Legendre polynomials

In this approach, the polynomials are defined as an orthogonal system with respect to the weight function ${\displaystyle w(x)=1}$ over the interval ${\displaystyle [-1,1]}$. That is, ${\displaystyle P_{n}(x)}$ is a polynomial of degree ${\displaystyle n}$, such that

$${\displaystyle \int _{-1}^{1}P_{m}(x)P_{n}(x)\,dx=0\quad {\text{if }}n\neq m.}$$

With the additional standardization condition ${\displaystyle P_{n}(1)=1}$, all the polynomials can be uniquely determined. We then start the construction process: ${\displaystyle P_{0}(x)=1}$ is the only correctly standardized polynomial of degree 0. ${\displaystyle P_{1}(x)}$ must be orthogonal to ${\displaystyle P_{0}}$, leading to ${\displaystyle P_{1}(x)=x}$, and ${\displaystyle P_{2}(x)}$ is determined by demanding orthogonality to ${\displaystyle P_{0}}$ and ${\displaystyle P_{1}}$, and so on. ${\displaystyle P_{n}}$ is fixed by demanding orthogonality to all ${\displaystyle P_{m}}$ with ${\displaystyle m<n}$. This gives ${\displaystyle n}$ conditions, which, along with the standardization ${\displaystyle P_{n}(1)=1}$ fixes all ${\displaystyle n+1}$ coefficients in ${\displaystyle P_{n}(x)}$. With work, all the coefficients of every polynomial can be systematically determined, leading to the explicit representation in powers of ${\displaystyle x}$ given below.

This definition of the ${\displaystyle P_{n}}$'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, ${\displaystyle x,x^{2},x^{3},\ldots }$. Finally, by defining them via orthogonality with respect to the most obvious weight function on a finite interval, it sets up the Legendre polynomials as one of the three classical orthogonal polynomial systems. The other two are the Laguerre polynomials, which are orthogonal over the half line ${\displaystyle [0,\infty )}$, and the Hermite polynomials, orthogonal over the full line ${\displaystyle (-\infty ,\infty )}$, with weight functions that are the most natural analytic functions that ensure convergence of all integrals.

The Legendre polynomials can also be defined as the coefficients in a formal expansion in powers of ${\displaystyle t}$ of the generating function[1]

$${\displaystyle {\frac {1}{\sqrt {1-2xt+t^{2}}}}=\sum _{n=0}^{\infty }P_{n}(x)t^{n}\,.}$$

(2)

The coefficient of ${\displaystyle t^{n}}$ is a polynomial in ${\displaystyle x}$ of degree ${\displaystyle n}$ with ${\displaystyle |x|\leq 1}$. Expanding up to ${\displaystyle t^{1}}$ gives

$${\displaystyle P_{0}(x)=1\,,\quad P_{1}(x)=x.}$$

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 ${\displaystyle P_{n}}$'s without resorting to direct expansion of the Taylor series, however. Equation 2 is differentiated with respect to t on both sides and rearranged to obtain

$${\displaystyle {\frac {x-t}{\sqrt {1-2xt+t^{2}}}}=\left(1-2xt+t^{2}\right)\sum _{n=1}^{\infty }nP_{n}(x)t^{n-1}\,.}$$

Replacing the quotient of the square root with its definition in Eq. 2, and equating the coefficients of powers of t in the resulting expansion gives Bonnet’s recursion formula

$${\displaystyle (n+1)P_{n+1}(x)=(2n+1)xP_{n}(x)-nP_{n-1}(x)\,.}$$

This relation, along with the first two polynomials P₀ and P₁, 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.

A third definition is in terms of solutions to Legendre's differential equation:

$${\displaystyle (1-x^{2})P_{n}''(x)-2xP_{n}'(x)+n(n+1)P_{n}(x)=0.}$$

(1)

This differential equation has regular singular points at x = ±1 so if a solution is sought using the standard Frobenius or power series method, a series about the origin will only converge for |x| < 1 in general. When n is an integer, the solution P_n(x) that is regular at x = 1 is also regular at 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,

$${\displaystyle {\frac {d}{dx}}\left(\left(1-x^{2}\right){\frac {d}{dx}}\right)P(x)=-\lambda P(x)\,,}$$

with the eigenvalue ${\displaystyle \lambda }$ in lieu of ${\displaystyle n(n+1)}$. If we demand that the solution be regular at ${\displaystyle x=\pm 1}$, the differential operator on the left is Hermitian. The eigenvalues are found to be of the form n(n + 1), with ${\displaystyle n=0,1,2,\ldots }$ and the eigenfunctions are the ${\displaystyle P_{n}(x)}$. 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 functions of the second kind ${\displaystyle Q_{n}}$. A two-parameter generalization of (Eq. 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 equations) 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 ${\displaystyle P_{n}(\cos \theta )}$ where ${\displaystyle \theta }$ 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.

The standardization ${\displaystyle P_{n}(1)=1}$ fixes the normalization of the Legendre polynomials (with respect to the L² norm on the interval −1 ≤ x ≤ 1). Since they are also orthogonal with respect to the same norm, the two statements can be combined into the single equation,

$${\displaystyle \int _{-1}^{1}P_{m}(x)P_{n}(x)\,dx={\frac {2}{2n+1}}\delta _{mn},}$$

(where δ_mn denotes the Kronecker delta, equal to 1 if m = n and to 0 otherwise). This normalization is most readily found by employing Rodrigues' formula, given below.

That the polynomials are complete means the following. Given any piecewise continuous function ${\displaystyle f(x)}$ with finitely many discontinuities in the interval [−1, 1], the sequence of sums

$${\displaystyle f_{n}(x)=\sum _{\ell =0}^{n}a_{\ell }P_{\ell }(x)}$$

converges in the mean to ${\displaystyle f(x)}$ as ${\displaystyle n\to \infty }$, provided we take

$${\displaystyle a_{\ell }={\frac {2\ell +1}{2}}\int _{-1}^{1}f(x)P_{\ell }(x)\,dx.}$$

This completeness property underlies all the expansions discussed in this article, and is often stated in the form

$${\displaystyle \sum _{\ell =0}^{\infty }{\frac {2\ell +1}{2}}P_{\ell }(x)P_{\ell }(y)=\delta (x-y),}$$

with −1 ≤ x ≤ 1 and −1 ≤ y ≤ 1.

An especially compact expression for the Legendre polynomials is given by Rodrigues' formula:

$${\displaystyle P_{n}(x)={\frac {1}{2^{n}n!}}{\frac {d^{n}}{dx^{n}}}(x^{2}-1)^{n}\,.}$$

This formula enables derivation of a large number of properties of the ${\displaystyle P_{n}}$'s. Among these are explicit representations such as

$${\displaystyle {\begin{aligned}P_{n}(x)&=[t^{n}]{\frac {((t+x)^{2}-1)^{n}}{2^{n}}}=[t^{n}]{\frac {(t+x+1)^{n}(t+x-1)^{n}}{2^{n}}},\\P_{n}(x)&={\frac {1}{2^{n}}}\sum _{k=0}^{n}{\binom {n}{k}}^{2}(x-1)^{n-k}(x+1)^{k},\\P_{n}(x)&=\sum _{k=0}^{n}{\binom {n}{k}}{\binom {n+k}{k}}\left({\frac {x-1}{2}}\right)^{k},\\P_{n}(x)&={\frac {1}{2^{n}}}\sum _{k=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }(-1)^{k}{\binom {n}{k}}{\binom {2n-2k}{n}}x^{n-2k},\\P_{n}(x)&=2^{n}\sum _{k=0}^{n}x^{k}{\binom {n}{k}}{\binom {\frac {n+k-1}{2}}{n}}.\end{aligned}}}$$

In the fourth representation, ⌊n/2⌋ stands for the largest integer less than or equal to n/2. The last representation, which is also immediate from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the generalized form of the binomial coefficient.

The first few Legendre polynomials are:

${\displaystyle n}$	${\displaystyle P_{n}(x)}$
0	${\textstyle 1}$
1	${\textstyle x}$
2	${\textstyle {\tfrac {1}{2}}\left(3x^{2}-1\right)}$
3	${\textstyle {\tfrac {1}{2}}\left(5x^{3}-3x\right)}$
4	${\textstyle {\tfrac {1}{8}}\left(35x^{4}-30x^{2}+3\right)}$
5	${\textstyle {\tfrac {1}{8}}\left(63x^{5}-70x^{3}+15x\right)}$
6	${\textstyle {\tfrac {1}{16}}\left(231x^{6}-315x^{4}+105x^{2}-5\right)}$
7	${\textstyle {\tfrac {1}{16}}\left(429x^{7}-693x^{5}+315x^{3}-35x\right)}$
8	${\textstyle {\tfrac {1}{128}}\left(6435x^{8}-12012x^{6}+6930x^{4}-1260x^{2}+35\right)}$
9	${\textstyle {\tfrac {1}{128}}\left(12155x^{9}-25740x^{7}+18018x^{5}-4620x^{3}+315x\right)}$
10	${\textstyle {\tfrac {1}{256}}\left(46189x^{10}-109395x^{8}+90090x^{6}-30030x^{4}+3465x^{2}-63\right)}$

The graphs of these polynomials (up to n = 5) are shown below:

Plot of the six first Legendre polynomials.

Legendre polynomials have definite parity. That is, they are even or odd,[5] according to

$${\displaystyle P_{n}(-x)=(-1)^{n}P_{n}(x)\,.}$$

Another useful property is

$${\displaystyle \int _{-1}^{1}P_{n}(x)\,dx=0{\text{ for }}n\geq 1,}$$

which follows from considering the orthogonality relation with ${\displaystyle P_{0}(x)=1}$. It is convenient when a Legendre series ${\textstyle \sum _{i}a_{i}P_{i}}$ is used to approximate a function or experimental data: the average of the series over the interval [−1, 1] is simply given by the leading expansion coefficient ${\displaystyle a_{0}}$.

Since the differential equation and the orthogonality property are independent of scaling, the Legendre polynomials' definitions are "standardized" (sometimes called "normalization", but the actual norm is not 1) by being scaled so that

$${\displaystyle P_{n}(1)=1\,.}$$

The derivative at the end point is given by

$${\displaystyle P_{n}'(1)={\frac {n(n+1)}{2}}\,.}$$

The Askey–Gasper inequality for Legendre polynomials reads

$${\displaystyle \sum _{j=0}^{n}P_{j}(x)\geq 0\quad {\text{for }}\quad x\geq -1\,.}$$

The Legendre polynomials of a scalar product of unit vectors can be expanded with spherical harmonics using

$${\displaystyle P_{\ell }\left(r\cdot r'\right)={\frac {4\pi }{2\ell +1}}\sum _{m=-\ell }^{\ell }Y_{\ell m}(\theta ,\varphi )Y_{\ell m}^{*}(\theta ',\varphi ')\,,}$$

where the unit vectors r and r′ have spherical coordinates (θ, φ) and (θ′, φ′), respectively.

Recurrence relations

As discussed above, the Legendre polynomials obey the three-term recurrence relation known as Bonnet's recursion formula given by

$${\displaystyle (n+1)P_{n+1}(x)=(2n+1)xP_{n}(x)-nP_{n-1}(x)}$$

and

$${\displaystyle {\frac {x^{2}-1}{n}}{\frac {d}{dx}}P_{n}(x)=xP_{n}(x)-P_{n-1}(x)}$$

or, with the alternative expression, which also holds at the endpoints

$${\displaystyle {\frac {d}{dx}}P_{n+1}(x)=(n+1)P_{n}(x)+x{\frac {d}{dx}}P_{n}(x)\,.}$$

Useful for the integration of Legendre polynomials is

$${\displaystyle (2n+1)P_{n}(x)={\frac {d}{dx}}{\bigl (}P_{n+1}(x)-P_{n-1}(x){\bigr )}\,.}$$

From the above one can see also that

$${\displaystyle {\frac {d}{dx}}P_{n+1}(x)=(2n+1)P_{n}(x)+{\bigl (}2(n-2)+1{\bigr )}P_{n-2}(x)+{\bigl (}2(n-4)+1{\bigr )}P_{n-4}(x)+\cdots }$$

or equivalently

$${\displaystyle {\frac {d}{dx}}P_{n+1}(x)={\frac {2P_{n}(x)}{\left\|P_{n}\right\|^{2}}}+{\frac {2P_{n-2}(x)}{\left\|P_{n-2}\right\|^{2}}}+\cdots }$$

where ‖P_n‖ is the norm over the interval −1 ≤ x ≤ 1

$${\displaystyle \|P_{n}\|={\sqrt {\int _{-1}^{1}{\bigl (}P_{n}(x){\bigr )}^{2}\,dx}}={\sqrt {\frac {2}{2n+1}}}\,.}$$

Asymptotics

Asymptotically, for ${\displaystyle \ell \to \infty }$, the Legendre polynomials can be written as [6]

$${\displaystyle {\begin{aligned}P_{\ell }(\cos \theta )&={\sqrt {\frac {\theta }{\sin \theta }}}\,J_{0}((\ell +1/2)\theta )+{\mathcal {O}}\left(\ell ^{-1}\right)\\&={\frac {2}{\sqrt {2\pi \ell \sin \theta }}}\cos \left(\left(\ell +{\tfrac {1}{2}}\right)\theta -{\frac {\pi }{4}}\right)+{\mathcal {O}}\left(\ell ^{-3/2}\right),\quad \theta \in (0,\pi ),\end{aligned}}}$$

and for arguments of magnitude greater than 1[7]

$${\displaystyle {\begin{aligned}P_{\ell }\left(\cosh \xi \right)&={\sqrt {\frac {\xi }{\sinh \xi }}}I_{0}\left(\left(\ell +{\frac {1}{2}}\right)\xi \right)\left(1+{\mathcal {O}}\left(\ell ^{-1}\right)\right)\,,\\P_{\ell }\left({\frac {1}{\sqrt {1-e^{2}}}}\right)&={\frac {1}{\sqrt {2\pi \ell e}}}{\frac {(1+e)^{\frac {\ell +1}{2}}}{(1-e)^{\frac {\ell }{2}}}}+{\mathcal {O}}\left(\ell ^{-1}\right)\end{aligned}}}$$

where J₀ and I₀ are Bessel functions.

Pointwise evaluations

The parity and normalization implicate the values at the boundaries ${\displaystyle x=\pm 1}$ to be

$${\displaystyle P_{n}(1)=1\,,\quad P_{n}(-1)=(-1)^{n}}$$

At the origin ${\displaystyle x=0}$ one can show that the values are given by

$${\displaystyle P_{2n}(0)={\frac {(-1)^{n}}{4^{n}}}{\binom {2n}{n}}={\frac {(-1)^{n}}{2^{2n}}}{\frac {(2n)!}{\left(n!\right)^{2}}}}$$

$${\displaystyle P_{2n+1}(0)=0}$$

Wikimedia Commons has media related to Legendre polynomials.

• A quick informal derivation of the Legendre polynomial in the context of the quantum mechanics of hydrogen
• "Legendre polynomials", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Wolfram MathWorld entry on Legendre polynomials
• Dr James B. Calvert's article on Legendre polynomials from his personal collection of mathematics
• The Legendre Polynomials by Carlyle E. Moore
• Legendre Polynomials from Hyperphysics