Editing Legendre polynomials

{{Short description|System of complete and orthogonal polynomials}}
{{distinguish|text = [[Lagrange polynomial]] (the interpolating polynomial)}}

[[File:Legendrepolynomials6.svg|360px|thumb|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 function]]s, Legendre functions of the second kind, [[big q-Legendre polynomials]], and [[associated Legendre function]]s.

== 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.]]

== Main properties ==

=== Orthogonality ===

The standardization <math>P_n(1) = 1</math> fixes the normalization of the Legendre polynomials (with respect to the [[L2-norm|{{math|''L''<sup>2</sup>}} norm]] on the interval {{math|−1 ≤ ''x'' ≤ 1}}). Since they are also [[orthogonal function|orthogonal]] with respect to the same norm, the two statements{{clarify|reason=unclear what two statements are being referred to|date=April 2022}} 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 {{math|''δ<sub>mn</sub>''}} denotes the [[Kronecker delta]], equal to 1 if {{math|1=''m'' = ''n''}} and to 0 otherwise).
This normalization is most readily found by employing [[Rodrigues' formula]], given below.

=== Completeness ===

That the polynomials are complete means the following. Given any [[piecewise]] continuous function <math> f(x) </math> with finitely many discontinuities in the interval {{closed-closed|−1, 1}}, 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 {{math|−1 ≤ ''x'' ≤ 1}} and {{math|−1 ≤ ''y'' ≤ 1}}.

==Applications==

===Expanding an inverse distance potential===
{{main|Laplace expansion (potential)}}

The Legendre polynomials were first introduced in 1782 by [[Adrien-Marie Legendre]]<ref>{{cite book |first1=A.-M. |last1=Legendre |chapter=Recherches sur l'attraction des sphéroïdes homogènes |title=Mémoires de Mathématiques et de Physique, présentés à l'Académie Royale des Sciences, par divers savans, et lus dans ses Assemblées |volume=X |pages=411–435 |location=Paris |date=1785 |orig-year=1782 |language=fr |chapter-url=http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf |url-status=dead |archive-url=https://web.archive.org/web/20090920070434/http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf |archive-date=2009-09-20 }}</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 {{math|''r''}} and {{math|''r''′}} are the lengths of the vectors {{math|'''x'''}} and {{math|'''x'''′}} respectively and {{math|''γ''}} is the angle between those two vectors. The series converges when {{math|''r'' > ''r''′}}. 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 [[electric potential|potential]], {{math|1=∇<sup>2</sup> Φ('''x''') = 0}}, 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 [[azimuth|azimuthal angle]]). Where {{math|'''ẑ'''}} is the axis of symmetry and {{math|''θ''}} is the angle between the position of the observer and the {{math|'''ẑ'''}} 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>

{{math|''A<sub>l</sub>''}} and {{math|''B<sub>l</sub>''}} are to be determined according to the boundary condition of each problem.<ref>{{cite book|last=Jackson |first=J. D. |title=Classical Electrodynamics |url=https://archive.org/details/classicalelectro00jack_449 |url-access=limited |edition= 3rd |location=Wiley & Sons |date=1999 |page=[https://archive.org/details/classicalelectro00jack_449/page/n102 103] |isbn=978-0-471-30932-1}}</ref>

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

=== In multipole expansions ===
[[File:Point axial multipole.svg|right|Diagram for the multipole expansion of electric potential.]]
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 expansion]]s.  The left-hand side of the equation is the [[generating function]] for the Legendre polynomials.

As an example, the [[electric potential]] {{math|Φ(''r'',''θ'')}} (in [[spherical coordinates]]) due to a [[point charge]] located on the {{math|''z''}}-axis at {{math|1=''z'' = ''a''}} (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 {{math|''r''}} of the observation point {{math|P}} is greater than {{math|''a''}}, 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 {{math|1=''η'' = {{sfrac|''a''|''r''}} < 1}} and {{math|1=''x'' = cos ''θ''}}. This expansion is used to develop the normal [[multipole expansion]].

Conversely, if the radius {{math|''r''}} of the observation point {{math|P}} is smaller than {{math|''a''}}, the potential may still be expanded in the Legendre polynomials as above, but with {{math|''a''}} and {{math|''r''}} exchanged. This expansion is the basis of [[interior multipole expansion]].

=== In trigonometry ===

The trigonometric functions {{math|cos ''nθ''}}, also denoted as the [[Chebyshev polynomials]] {{math|''T<sub>n</sub>''(cos ''θ'') ≡ cos ''nθ''}}, can also be multipole expanded by the Legendre polynomials {{math|''P<sub>n</sub>''(cos ''θ'')}}. 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 {{math|sin (''n'' + 1)''θ''}}, 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 networks ===

A [[recurrent neural network]] that contains a {{math|''d''}}-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 [[Approximation theory|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>{{cite conference |last1=Voelker |first1=Aaron R. |last2=Kajić |first2=Ivana |last3=Eliasmith |first3=Chris |title=Legendre Memory Units: Continuous-Time Representation in Recurrent Neural Networks |url=http://compneuro.uwaterloo.ca/files/publications/voelker.2019.lmu.pdf |conference=Advances in Neural Information Processing Systems |conference-url=https://neurips.cc |year=2019 }}</ref>

== Additional properties ==
Legendre polynomials have definite parity. That is, they are [[Even and odd functions|even or odd]],<ref>{{harvnb|Arfken|Weber|2005|loc=p.753}}</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 {{closed-closed|−1, 1}} is simply given by the leading expansion coefficient <math>a_0</math>.

The underivative is<ref>
{{cite journal|first1=Orion|last1=Ciftja|title=Integrals of Legendre Polynomials over half range and their relation to the electrostatic potential in hemispherical geometry|year=2022|journal=Results in Physics|volume=40|page=105838|doi=10.1016/j.rinp.2022.105838|bibcode=2022ResPh..4005838C |doi-access=free}}
</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>{{cite journal|first1=L.|last1=Carlitz|title=Some integrals containing products of legendre polynomials|year=1961|journal=Archiv Mathem.|volume=12|pages=334–340|doi=10.1007/BF01650571}}</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 {{math|''r''}} and {{math|''r''′}} have [[spherical coordinates]] {{math|(''θ'', ''φ'')}} and {{math|(''θ''′, ''φ''′)}}, respectively.

The product of two Legendre polynomials <ref>{{cite journal|author = Leonard C. Maximon|title = A generating function for the product of two Legendre polynomials|journal = Norske Videnskabers Selskab Forhandlinger | volume = 29 | year = 1957 | pages = 82–86 | url=https://www.researchgate.net/publication/269015726}}</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>{{Cite journal |date=1 July 1837 |title=Sur les séries dont le terme général dépend de deux angles, et qui servent à exprimer des fonctions arbitraires entre des limites donnée. |url=https://www.degruyter.com/document/doi/10.1515/crll.1837.17.35/html |journal=Journal für die reine und angewandte Mathematik (Crelles Journal) |volume=1837 |issue=17 |pages=35–56 |doi=10.1515/crll.1837.17.35 |issn=0075-4102}}</ref><ref>{{Cite journal |last=Mehler |first=F. G. |date=June 1881 |title=Ueber eine mit den Kugel- und Cylinderfunctionen verwandte Function und ihre Anwendung in der Theorie der Elektricitätsvertheilung |url=http://link.springer.com/10.1007/BF01445847 |journal=Mathematische Annalen |language=de |volume=18 |issue=2 |pages=161–194 |doi=10.1007/BF01445847 |issn=0025-5831}}</ref><ref name=":0" />{{Pg|page=86|location=Eq. 4.8.6, Eq. 4.8.7}}<ref>{{Cite web |title=DLMF: §18.10 Integral Representations ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials |url=https://dlmf.nist.gov/18.10 |access-date=18 March 2025 |website=dlmf.nist.gov}}</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>{{Cite journal |date=1896-12-31 |title=II. On a type of spherical harmonics of unrestricted degree, order, and argument |url=https://royalsocietypublishing.org/doi/10.1098/rspl.1895.0075 |journal=Proceedings of the Royal Society of London |language=en |volume=59 |issue=353–358 |pages=189–196 |doi=10.1098/rspl.1895.0075 |issn=0370-1662}}</ref><ref>{{Citation |last=Gasper |first=George |title=Formulas of the dirichlet-mehler type |date=1975 |work=Fractional Calculus and Its Applications |series=Lecture Notes in Mathematics |volume=457 |pages=207–215 |editor-last=Ross |editor-first=Bertram |url=http://link.springer.com/10.1007/BFb0067105 |access-date=2025-03-18 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |doi=10.1007/bfb0067105 |isbn=978-3-540-07161-7}}</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 relations===
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 {{math|{{norm|''P<sub>n</sub>''}}}} is the norm over the interval {{math|−1 ≤ ''x'' ≤ 1}}
<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>{{Cite journal |last=Doha |first=E. H. |date=1991-01-01 |title=The coefficients of differentiated expansions and derivatives of ultraspherical polynomials |url=https://dx.doi.org/10.1016/0898-1221%2891%2990089-M |journal=Computers & Mathematics with Applications |volume=21 |issue=2 |pages=115–122 |doi=10.1016/0898-1221(91)90089-M |issn=0898-1221}}</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>

===Asymptotics===

Asymptotically, for <math>\ell \to \infty</math>, the Legendre polynomials can be written as <ref name=":0">{{Cite book |last=Szegő |first=Gábor |title=Orthogonal polynomials |date=1975 |publisher=American Mathematical Society |isbn=0821810235 |edition=4th |location=Providence |oclc=1683237}}</ref>{{Pg|location=Theorem 8.21.2|page=194}}
<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>{{Cite web|url=https://dlmf.nist.gov/14.15.E13|title = DLMF: 14.15 Uniform Asymptotic Approximations}}</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 {{math|''J''<sub>0</sub>}}, {{math|''J''<sub>1</sub>}}, and {{math|''I''<sub>0</sub>}} are [[Bessel functions]].

=== Zeros ===
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>{{Cite journal |last=Askey |first=Richard |date=November 1969 |title=Mehler's Integral for P_n (cos θ) |url=https://www.tandfonline.com/doi/abs/10.1080/00029890.1969.12000407 |journal=The American Mathematical Monthly |language=en |volume=76 |issue=9 |pages=1046–1049 |doi=10.1080/00029890.1969.12000407 |issn=0002-9890}}</ref> This can be proved by looking at the first formula of Dirichlet-Mehler.<ref>{{Cite journal |last=Bruns |first=H. |date=1881 |title=Zur Theorie der Kugelfunctionen. |url=https://www.degruyter.com/document/doi/10.1515/crll.1881.90.322/html |journal=CRLL |language=en |volume=1881 |issue=90 |pages=322–328 |doi=10.1515/crll.1881.90.322 |issn=1435-5345}}</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 evaluations===
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 argument ==

=== Shifted Legendre polynomials ===

The '''shifted Legendre polynomials''' are defined as
<math display="block">\widetilde{P}_n(x) = P_n(2x-1) \,.</math>
Here the "shifting" function {{math|''x'' ↦ 2''x'' − 1}} is an [[affine transformation]] that [[bijection|bijectively maps]] the interval {{closed-closed|0, 1}} to the interval {{closed-closed|−1, 1}}, implying that the polynomials {{math|''P̃<sub>n</sub>''(''x'')}} are orthogonal on {{closed-closed|0, 1}}:
<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:
{| class="wikitable" style="text-align: right;"
! <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 functions ===
{{main|Legendre rational functions}}

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 [[eigenfunction]]s 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 also==
{{div col|colwidth=20em}}
* [[Gaussian quadrature]]
* [[Gegenbauer polynomials]]
* [[Turán's inequalities]]
* [[Legendre wavelet]]
* [[Legendre function]]
* [[Jacobi polynomials]]
* [[Romanovski polynomials]]
* [[Laplace expansion (potential)]]
{{div col end}}

== Notes ==
{{reflist|30em}}

== References ==
{{refbegin|30em}}
* {{Abramowitz_Stegun_ref2|8|332|22|773}}
* {{cite book|first1=George B.|last1=Arfken|author-link1=George B. Arfken|first2=Hans J.|last2=Weber|year=2005|title=Mathematical Methods for Physicists|publisher=Elsevier Academic Press|isbn=0-12-059876-0}}
* {{cite book|last=Bayin|first=S. S.|year=2006|title=Mathematical Methods in Science and Engineering|publisher=Wiley|isbn= 978-0-470-04142-0|at=ch. 2}}
* {{cite book|last=Belousov|first=S. L.|year=1962|title=Tables of Normalized Associated Legendre Polynomials|series=Mathematical Tables|volume=18|publisher=Pergamon Press|isbn=978-0-08-009723-7}}
* {{cite book|first1=Richard|last1=Courant|author-link1=Richard Courant|first2=David|last2=Hilbert|author-link2=David Hilbert|year=1953|title=Methods of Mathematical Physics| volume= 1|publisher=Interscience |location=New York, NY|isbn=978-0-471-50447-4}}
* {{dlmf|first=T. M. |last=Dunster|id=14|title=Legendre and Related Functions}}
* {{cite book| first=Refaat | last=El Attar | title= Legendre Polynomials and Functions | publisher= CreateSpace | year=2009 | isbn = 978-1-4414-9012-4}}
* {{dlmf|id=18|title=Orthogonal Polynomials|first=Tom H. |last=Koornwinder|authorlink=Tom H. Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek|first4=René F. |last4=Swarttouw}}
{{refend}}

==External links==
{{Commons category|Legendre polynomials}}
* [http://www.physics.drexel.edu/~tim/open/hydrofin A quick informal derivation of the Legendre polynomial in the context of the quantum mechanics of hydrogen]
* {{springer|title=Legendre polynomials|id=p/l058050}}
* [http://mathworld.wolfram.com/LegendrePolynomial.html Wolfram MathWorld entry on Legendre polynomials]
* [https://web.archive.org/web/20060427014500/http://www.du.edu/~jcalvert/math/legendre.htm Dr James B. Calvert's article on Legendre polynomials from his personal collection of mathematics]
* [https://web.archive.org/web/20181009221546/http://www.morehouse.edu/facstaff/cmoore/Legendre%20Polynomials.htm The Legendre Polynomials by Carlyle E. Moore]
* [http://hyperphysics.phy-astr.gsu.edu/hbase/math/legend.html Legendre Polynomials from Hyperphysics]

{{Authority control}}

{{DEFAULTSORT:Legendre Polynomials}}
[[Category:Special hypergeometric functions]]
[[Category:Orthogonal polynomials]]
[[Category:Polynomials]]