Legendre polynomials#Applications of Legendre polynomials in physics

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

$$\backslash int\_\{-1\}^1\; P\_m(x)\; P\_n(x)\; \backslash ,dx\; =\; 0\; \backslash quad\; \backslash text\{if\; \}\; n\; \backslash ne\; m.$$

With the additional standardization condition $P\_n(1)\; =\; 1$, all the polynomials can be uniquely determined. We then start the construction process: $P\_0(x)\; =\; 1$ is the only correctly standardized polynomial of degree 0. $P\_1(x)$ must be orthogonal to $P\_0$, leading to $P\_1(x)\; =\; x$, and $P\_2(x)$ is determined by demanding orthogonality to $P\_0$ and $P\_1$, and so on. $P\_n$ is fixed by demanding orthogonality to all $P\_m$ with . This gives $n$ conditions, which, along with the standardization $P\_n(1)\; =\; 1$ fixes all $n+1$ coefficients in $P\_n(x)$. With work, all the coefficients of every polynomial can be systematically determined, leading to the explicit representation in powers of $x$ given below.

This definition of the $P\_n$'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, $x,\; x^2,\; x^3,\; \backslash ldots$. Finally, by defining them via orthogonality with respect to the Lebesgue measure on $[-1,\; 1]$, 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 $[0,\backslash infty)$ with the weight $e^\{-x\}$, and the Hermite polynomials, orthogonal over the full line $(-\backslash infty,\backslash infty)$ with weight $e^\{-x^2\}$.

The Legendre polynomials can also be defined as the coefficients in a formal expansion in powers of $t$ of the generating function{{harvnb|Arfken|Weber|2005|loc=p.743}}

{{NumBlk||$$\backslash frac\{1\}\{\backslash sqrt\{1-2xt+t^2\}\}\; =\; \backslash sum\_\{n=0\}^\backslash infty\; P\_n(x)\; t^n\; \backslash ,.$$|{{EquationRef|2}}}}

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

$$P\_0(x)\; =\; 1\; \backslash ,,\backslash quad\; P\_1(x)\; =\; x.$$

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 $P\_n$'s without resorting to direct expansion of the Taylor series, however. Equation {{EquationNote|2}} is differentiated with respect to {{mvar|t}} on both sides and rearranged to obtain

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

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

$$(n+1)\; P\_\{n+1\}(x)\; =\; (2n+1)\; x\; P\_n(x)\; -\; n\; P\_\{n-1\}(x)\backslash ,.$$

This relation, along with the first two polynomials {{math|P₀}} and {{math|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:

{{NumBlk||$$(1\; -\; x^2)\; P\_n\text{'}\text{'}(x)\; -\; 2\; x\; P\_n\text{'}(x)\; +\; n\; (n\; +\; 1)\; P\_n(x)\; =\; 0.$$|{{EquationRef|1}}}}

This differential equation has regular singular points at {{math|1=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 {{math|{{abs|x}} < 1}} in general. When {{math|n}} is an integer, the solution {{math|P_n(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,

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

with the eigenvalue $\backslash lambda$ in lieu of $n(n+1)$. If we demand that the solution be regular at

$x\; =\; \backslash pm\; 1$, the differential operator on the left is Hermitian. The eigenvalues are found to be of the form

{{math|n(n + 1)}}, with $n\; =\; 0,\; 1,\; 2,\; \backslash ldots$ and the eigenfunctions are the $P\_n(x)$. 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 $Q\_n$.

A two-parameter generalization of (Eq. {{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 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 $P\_n(\backslash cos\backslash theta)$ where $\backslash theta$ 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.

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

$$P\_n(x)\; =\; \backslash frac\{1\}\{2^n\; n!\}\; \backslash frac\{d^n\}\{dx^n\}\; (x^2\; -1)^n\; \backslash ,.$$

This formula enables derivation of a large number of properties of the $P\_n$'s. Among these are explicit representations such as

$$\backslash begin\{align\}$$

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_

\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}

Expressing the polynomial as a power series, $P\_n(x)\; =\; \backslash sum\; a\_\{n,k\}\; x^k$, the coefficients of powers of $x$ can also be calculated using the recurrences

$$a\_\{n,k\}\; =\; -\; \backslash frac\{(n-k+2)(n+k-1)\}\{k(k-1)\}a\_\{n,k-2\}.$$ or

a_{n,k}=-\frac{n+k-1}{n-k}a_{n-2,k}.

The Legendre polynomial is determined by the values used for the two constants $a\_\{n,0\}$ and $a\_\{n,1\}$, where $a\_\{n,0\}=0$ if $n$ is odd and $a\_\{n,1\}=0$ if $n$ is even.{{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}}

In the fourth representation, $\backslash lfloor\; n/2\; \backslash rfloor$ 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 reversal of the representation as a power series is

{{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}}{{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}}

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).

for $m=0,1,2,\backslash ldots$, where an empty product in the numerator (last factor less than the first factor) evaluates to 1.

The first few Legendre polynomials are:

The graphs of these polynomials (up to {{math|1=n = 5}}) are shown below:

File:Legendrepolynomials6.svg

The standardization $P\_n(1)\; =\; 1$ fixes the normalization of the Legendre polynomials (with respect to the L2-norm on the interval {{math|−1 ≤ x ≤ 1}}). Since they are also 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,

$$\backslash int\_\{-1\}^1\; P\_m(x)\; P\_n(x)\backslash ,dx\; =\; \backslash frac\{2\}\{2n\; +\; 1\}\; \backslash delta\_\{mn\},$$

(where {{math|δ_mn}} 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.

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

$$f\_n(x)\; =\; \backslash sum\_\{\backslash ell=0\}^n\; a\_\backslash ell\; P\_\backslash ell(x)$$

converges in the mean to $f(x)$ as $n\; \backslash to\; \backslash infty$, provided we take

$$a\_\backslash ell\; =\; \backslash frac\{2\backslash ell\; +\; 1\}\{2\}\; \backslash int\_\{-1\}^1\; f(x)\; P\_\backslash ell(x)\backslash ,dx.$$

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

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

with {{math|−1 ≤ x ≤ 1}} and {{math|−1 ≤ y ≤ 1}}.

{{main|Laplace expansion (potential)}}

The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre{{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 }} as the coefficients in the expansion of the Newtonian potential

$$\backslash frac\{1\}\{\backslash left|\; \backslash mathbf\{x\}-\backslash mathbf\{x\}\text{'}\; \backslash right|\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{r^2+\{r\text{'}\}^2-2r\{r\text{'}\}\backslash cos\backslash gamma\}\}\; =\; \backslash sum\_\{\backslash ell=0\}^\backslash infty\; \backslash frac\{\{r\text{'}\}^\backslash ell\}\{r^\{\backslash ell+1\}\}\; P\_\backslash ell(\backslash cos\; \backslash gamma),$$

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 potential, {{math|1=∇² 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 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

$$\backslash Phi(r,\backslash theta)\; =\; \backslash sum\_\{\backslash ell=0\}^\backslash infty\; \backslash left(\; A\_\backslash ell\; r^\backslash ell\; +\; B\_\backslash ell\; r^\{-(\backslash ell+1)\}\; \backslash right)\; P\_\backslash ell(\backslash cos\backslash theta)\; \backslash ,.$$

{{math|A_l}} and {{math|B_l}} are to be determined according to the boundary condition of each problem.{{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}}

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

File:Point axial multipole.svg

Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently):

$$\backslash frac\{1\}\{\backslash sqrt\{1\; +\; \backslash eta^2\; -\; 2\backslash eta\; x\}\}\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash eta^k\; P\_k(x),$$

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 {{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

$$\backslash Phi\; (r,\; \backslash theta\; )\; \backslash propto\; \backslash frac\{1\}\{R\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{r^2\; +\; a^2\; -\; 2ar\; \backslash cos\backslash theta\}\}.$$

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

$$\backslash Phi(r,\; \backslash theta)\; \backslash propto\; \backslash frac\{1\}\{r\}\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash left(\; \backslash frac\{a\}\{r\}\; \backslash right)^k\; P\_k(\backslash cos\; \backslash theta),$$

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.

The trigonometric functions {{math|cos nθ}}, also denoted as the Chebyshev polynomials {{math|T_n(cos θ) ≡ cos nθ}}, can also be multipole expanded by the Legendre polynomials {{math|P_n(cos θ)}}. The first several orders are as follows:

$$\backslash begin\{alignat\}\{2\}$$

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}

This can be summarized for $n>0$ as

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)

.

where $\backslash hat\; n\backslash equiv\; \backslash lfloor\; n/2\backslash rfloor$,

$n\text{'}\backslash equiv\; \backslash lfloor\; (n+1)/2\backslash rfloor$, 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

$$\backslash frac\{\backslash sin\; (n+1)\backslash theta\}\{\backslash sin\backslash theta\}=\backslash sum\_\{\backslash ell=0\}^n\; P\_\backslash ell(\backslash cos\backslash theta)\; P\_\{n-\backslash ell\}(\backslash cos\backslash theta).$$

A recurrent neural network that contains a {{math|d}}-dimensional memory vector, $\backslash mathbf\{m\}\; \backslash in\; \backslash R^d$, can be optimized such that its neural activities obey the linear time-invariant system given by the following state-space representation:

$$\backslash theta\; \backslash dot\{\backslash mathbf\{m\}\}(t)\; =\; A\backslash mathbf\{m\}(t)\; +\; Bu(t),$$

$$\backslash begin\{align\}$$

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}

In this case, the sliding window of $u$ across the past $\backslash theta$ units of time is best approximated by a linear combination of the first $d$ shifted Legendre polynomials, weighted together by the elements of $\backslash mathbf\{m\}$ at time $t$:

$$u(t\; -\; \backslash theta\text{'})\; \backslash approx\; \backslash sum\_\{\backslash ell=0\}^\{d-1\}\; \backslash widetilde\{P\}\_\backslash ell\; \backslash left(\backslash frac\{\backslash theta\text{'}\}\{\backslash theta\}\; \backslash right)\; \backslash ,\; m\_\{\backslash ell\}(t)\; ,\; \backslash quad\; 0\; \backslash le\; \backslash theta\text{'}\; \backslash le\; \backslash theta\; .$$

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.{{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 }}

Legendre polynomials have definite parity. That is, they are even or odd,{{harvnb|Arfken|Weber|2005|loc=p.753}} according to

$$P\_n(-x)\; =\; (-1)^n\; P\_n(x)\; \backslash ,.$$

Another useful property is

$$\backslash int\_\{-1\}^1\; P\_n(x)\backslash ,dx\; =\; 0\; \backslash text\{\; for\; \}\; n\backslash ge1,$$

which follows from considering the orthogonality relation with $P\_0(x)\; =\; 1$. It is convenient when a Legendre series $\backslash sum\_i\; a\_i\; P\_i$ 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 $a\_0$.

The underivative is

{{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}}

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

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

$$P\_n(1)\; =\; 1\; \backslash ,.$$

The derivative at the end point is given by

$$P\_n\text{'}(1)\; =\; \backslash frac\{n(n+1)\}\{2\}\; \backslash ,.$$

The product expansion is {{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}}

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)

where $A\_r\backslash equiv\; (2r-1)!!/r!$.

The Askey–Gasper inequality for Legendre polynomials reads

$$\backslash sum\_\{j=0\}^n\; P\_j(x)\; \backslash ge\; 0\; \backslash quad\; \backslash text\{for\; \}\backslash quad\; x\backslash ge\; -1\; \backslash ,.$$

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

$$P\_\backslash ell\; \backslash left(r\; \backslash cdot\; r\text{'}\backslash right)\; =\; \backslash frac\{4\backslash pi\}\{2\backslash ell\; +\; 1\}\; \backslash sum\_\{m=-\backslash ell\}^\backslash ell\; Y\_\{\backslash ell\; m\}(\backslash theta,\backslash varphi)\; Y\_\{\backslash ell\; m\}^*(\backslash theta\text{'},\backslash varphi\text{'})\backslash ,,$$

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 {{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}}

\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}}\,,

where $K(\backslash cdot)$ is the complete elliptic integral of the first kind.

The formulas of Dirichlet-Mehler:{{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}}{{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}}{{Pg|page=86|location=Eq. 4.8.6, Eq. 4.8.7}}{{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}}$$P\_n(\backslash cos\; \backslash theta)$$

= \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 \phiwhich has generalizations for associated Legendre polynomials.{{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}}{{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}}

The Fourier-Legendre series:Lord Rayleigh, Theory of sound, Volume II, p. 273$$e^\{i\; t\; x\}=\backslash sum\_\{n=0\}^\{\backslash infty\}(2\; n+1)\; i^n\; \backslash sqrt\{\backslash frac\{\backslash pi\}\{2\; t\}\}\; J\_\{n+\backslash frac\{1\}\{2\}\}(t)\; P\_n(x)$$where $J$ is the Bessel function of the first kind.

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

$$(n+1)\; P\_\{n+1\}(x)\; =\; (2n+1)\; x\; P\_n(x)\; -\; n\; P\_\{n-1\}(x)$$

and

$$\backslash frac\{x^2-1\}\{n\}\; \backslash frac\{d\}\{dx\}\; P\_n(x)\; =\; xP\_n(x)\; -\; P\_\{n-1\}(x)$$

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

$$\backslash frac\{d\}\{dx\}\; P\_\{n+1\}(x)\; =\; (n+1)P\_n(x)\; +\; x\; \backslash frac\{d\}\{dx\}P\_\{n\}(x)\; \backslash ,.$$

Useful for the integration of Legendre polynomials is

$$(2n+1)\; P\_n(x)\; =\; \backslash frac\{d\}\{dx\}\; \backslash bigl(\; P\_\{n+1\}(x)\; -\; P\_\{n-1\}(x)\; \backslash bigr)\; \backslash ,.$$

From the above one can see also that

$$\backslash frac\{d\}\{dx\}\; P\_\{n+1\}(x)\; =\; (2n+1)\; P\_n(x)\; +\; \backslash bigl(2(n-2)+1\backslash bigr)\; P\_\{n-2\}(x)\; +\; \backslash bigl(2(n-4)+1\backslash bigr)\; P\_\{n-4\}(x)\; +\; \backslash cdots$$

or equivalently

$$\backslash frac\{d\}\{dx\}\; P\_\{n+1\}(x)\; =\; \backslash frac\{2\; P\_n(x)\}\{\backslash left\backslash |\; P\_n\; \backslash right\backslash |^2\}\; +\; \backslash frac\{2\; P\_\{n-2\}(x)\}\{\backslash left\backslash |\; P\_\{n-2\}\; \backslash right\backslash |^2\}\; +\; \backslash cdots$$

where {{math|{{norm|P_n}}}} is the norm over the interval {{math|−1 ≤ x ≤ 1}}

$$\backslash |\; P\_n\; \backslash |\; =\; \backslash sqrt\{\backslash int\_\{-1\}^1\; \backslash bigl(P\_n(x)\backslash bigr)^2\; \backslash ,dx\}\; =\; \backslash sqrt\{\backslash frac\{2\}\{2\; n\; +\; 1\}\}\; \backslash ,.$$More generally, all orders of derivatives are expressible as a sum of Legendre polynomials:{{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}}$$\backslash begin\{aligned\}$$

&\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}

Asymptotically, for $\backslash ell\; \backslash to\; \backslash infty$, the Legendre polynomials can be written as {{Cite book |last=Szegő |first=Gábor |title=Orthogonal polynomials |date=1975 |publisher=American Mathematical Society |isbn=0821810235 |edition=4th |location=Providence |oclc=1683237}}{{Pg|location=Theorem 8.21.2|page=194}}

$$\backslash begin\{align\}$$

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}

and for arguments of magnitude greater than 1{{Cite web|url=https://dlmf.nist.gov/14.15.E13|title = DLMF: 14.15 Uniform Asymptotic Approximations}}

$$\backslash begin\{align\}$$

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}

where {{math|J₀}}, {{math|J₁}}, and {{math|I₀}} are Bessel functions.

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

The zeros of $P\_n(\backslash cos\; \backslash theta)$ are distributed nearly uniformly over the range of $\backslash theta\; \backslash in\; (0,\; \backslash pi)$, in the sense that there is one zero $\backslash theta\; \backslash in\; \backslash left(\backslash frac\{\backslash pi(k\; +\; 1/2)\}\{n\; +\; 1/2\},\; \backslash frac\{\backslash pi(k\; +\; 1)\}\{n\; +\; 1/2\}\backslash right)$ per $k\; =\; 0,\; 1,\; \backslash dots,\; n-1$.{{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}} This can be proved by looking at the first formula of Dirichlet-Mehler.{{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}}

From this property and the facts that $P\_n(\backslash pm\; 1)\; \backslash ne\; 0$, it follows that $P\_n(x)$ has $n-1$ local minima and maxima in $(-1,1)$. Equivalently, $dP\_n(x)/dx$ has $n\; -1$ zeros in $(-1,1)$.

The parity and normalization implicate the values at the boundaries $x=\backslash pm\; 1$ to be

P_n(1) = 1

\,, \quad

P_n(-1) = (-1)^n

At the origin $x=0$ one can show that the values are given by

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)!!}

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

The shifted Legendre polynomials are defined as

$$\backslash widetilde\{P\}\_n(x)\; =\; P\_n(2x-1)\; \backslash ,.$$

Here the "shifting" function {{math|x ↦ 2x − 1}} is an affine transformation that bijectively maps the interval {{closed-closed|0, 1}} to the interval {{closed-closed|−1, 1}}, implying that the polynomials {{math|P̃_n(x)}} are orthogonal on {{closed-closed|0, 1}}:

$$\backslash int\_0^1\; \backslash widetilde\{P\}\_m(x)\; \backslash widetilde\{P\}\_n(x)\backslash ,dx\; =\; \backslash frac\{1\}\{2n\; +\; 1\}\; \backslash delta\_\{mn\}\; \backslash ,.$$

An explicit expression for the shifted Legendre polynomials is given by

$$\backslash widetilde\{P\}\_n(x)\; =\; (-1)^n\; \backslash sum\_\{k=0\}^n\; \backslash binom\{n\}\{k\}\; \backslash binom\{n+k\}\{k\}\; (-x)^k\; \backslash ,.$$

The analogue of Rodrigues' formula for the shifted Legendre polynomials is

$$\backslash widetilde\{P\}\_n(x)\; =\; \backslash frac\{1\}\{n!\}\; \backslash frac\{d^n\}\{dx^n\}\; \backslash left(x^2\; -x\; \backslash right)^n\; \backslash ,.$$

The first few shifted Legendre polynomials are:

{{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:

$$R\_n(x)\; =\; \backslash frac\{\backslash sqrt\{2\}\}\{x+1\}\backslash ,P\_n\backslash left(\backslash frac\{x-1\}\{x+1\}\backslash right)\backslash ,.$$

They are eigenfunctions of the singular Sturm–Liouville problem:

$$\backslash left(x+1\backslash right)\; \backslash frac\{d\}\{dx\}\; \backslash left(x\; \backslash frac\{d\}\{dx\}\; \backslash left[\backslash left(x+1\backslash right)\; v(x)\backslash right]\backslash right)\; +\; \backslash lambda\; v(x)\; =\; 0$$

with eigenvalues $$\backslash lambda\_n=n(n+1)\backslash ,.$$

{{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}}

{{reflist|30em}}

{{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}}

{{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]

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{{DEFAULTSORT:Legendre Polynomials}}

Category:Special hypergeometric functions

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