <math>n = 0, 1, 2, \ldots</math> and the eigenfunctions are the <math>P_n(x)</math>. The orthogonality and completeness of this set of solutions follows at once from the larger framework of Sturm–Liouville theory.

The differential equation admits another, non-polynomial solution, the [[Legendre function#Legendre functions of the second kind (Qn)|Legendre functions of the second kind]] <math>Q_n</math>.
A two-parameter generalization of (Eq.&nbsp;{{EquationNote|1}}) is called Legendre's ''general'' differential equation, solved by the [[Associated Legendre polynomials]]. [[Legendre functions]] are solutions of Legendre's differential equation (generalized or not) with ''non-integer'' parameters.

In physical settings, Legendre's differential equation arises naturally whenever one solves [[Laplace's equation]] (and related [[partial differential equation]]s) by separation of variables in [[spherical coordinates]]. From this standpoint, the eigenfunctions of the angular part of the Laplacian operator are the [[spherical harmonics]], of which the Legendre polynomials are (up to a multiplicative constant) the subset that is left invariant by rotations about the polar axis. The polynomials appear as <math>P_n(\cos\theta)</math> where <math>\theta</math> is the polar angle. This approach to the Legendre polynomials provides a deep connection to rotational symmetry. Many of their properties which are found laboriously through the methods of analysis — for example the addition theorem — are more easily found using the methods of symmetry and [[group theory]], and acquire profound physical and geometrical meaning.

=== Rodrigues' formula and other explicit formulas ===
An especially compact expression for the Legendre polynomials is given by [[Rodrigues' formula]]:
<math display="block">P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 -1)^n \,.</math>

This formula enables derivation of a large number of properties of the <math>P_n</math>'s. Among these are explicit representations such as
<math display="block">\begin{align}
P_n(x) & = [t^n] \frac{\left((t+x)^2 - 1\right)^n}{2^n} = [t^n] \frac{\left(t+x+1\right)^n \left(t+x-1\right)^n}{2^n}, \\[1ex]
P_n(x)&= \frac{1}{2^n} \sum_{k=0}^n \binom{n}{k}^{\!2} (x-1)^{n-k}(x+1)^k, \\[1ex]
P_n(x)&= \sum_{k=0}^n \binom{n}{k} \binom{n+k}{k} \left( \frac{x-1}{2} \right)^{\!k}, \\[1ex]
P_n(x)&= \frac{1}{2^n}\sum_{k=0}^{\left\lfloor n/2 \right\rfloor} \left(-1\right)^k \binom{n}{k}\binom{2n-2k}n x^{n-2k},\\[1ex]
P_n(x)&= 2^n \sum_{k=0}^n x^k \binom{n}{k} \binom{\frac{n+k-1}{2}}{n}, \\[1ex]
P_n(x)&=\frac{1}{2^n n!}\sum_{k=\lceil n/2 \rceil}^{n}\frac{(-1)^{k+n}(2k)!}{(2k-n)!(n-k)!k!}x^{2k-n}, \\[1ex]
P_n(x)&= \begin{cases}
 \displaystyle\frac{1}{\pi}\int_0^\pi {\left(x+\sqrt{x^2-1}\cdot\cos (t) \right)}^n\,dt & \text{if } |x|>1, \\
x^n & \text{if } |x|=1, \\
 \displaystyle\frac{2}{\pi}\cdot x^n\cdot |x|\cdot \int_{|x|}^1 \frac{t^{-n-1}}{\sqrt{t^2-x^2}}\cdot \frac{\cos\left(n\cdot \arccos(t)\right)}{\sin\left(\arccos(t)\right)}\,dt & \text{if } 0<|x|<1, \\
 \displaystyle(-1)^{n/2}\cdot2^{-n}\cdot \binom{n}{n/2} & \text{if } x=0 \text{ and }n\text{ even}, \\
0 & \text{if } x=0 \text{ and }n\text{ odd}.
\end{cases}
\end{align}</math>

Expressing the polynomial as a power series, <math display="inline">P_n(x) = \sum a_{n,k} x^k  </math>, the coefficients of powers of <math>x</math> can also be calculated using the recurrences

<math display="block">a_{n,k} = - \frac{(n-k+2)(n+k-1)}{k(k-1)}a_{n,k-2}. </math> or

<math>
a_{n,k}=-\frac{n+k-1}{n-k}a_{n-2,k}.
</math>
 
The Legendre polynomial is determined by the values used for the two constants <math display="inline">a_{n,0}</math> and <math display="inline">a_{n,1} </math>, where <math display="inline">a_{n,0}=0 </math> if <math>n</math> is odd and <math display="inline">a_{n,1}=0 </math> if <math>n</math> is even.<ref>{{Cite book |last=Boas |first=Mary L. |title=Mathematical methods in the physical sciences |date=2006 |publisher=Wiley |isbn=978-0-471-19826-0 |edition=3rd |location=Hoboken, NJ}}</ref>

In the fourth representation, <math>\lfloor n/2 \rfloor</math> stands for the [[floor function|largest integer less than or equal to]] <math>n/2</math>. The last representation, which is also immediate from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the [[Binomial coefficient#Generalization and connection to the binomial series|generalized form of the binomial coefficient]].

The reversal of the representation as a power series is
<ref>{{cite book|first1=Wilhelm|last1=Magnus|first2=Fritz|last2=Oberhettinger|year=1943|title=Formeln und Satze fur die speziellen Funktionen der Mathematischen Physik|publisher=Springer|series=Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen|volume=52|isbn=978-3-662-41656-3|oclc=1026897547|mr=0022272}}</ref><ref>{{cite book|first1=I. S.|last1=Gradshteyn|first2=I. M.|last2=Ryzhik|year=2015|title=Table of Integrals, Series, and Products|publisher=Elsevier|isbn=978-0-12-384933-5|mr=3307944}}</ref>

<math>
x^m
=\sum_{s= 0}^{\lfloor m/2\rfloor} (2m-4s+1)
\frac{(2s+2)(2s+4)\cdots 2\lfloor m/2\rfloor}{(2m-2s+1)(2m-2s-1)(2m-2s-3)\cdots (1+2\lfloor (m+1)/2\rfloor)}P_{m-2s}(x).
</math>

for <math>m=0,1,2,\ldots</math>, where an empty product in the numerator (last factor less than the first factor) evaluates to 1.

The first few Legendre polynomials are:
{| class="wikitable" style="text-align: right;"
! <math>n</math> !! <math>P_n(x)</math>
|-
|0 || <math display="inline">1</math>
|-
|1 || <math display="inline">x</math>
|-
|2 || <math display="inline">\tfrac12 \left(3x^2-1\right)</math>
|-
|3 || <math display="inline">\tfrac12 \left(5x^3-3x\right)</math>
|-
|4 || <math display="inline">\tfrac18 \left(35x^4-30x^2+3\right)</math>
|-
|5 || <math display="inline">\tfrac18 \left(63x^5-70x^3+15x\right)</math>
|-
|6 || <math display="inline">\tfrac1{16} \left(231x^6-315x^4+105x^2-5\right)</math>
|-
|7 || <math display="inline">\tfrac1{16} \left(429x^7-693x^5+315x^3-35x\right)</math>
|-
|8 || <math display="inline">\tfrac1{128} \left(6435x^8-12012x^6+6930x^4-1260x^2+35\right)</math>
|-
|9 || <math display="inline">\tfrac1{128} \left(12155x^9-25740x^7+18018x^5-4620x^3+315x\right)</math>
|-
|10 || <math display="inline">\tfrac1{256} \left(46189x^{10}-109395x^8+90090x^6-30030x^4+3465x^2-63\right)</math>
|}

The graphs of these polynomials (up to {{math|1=''n'' = 5}}) are shown below:
[[File:Legendrepolynomials6.svg|640px|none|Plot of the six first Legendre polynomials.]]