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File:Legendrepolynomials6.svg
The first six Legendre polynomials

In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications.

Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, big q-Legendre polynomials, and associated Legendre functions.

Definition and representationEdit

Definition by construction as an orthogonal systemEdit

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 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 functionEdit

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>Template:Harvnb</ref> Template:NumBlk = \sum_{n=0}^\infty P_n(x) t^n \,.</math>|Template:EquationRef}}

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 Template:EquationNote is differentiated with respect to Template:Mvar 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. Template:EquationNote, and equating the coefficients of powers of Template:Math 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 Template:Math and Template:Math, 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 equationEdit

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

This differential equation has regular singular points at Template:Math so if a solution is sought using the standard Frobenius or power series method, a series about the origin will only converge for Template:Math in general. When Template:Math is an integer, the solution Template:Math that is regular at Template:Math is also regular at Template:Math, 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 Template:Math, 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 functions of the second kind <math>Q_n</math>. A two-parameter generalization of (Eq. Template:EquationNote) 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 <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 formulasEdit

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>Template:Cite book</ref>

In the fourth representation, <math>\lfloor n/2 \rfloor</math> stands for the 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 generalized form of the binomial coefficient.

The reversal of the representation as a power series is <ref>Template:Cite book</ref><ref>Template:Cite book</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:

<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 Template:Math) are shown below:

Main propertiesEdit

OrthogonalityEdit

The standardization <math>P_n(1) = 1</math> fixes the normalization of the Legendre polynomials (with respect to the [[L2-norm|Template:Math norm]] on the interval Template:Math). Since they are also orthogonal with respect to the same norm, the two statementsTemplate:Clarify can be combined into the single equation, <math display="block">\int_{-1}^1 P_m(x) P_n(x)\,dx = \frac{2}{2n + 1} \delta_{mn},</math> (where Template:Math denotes the Kronecker delta, equal to 1 if Template:Math and to 0 otherwise). This normalization is most readily found by employing Rodrigues' formula, given below.

CompletenessEdit

That the polynomials are complete means the following. Given any piecewise continuous function <math> f(x) </math> with finitely many discontinuities in the interval Template:Closed-closed, the sequence of sums <math display="block"> f_n(x) = \sum_{\ell=0}^n a_\ell P_\ell(x)</math> converges in the mean to <math> f(x) </math> as <math> n \to \infty </math>, provided we take <math display="block"> a_\ell = \frac{2\ell + 1}{2} \int_{-1}^1 f(x) P_\ell(x)\,dx.</math>

This completeness property underlies all the expansions discussed in this article, and is often stated in the form <math display="block">\sum_{\ell=0}^\infty \frac{2\ell + 1}{2} P_\ell(x)P_\ell(y) = \delta(x-y), </math> with Template:Math and Template:Math.

ApplicationsEdit

Expanding an inverse distance potentialEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre<ref>Template:Cite book</ref> as the coefficients in the expansion of the Newtonian potential <math display="block">\frac{1}{\left| \mathbf{x}-\mathbf{x}' \right|} = \frac{1}{\sqrt{r^2+{r'}^2-2r{r'}\cos\gamma}} = \sum_{\ell=0}^\infty \frac{{r'}^\ell}{r^{\ell+1}} P_\ell(\cos \gamma),</math> where Template:Math and Template:Math are the lengths of the vectors Template:Math and Template:Math respectively and Template:Math is the angle between those two vectors. The series converges when Template:Math. The expression gives the gravitational potential associated to a point mass or the Coulomb potential associated to a point charge. The expansion using Legendre polynomials might be useful, for instance, when integrating this expression over a continuous mass or charge distribution.

Legendre polynomials occur in the solution of Laplace's equation of the static potential, Template:Math, in a charge-free region of space, using the method of separation of variables, where the boundary conditions have axial symmetry (no dependence on an azimuthal angle). Where Template:Math is the axis of symmetry and Template:Math is the angle between the position of the observer and the Template:Math axis (the zenith angle), the solution for the potential will be <math display="block">\Phi(r,\theta) = \sum_{\ell=0}^\infty \left( A_\ell r^\ell + B_\ell r^{-(\ell+1)} \right) P_\ell(\cos\theta) \,.</math>

Template:Math and Template:Math are to be determined according to the boundary condition of each problem.<ref>Template:Cite book</ref>

They also appear when solving the Schrödinger equation in three dimensions for a central force.

In multipole expansionsEdit

Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently): <math display="block">\frac{1}{\sqrt{1 + \eta^2 - 2\eta x}} = \sum_{k=0}^\infty \eta^k P_k(x),</math> which arise naturally in multipole expansions. The left-hand side of the equation is the generating function for the Legendre polynomials.

As an example, the electric potential Template:Math (in spherical coordinates) due to a point charge located on the Template:Math-axis at Template:Math (see diagram right) varies as <math display="block">\Phi (r, \theta ) \propto \frac{1}{R} = \frac{1}{\sqrt{r^2 + a^2 - 2ar \cos\theta}}.</math>

If the radius Template:Math of the observation point Template:Math is greater than Template:Math, the potential may be expanded in the Legendre polynomials <math display="block">\Phi(r, \theta) \propto \frac{1}{r} \sum_{k=0}^\infty \left( \frac{a}{r} \right)^k P_k(\cos \theta),</math> where we have defined Template:Math and Template:Math. This expansion is used to develop the normal multipole expansion.

Conversely, if the radius Template:Math of the observation point Template:Math is smaller than Template:Math, the potential may still be expanded in the Legendre polynomials as above, but with Template:Math and Template:Math exchanged. This expansion is the basis of interior multipole expansion.

In trigonometryEdit

The trigonometric functions Template:Math, also denoted as the Chebyshev polynomials Template:Math, can also be multipole expanded by the Legendre polynomials Template:Math. The first several orders are as follows: <math display="block">\begin{alignat}{2} T_0(\cos\theta)&=1 &&=P_0(\cos\theta),\\[4pt] T_1(\cos\theta)&=\cos \theta&&=P_1(\cos\theta),\\[4pt] T_2(\cos\theta)&=\cos 2\theta&&=\tfrac{1}{3}\bigl(4P_2(\cos\theta)-P_0(\cos\theta)\bigr),\\[4pt] T_3(\cos\theta)&=\cos 3\theta&&=\tfrac{1}{5}\bigl(8P_3(\cos\theta)-3P_1(\cos\theta)\bigr),\\[4pt] T_4(\cos\theta)&=\cos 4\theta&&=\tfrac{1}{105}\bigl(192P_4(\cos\theta)-80P_2(\cos\theta)-7P_0(\cos\theta)\bigr),\\[4pt] T_5(\cos\theta)&=\cos 5\theta&&=\tfrac{1}{63}\bigl(128P_5(\cos\theta)-56P_3(\cos\theta)-9P_1(\cos\theta)\bigr),\\[4pt] T_6(\cos\theta)&=\cos 6\theta&&=\tfrac{1}{1155}\bigl(2560P_6(\cos\theta)-1152P_4(\cos\theta)-220P_2(\cos\theta)-33P_0(\cos\theta)\bigr). \end{alignat}</math>

This can be summarized for <math>n>0</math> as

<math> T_n(x)=2^{2n-n'}\hat n!\sum_{t=0}^{\hat n} (n-2t+1/2) \frac{(n-t-1)!}{2^{2t}t!(n-1)!} \times \frac{(-1)\cdot 1\cdot 3\cdots (2t-3)}{(1+2n')(3+2n')\cdots (2n-2t+1)}P_{n-2t}(x) . </math>

where <math>\hat n\equiv \lfloor n/2\rfloor</math>, <math>n'\equiv \lfloor (n+1)/2\rfloor</math>, and where the products with the steps of two in the numerator and denominator are to be interpreted as 1 if the are empty, i.e., if the last factor is smaller than the first factor.


Another property is the expression for Template:Math, which is <math display="block">\frac{\sin (n+1)\theta}{\sin\theta}=\sum_{\ell=0}^n P_\ell(\cos\theta) P_{n-\ell}(\cos\theta).</math>

In recurrent neural networksEdit

A recurrent neural network that contains a Template:Math-dimensional memory vector, <math>\mathbf{m} \in \R^d</math>, can be optimized such that its neural activities obey the linear time-invariant system given by the following state-space representation: <math display="block">\theta \dot{\mathbf{m}}(t) = A\mathbf{m}(t) + Bu(t),</math> <math display="block">\begin{align} A &= \left[ a \right]_{ij} \in \R^{d \times d} \text{,} \quad && a_{ij} = \left(2i + 1\right) \begin{cases}

 -1 & i < j \\
 (-1)^{i-j+1} & i \ge j

\end{cases},\\ B &= \left[ b \right]_i \in \R^{d \times 1} \text{,} \quad && b_i = (2i + 1) (-1)^i . \end{align}</math>

In this case, the sliding window of <math>u</math> across the past <math>\theta</math> units of time is best approximated by a linear combination of the first <math>d</math> shifted Legendre polynomials, weighted together by the elements of <math>\mathbf{m}</math> at time <math>t</math>: <math display="block">u(t - \theta') \approx \sum_{\ell=0}^{d-1} \widetilde{P}_\ell \left(\frac{\theta'}{\theta} \right) \, m_{\ell}(t) , \quad 0 \le \theta' \le \theta .</math>

When combined with deep learning methods, these networks can be trained to outperform long short-term memory units and related architectures, while using fewer computational resources.<ref>Template:Cite conference</ref>

Additional propertiesEdit

Legendre polynomials have definite parity. That is, they are even or odd,<ref>Template:Harvnb</ref> according to <math display="block">P_n(-x) = (-1)^n P_n(x) \,.</math>

Another useful property is <math display="block">\int_{-1}^1 P_n(x)\,dx = 0 \text{ for } n\ge1,</math> which follows from considering the orthogonality relation with <math>P_0(x) = 1</math>. It is convenient when a Legendre series <math display="inline">\sum_i a_i P_i</math> is used to approximate a function or experimental data: the average of the series over the interval Template:Closed-closed is simply given by the leading expansion coefficient <math>a_0</math>.

The underivative is<ref> Template:Cite journal </ref>

<math> \int P_n(x)dx=\frac{1}{2n+1}[P_{n+1}(x)-P_{n-1}(x)],\quad n\ge 1. </math>

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 <math display="block">P_n(1) = 1 \,.</math>

The derivative at the end point is given by <math display="block">P_n'(1) = \frac{n(n+1)}{2} \,. </math>

The product expansion is <ref>Template:Cite journal</ref>

<math> P_m(x)P_n(x)=\sum_{r=0}^{\min(m,n)}\frac{A_rA_{m-r}A_{n-r}}{A_{m+n-r}}\frac{2m+2n-4r+1}{2m+2n-2r+1}P_{m+n-2r}(x) </math>

where <math>A_r\equiv (2r-1)!!/r!</math>.

The Askey–Gasper inequality for Legendre polynomials reads <math display="block">\sum_{j=0}^n P_j(x) \ge 0 \quad \text{for }\quad x\ge -1 \,.</math>

The Legendre polynomials of a scalar product of unit vectors can be expanded with spherical harmonics using <math display="block">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')\,,</math> where the unit vectors Template:Math and Template:Math have spherical coordinates Template:Math and Template:Math, respectively.

The product of two Legendre polynomials <ref>Template:Cite journal</ref> <math display="block">

\sum_{p=0}^\infty t^{p}P_p(\cos\theta_1)P_p(\cos\theta_2)=\frac2\pi\frac{\mathbf K\left( 2\sqrt{\frac{t\sin\theta_1\sin\theta_2}{t^2-2t\cos\left( \theta_1+\theta_2 \right)+1}} \right)}{\sqrt{t^2-2t\cos\left( \theta_1+\theta_2 \right)+1}}\,,</math>

where <math>K(\cdot)</math> is the complete elliptic integral of the first kind.

The formulas of Dirichlet-Mehler:<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref name=":0" />Template:Pg<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><math display="block">P_n(\cos \theta) = \frac{2}{\pi} \int_0^\theta \frac{\cos \left(n+\frac{1}{2}\right) \phi}{(2 \cos \phi - 2 \cos \theta)^{\frac{1}{2}}} d \phi = \frac{2}{\pi} \int_\theta^\pi \frac{\sin \left(n+\frac{1}{2}\right) \phi}{(2 \cos \theta-2 \cos \phi)^{\frac{1}{2}}} d \phi</math>which has generalizations for associated Legendre polynomials.<ref>Template:Cite journal</ref><ref>Template:Citation</ref>

The Fourier-Legendre series:<ref>Lord Rayleigh, Theory of sound, Volume II, p. 273</ref><math display="block">e^{i t x}=\sum_{n=0}^{\infty}(2 n+1) i^n \sqrt{\frac{\pi}{2 t}} J_{n+\frac{1}{2}}(t) P_n(x)</math>where <math>J</math> is the Bessel function of the first kind.

Recurrence relationsEdit

As discussed above, the Legendre polynomials obey the three-term recurrence relation known as Bonnet's recursion formula given by <math display="block"> (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x)</math> and <math display="block"> \frac{x^2-1}{n} \frac{d}{dx} P_n(x) = xP_n(x) - P_{n-1}(x) </math> or, with the alternative expression, which also holds at the endpoints <math display="block"> \frac{d}{dx} P_{n+1}(x) = (n+1)P_n(x) + x \frac{d}{dx}P_{n}(x) \,.</math>

Useful for the integration of Legendre polynomials is <math display="block">(2n+1) P_n(x) = \frac{d}{dx} \bigl( P_{n+1}(x) - P_{n-1}(x) \bigr) \,.</math>

From the above one can see also that <math display="block">\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</math> or equivalently <math display="block">\frac{d}{dx} P_{n+1}(x) = \frac{2 P_n(x)}{\left\| P_n \right\|^2} + \frac{2 P_{n-2}(x)}{\left\| P_{n-2} \right\|^2} + \cdots</math> where Template:Math is the norm over the interval Template:Math <math display="block">\| P_n \| = \sqrt{\int_{-1}^1 \bigl(P_n(x)\bigr)^2 \,dx} = \sqrt{\frac{2}{2 n + 1}} \,.</math>More generally, all orders of derivatives are expressible as a sum of Legendre polynomials:<ref>Template:Cite journal</ref><math display="block">\begin{aligned} &\begin{aligned} & \frac{d^q}{dx^q} P_{q+2 j}(x)=\frac{2^{q-1}}{(q-1)!} \sum_{i=0}^j(4 i+1) \frac{(q+j-i-1)!\Gamma\left(q+j+i+\frac{1}{2}\right)}{(j-i)!\Gamma(j+i+3 / 2)} P_{2 i}(x) \\ & \quad=\frac{1}{2^{q-2}(q-1)!} \sum_{i=0}^j(4 i+1) \frac{(q+j-i-1)!(2 q+2 j+2 i-1)!}{(j-i)!(2 j+2 i+2)!} \frac{(j+i+1)!}{(q+j+i-1)!} P_{2 i}(x) \end{aligned}\\ &\begin{aligned} & \frac{d^q}{dx^q} P_{q+2 j+1}(x)=\frac{2^{q-1}}{(q-1)!} \sum_{i=0}^j(4 i+3) \frac{(q+j-i-1)!\Gamma(q+j+i+3 / 2)}{(j-i)!\Gamma(j+i+5 / 2)} P_{2 i+1}(x) \\ & \quad=\frac{1}{2^{q-2}(q-1)!} \sum_{i=0}^j(4 i+3) \frac{(q+j-i-1)!(2 q+2 j+2 i+1)!}{(j-i)!(2 j+2 i+4)!} \frac{(j+i+2)!}{(q+j+i)!} P_{2 i+1}(x) \end{aligned} \end{aligned}</math>

AsymptoticsEdit

Asymptotically, for <math>\ell \to \infty</math>, the Legendre polynomials can be written as <ref name=":0">Template:Cite book</ref>Template:Pg <math display="block">\begin{align} P_\ell (\cos \theta) &= \sqrt{\frac{\theta}{\sin\left(\theta\right)}} \left\{J_0{\left[\left(\ell+\tfrac{1}{2}\right)\theta\right]} - \frac{\left(\frac{1}{\theta}-\cot\theta\right)}{8(\ell+\frac{1}{2})} J_1{\left[\left(\ell+\tfrac{1}{2}\right)\theta\right]} \right\} + \mathcal{O}\left(\ell^{-2}\right) \\[1ex] &= \sqrt{\frac{2}{\pi \ell\sin\left(\theta\right)}}\cos\left[\left(\ell + \tfrac{1}{2} \right)\theta - \tfrac{\pi}{4}\right] + \mathcal{O}\left(\ell^{-3/2}\right), \quad \theta \in (0,\pi), \end{align}</math> and for arguments of magnitude greater than 1<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">\begin{align} 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{align}</math> where Template:Math, Template:Math, and Template:Math are Bessel functions.

ZerosEdit

All <math> n</math> zeros of <math>P_n(x)</math> are real, distinct from each other, and lie in the interval <math>(-1,1)</math>. Furthermore, if we regard them as dividing the interval <math>[-1,1]</math> into <math> n+1 </math> subintervals, each subinterval will contain exactly one zero of <math>P_{n+1}</math>. This is known as the interlacing property. Because of the parity property it is evident that if <math>x_k</math> is a zero of <math>P_n(x)</math>, so is <math>-x_k</math>. These zeros play an important role in numerical integration based on Gaussian quadrature. The specific quadrature based on the <math>P_n</math>'s is known as Gauss-Legendre quadrature.

The zeros of <math>P_n(\cos \theta)</math> are distributed nearly uniformly over the range of <math>\theta \in (0, \pi)</math>, in the sense that there is one zero <math>\theta \in \left(\frac{\pi(k + 1/2)}{n + 1/2}, \frac{\pi(k + 1)}{n + 1/2}\right)</math> per <math>k = 0, 1, \dots, n-1</math>.<ref>Template:Cite journal</ref> This can be proved by looking at the first formula of Dirichlet-Mehler.<ref>Template:Cite journal</ref>

From this property and the facts that <math> P_n(\pm 1) \ne 0 </math>, it follows that <math> P_n(x) </math> has <math> n-1 </math> local minima and maxima in <math> (-1,1) </math>. Equivalently, <math> dP_n(x)/dx </math> has <math> n -1 </math> zeros in <math> (-1,1) </math>.

Pointwise evaluationsEdit

The parity and normalization implicate the values at the boundaries <math> x=\pm 1 </math> to be <math display="block">

 P_n(1) = 1
 \,, \quad
 P_n(-1) =  (-1)^n

</math> At the origin <math> x=0 </math> one can show that the values are given by <math display="block">

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

= (-1)^n\frac{(2n-1)!!}{(2n)!!} </math><math display="block">

 P_{2n+1}(0) = 0

</math>

Variants with transformed argumentEdit

Shifted Legendre polynomialsEdit

The shifted Legendre polynomials are defined as <math display="block">\widetilde{P}_n(x) = P_n(2x-1) \,.</math> Here the "shifting" function Template:Math is an affine transformation that bijectively maps the interval Template:Closed-closed to the interval Template:Closed-closed, implying that the polynomials Template:Math are orthogonal on Template:Closed-closed: <math display="block">\int_0^1 \widetilde{P}_m(x) \widetilde{P}_n(x)\,dx = \frac{1}{2n + 1} \delta_{mn} \,.</math>

An explicit expression for the shifted Legendre polynomials is given by <math display="block">\widetilde{P}_n(x) = (-1)^n \sum_{k=0}^n \binom{n}{k} \binom{n+k}{k} (-x)^k \,.</math>

The analogue of Rodrigues' formula for the shifted Legendre polynomials is <math display="block">\widetilde{P}_n(x) = \frac{1}{n!} \frac{d^n}{dx^n} \left(x^2 -x \right)^n \,.</math>

The first few shifted Legendre polynomials are:

<math>n</math> <math>\widetilde{P}_n(x)</math>
0 <math>1</math>
1 <math>2x-1</math>
2 <math>6x^2-6x+1</math>
3 <math>20x^3-30x^2+12x-1</math>
4 <math>70x^4-140x^3+90x^2-20x+1</math>
5 <math>252x^5 -630x^4 +560x^3 - 210 x^2 + 30 x - 1</math>

Legendre rational functionsEdit

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The Legendre rational functions are a sequence of orthogonal functions on [0, ∞). They are obtained by composing the Cayley transform with Legendre polynomials.

A rational Legendre function of degree n is defined as: <math display="block">R_n(x) = \frac{\sqrt{2}}{x+1}\,P_n\left(\frac{x-1}{x+1}\right)\,.</math>

They are eigenfunctions of the singular Sturm–Liouville problem: <math display="block">\left(x+1\right) \frac{d}{dx} \left(x \frac{d}{dx} \left[\left(x+1\right) v(x)\right]\right) + \lambda v(x) = 0</math> with eigenvalues <math display="block">\lambda_n=n(n+1)\,.</math>

See alsoEdit

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NotesEdit

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ReferencesEdit

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External linksEdit

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