Editing Legendre polynomials (section)

=== 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.]]