# nLab Legendre polynomial

The Legendre polynomial ${P}_{l}$ (for $l=0,1,2,\dots$) is the polynomial in one variable given by the formula

${P}_{l}\left(x\right)=\frac{1}{{2}^{l}l!}\frac{{d}^{l}}{d{x}^{l}}\left({x}^{2}-1{\right)}^{l}$P_l(x) = \frac{1}{2^l l!} \frac{d^l}{d x^l}(x^2-1)^l

Alternatively they can be defined via a generating function:

$\frac{1}{\sqrt{1-2\mathrm{tx}+{t}^{2}}}=\sum _{n\ge 0}{P}_{n}\left(x\right){t}^{n}$\frac{1}{\sqrt{1-2tx+t^2}} = \sum_{n\geq 0} P_n(x) t^n

The Legendre polynomials satisfy:

• the following differential equation of the second order

$\frac{d}{dx}\left[\left(1-{x}^{2}\right)\frac{d{P}_{l}}{dx}\right]+l\left(l+1\right){P}_{l}=0$\frac{d}{d x}[(1-x^2)\frac{d P_l}{d x}] + l(l+1) P_l = 0
• the recursion relations

$\left(l+1\right){P}_{l+1}-\left(2l+1\right)x{P}_{l}+l{P}_{l-1}=0$(l+1)P_{l+1}-(2l+1)x P_l+l P_{l-1}=0
• the mixed differential recursion relations

$\begin{array}{c}P{\prime }_{l+1}-P{\prime }_{l-1}=\left(2l+1\right){P}_{l}\\ P{\prime }_{l+1}-xP{\prime }_{l}=\left(l+1\right){P}_{l}\\ \left({x}^{2}-1\right){P}_{l}\prime -lx{P}_{l}+l{P}_{l-1}=0\end{array}$\array { P'_{l+1}-P'_{l-1} = (2l+1)P_l \\ P'_{l+1}-x P'_l = (l+1)P_l \\ (x^2-1)P_l'-l x P_l+l P_{l-1} = 0 }

The Legendre polynomials form a complete system of orthogonal polynomials on the interval $\left[-1,1\right]$, that is an unnormalised basis of the Hilbert space ${L}^{2}\left[-1,1\right]$.

The first few Legendre polynomials are ${P}_{0}\left(x\right)=1$, ${P}_{1}\left(x\right)=x$, ${P}_{2}\left(x\right)=\frac{1}{2}\left(3{x}^{2}-1\right)$, ${P}_{3}\left(x\right)=\frac{1}{2}\left(5{x}^{2}-3\right)$, ${P}_{4}\left(x\right)=\frac{1}{8}\left(35{x}^{4}-30{x}^{2}+3\right)$. Their values at $0$ are

${P}_{2n+1}\left(0\right)=0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{P}_{2n}\left(0\right)=\left(-1{\right)}^{n}\frac{\left(2n-1\right)!!}{\left(2n\right)!}=\frac{\left(-1{\right)}^{n}\left(2n\right)!}{{2}^{2n}\left(n!{\right)}^{2}}$P_{2n+1}(0)=0,\,\,\,\,\,P_{2n}(0)=(-1)^n\frac{(2n-1)!!}{(2n)!} = \frac{(-1)^n (2n)!}{2^{2n}(n!)^2}

and ${P}_{l}\left(±1\right)=\left(±1{\right)}^{l}$.

One also has the integral formulas

$\begin{array}{c}{\int }_{0}^{1}{P}_{2k+1}\left(x\right)dx=\frac{\left(-1{\right)}^{k}\left(2k\right)!}{{2}^{2k+1}k!\left(k+1\right)!}\\ {\int }_{0}^{1}{P}_{2k}\left(x\right)dx={\delta }_{k0}\\ {\int }_{-1}^{1}x{P}_{l}{P}_{k}=\left\{\begin{array}{cc}\frac{2\left(l+1\right)}{\left(2l+1\right)\left(2l+3\right)},& k=l+1\\ \frac{\left(2l\right)}{\left(2l-1\right)\left(2l+1\right)},& k=l-1\\ 0,& \mathrm{otherwise}\end{array}\end{array}$\array { \int^1_0 P_{2k+1}(x) d x = \frac{(-1)^k (2k)!}{2^{2k+1} k! (k+1)!} \\ \int^1_0 P_{2k}(x) d x = \delta_{k0} \\ \int^1_{-1} x P_l P_k = \left\lbrace \array{ \frac{2(l+1)}{(2l+1)(2l+3)},&k=l+1 \\ \frac{(2l)}{(2l-1)(2l+1)},&k=l-1\\ 0,& otherwise} \right. }

A generalization of Legendre polynomials are the Legendre functions ${P}_{\nu }$ where $\nu$ is not necessarily an integer and ${P}_{l}^{m}$ which are given by

${P}_{l}^{m}\left(x\right)=\frac{\left(-1{\right)}^{m}}{{2}^{l}l!}\left(1-{x}^{2}{\right)}^{m/2}\frac{{d}^{l+m}}{d{x}^{l+m}}\left({x}^{2}-1{\right)}^{l}=\left(-1{\right)}^{m}\left(1-{x}^{2}{\right)}^{m/2}\frac{{d}^{l}}{d{x}^{l}}{P}_{l}\left(x\right)$P^m_l(x) = \frac{(-1)^m}{2^l l!} (1-x^2)^{m/2} \frac{d^{l+m}}{d x^{l+m}} (x^2-1)^l = (-1)^m (1-x^2)^{m/2} \frac{d^{l}}{d x^{l}} P_l(x)

for $m\ge 0$ and also

${P}_{l}^{-m}\left(x\right)=\left(-1{\right)}^{m}\frac{\left(l-m\right)!}{\left(l+m\right)!}{P}_{l}^{m}\left(x\right)$P^{-m}_l(x)= (-1)^m\frac{(l-m)!}{(l+m)!} P^m_l(x)

These ${P}_{l}^{m}$ are satisfying the orthogonality relations

${\int }_{-1}^{1}{P}_{l}^{m}\left(x\right){P}_{k}^{m}\left(x\right)dx=\frac{2}{2l+1}\frac{\left(l+m\right)!}{\left(l-m\right)!}{\delta }_{\mathrm{lk}}$\int_{-1}^1 P^m_l(x) P^m_k(x) d x = \frac{2}{2l+1}\frac{(l+m)!}{(l-m)!}\delta_{lk}

recursion relation

$x{P}_{l}^{m}\left(x\right)=\frac{l+\mid m\mid }{2l+1}{P}_{l-1}^{m}\left(x\right)+\frac{l-\mid m\mid +1}{2l+1}{P}_{l+1}^{m}\left(x\right)$x P^m_l(x) = \frac{l+|m|}{2l+1}P^m_{l-1}(x)+\frac{l-|m|+1}{2l+1} P^m_{l+1}(x)

and the differential equation

$\frac{d}{dx}\left[\left(1-{x}^{2}\right)\frac{d{P}_{l}^{m}}{dx}\right]+\left[\left(l\left(l+1\right)-\frac{{m}^{2}}{1-{x}^{2}}\right]{P}_{l}^{m}=0$\frac{d}{d x}[(1-x^2)\frac{d P^m_l}{d x}]+[(l(l+1)-\frac{m^2}{1-x^2}]P^m_l = 0

${P}_{\nu }\left(x\right)$ is a special case of a hypergeometric function, namely

${P}_{\nu }\left(x\right)={}_{2}{F}_{1}\left(-\nu ,\nu +1;1;\frac{1-x}{2}\right)$P_\nu(x)={}_2 F_1(-\nu,\nu+1;1;\frac{1-x}{2})

Legendre polynomials enter the expressions for the spherical function?s for sphere ${S}^{2}$ in 3d:

${Y}_{lm}\left(\theta ,\varphi \right)=\sqrt{\frac{2l+1}{4\pi }\frac{\left(l-m\right)!}{\left(l+m\right)!}}{P}_{l}^{m}\left(\mathrm{cos}\theta \right){e}^{\mathrm{im}\varphi }$Y_{l m}(\theta,\phi) = \sqrt{ \frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!} } P^m_l(cos\theta) e^{im\phi}

(If it is not clear from mathML rendering – both fractions are under square root – including both the numerators and denominators).

If $\left(\theta ,\varphi \right)$ and $\left(\theta \prime ,\varphi \prime \right)$ are two points of the unit sphere in spherical coordinates (polar angle, azimuth), and $\gamma$ is the angle between the two corresponding rays from the origin then

${P}_{l}\left(\mathrm{cos}\gamma \right)=\sum _{m=-l}^{l}\left(-1{\right)}^{m}{P}_{l}^{m}\left(\mathrm{cos}\theta \right){P}_{l}^{-m}\left(\mathrm{cos}\theta \prime \right)\mathrm{cos}\left(m\left(\varphi -\varphi \prime \right)\right)=\frac{4\pi }{2l+1}\sum _{m=-l}^{l}{Y}_{\mathrm{lm}}\left(\theta ,\varphi \right){Y}_{\mathrm{lm}}^{*}\left(\theta \prime ,\varphi \prime \right)$P_l(cos \gamma) = \sum_{m=-l}^l (-1)^m P_l^m(cos \theta)P_l^{-m}(cos \theta') cos(m(\phi-\phi')) = \frac{4\pi}{2l+1}\sum_{m=-l}^l Y_{lm}(\theta,\phi) Y_{lm}^*(\theta',\phi')

what for $l=1$ reduces to the spherical law of cosine from spherical trigonometry:

$\mathrm{cos}\gamma ={P}_{1}\left(\mathrm{cos}\gamma \right)=\mathrm{cos}\theta \mathrm{cos}\theta \prime +\mathrm{sin}\theta \mathrm{sin}\theta \prime \mathrm{cos}\left(\varphi -\varphi \prime \right)$cos \gamma = P_1(cos \gamma) = cos\theta cos\theta' + sin\theta sin\theta' cos(\phi-\phi')

The orthogonality relation for Legendre polynomials gives Laplace’s formula

${\int }_{{S}^{2}}d{\Omega }_{\stackrel{^}{k}}{Y}_{l}\left(\stackrel{^}{k}\right){P}_{l}\left(\stackrel{^}{k}\stackrel{^}{p}\right)=\frac{4\pi }{2l+1}{\delta }_{\mathrm{ll}\prime }{Y}_{l}\left(\stackrel{^}{p}\right)$\int_{S^2} d\Omega_{\hat{k}} Y_l(\hat{k}) P_l(\hat{k}\hat{p}) = \frac{4\pi}{2l+1} \delta_{ll'} Y_l(\hat{p})

where $\stackrel{^}{k},\stackrel{^}{p}$ are unit vectors and ${Y}_{l}=\sum {a}_{m}{Y}_{\mathrm{lm}}$ is some spherical function.

The following orthogonality integral relation is over product of unit spheres in ${ℝ}^{3}$:

${\int }_{\stackrel{^}{k}}{\int }_{\stackrel{^}{p}}d{\Omega }_{\stackrel{^}{k}}d{\Omega }_{\stackrel{^}{p}}{P}_{l}\left(\stackrel{^}{k}\stackrel{^}{p}\right){P}_{l\prime }\left(\stackrel{^}{k}\stackrel{^}{q}\right){P}_{l″}\left(\stackrel{^}{p}\stackrel{^}{q}\right)=\left(\frac{4\pi }{2l+1}\right){\delta }_{\mathrm{ll}\prime }{\delta }_{\mathrm{ll}″},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left(*\right)$\int_{\hat{k}}\int_{\hat{p}} d\Omega_{\hat{k}}d\Omega_{\hat{p}} P_l(\hat{k}\hat{p}) P_{l'}(\hat{k}\hat{q}) P_{l''}(\hat{p}\hat{q}) = \left(\frac{4\pi}{2l+1}\right)\delta_{ll'}\delta_{ll''}, \,\,\,\,\,(*)

where the arguments of Legendre polynomials are the inner products of the unit vectors.

Zoran: the last formula $\left(*\right)$ is in my own formula notes which I have written as a student many years ago and used hundreds of times, but it now looks to me suspicious; I have no time to check it right now.

Revised on August 24, 2009 19:31:15 by Toby Bartels (71.104.230.172)