Legendre polynomials explained

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 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

w(x) = 1 over the interval [ -1 , 1 ] . That is, P _{n(x)} is a polynomial of degree n , such that$$\backslash int\_^1\; P\_m(x)\; P\_n(x)\; \backslash ,dx\; =\; 0\; \backslash quad\; \backslash text\; n\; \backslash ne\; m.$$

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 m < n . 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,} \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 , inf ty) with the weight e ^{-x} , and the Hermite polynomials, orthogonal over the full line ( -inf ty, inf ty) with weight e ^{- x ^{2}} .

Definition via generating function

The Legendre polynomials can also be defined as the coefficients in a formal expansion in powers of

t of the generating function

The coefficient of

t ^{n} is a polynomial in x of degree n with |x| \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  is differentiated with respect to on both sides and rearranged to obtain$$\backslash frac\; =\; \backslash left(1-2xt+t^2\backslash right)\; \backslash sum\_^\backslash infty\; n\; P\_n(x)\; t^\; \backslash ,.$$Replacing the quotient of the square root with its definition in Eq. , and equating the coefficients of powers of in the resulting expansion gives Bonnet’s recursion formula$$(n+1)\; P\_(x)\; =\; (2n+1)\; x\; P\_n(x)\; -\; n\; P\_(x)\backslash ,.$$This relation, along with the first two polynomials and, 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:

This differential equation has regular singular points at so if a solution is sought using the standard Frobenius or power series method, a series about the origin will only converge for in general. When is an integer, the solution that is regular at is also regular at, 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,$$\backslash frac\; \backslash left(\backslash left(1-x^2\backslash right)\; \backslash frac\; \backslash right)\; P(x)\; =\; -\backslash lambda\; P(x)\; \backslash ,,$$with the eigenvalue

λ in lieu of n(n +1 ) . If we demand that the solution be regular atx = \pm 1 , the differential operator on the left is Hermitian. The eigenvalues are found to be of the form, with n = 0 , 1 , 2 , \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.Q _{n} .A two-parameter generalization of (Eq. ) 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(\cos\theta)} where \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.

Rodrigues' formula and other explicit formulas

An especially compact expression for the Legendre polynomials is given by Rodrigues' formula:$$P\_n(x)\; =\; \backslash frac\; \backslash frac\; (x^2\; -1)^n\; \backslash ,.$$

This formula enables derivation of a large number of properties of the

P _{n} 's. Among these are explicit representations such as^1 \frac\cdot \frac\,dt & \text 0<|x|<1, \\ \displaystyle(-1)^\cdot2^\cdot \binom & \text x=0 \textn\text, \\0 & \text x=0 \textn\text.\end\end

Expressing the polynomial as a power series, $P\_n(x)\; =\; \backslash sum\; a\_k\; x^k$, the coefficients of powers of

x can also be calculated using a general formula:$$a\_\; =\; -\; \backslash fraca\_k.$$The Legendre polynomial is determined by the values used for the two constants $a\_0$ and $a\_1$, where $a\_0=0$ if n is odd and $a\_1=0$ if n is even.^[1]

In the fourth representation,

\lfloor n/ 2 \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 first few Legendre polynomials are:

n	P _{n(x)}
0	$1$
1	$x$
2	$\backslash tfrac12\; \backslash left(3x^2-1\backslash right)$
3	$\backslash tfrac12\; \backslash left(5x^3-3x\backslash right)$
4	$\backslash tfrac18\; \backslash left(35x^4-30x^2+3\backslash right)$
5	$\backslash tfrac18\; \backslash left(63x^5-70x^3+15x\backslash right)$
6	$\backslash tfrac1\; \backslash left(231x^6-315x^4+105x^2-5\backslash right)$
7	$\backslash tfrac1\; \backslash left(429x^7-693x^5+315x^3-35x\backslash right)$
8	$\backslash tfrac1\; \backslash left(6435x^8-12012x^6+6930x^4-1260x^2+35\backslash right)$
9	$\backslash tfrac1\; \backslash left(12155x^9-25740x^7+18018x^5-4620x^3+315x\backslash right)$
10	$\backslash tfrac1\; \backslash left(46189x^-109395x^8+90090x^6-30030x^4+3465x^2-63\backslash right)$

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

Main properties

Orthogonality

The standardization

P _{n(1)} = 1 fixes the normalization of the Legendre polynomials (with respect to the norm on the interval). Since they are also orthogonal with respect to the same norm, the two statements can be combined into the single equation,$$\backslash int\_^1\; P\_m(x)\; P\_n(x)\backslash ,dx\; =\; \backslash frac\; \backslash delta\_,$$(where denotes the Kronecker delta, equal to 1 if 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

f(x) with finitely many discontinuities in the interval, the sequence of sums$$f\_n(x)\; =\; \backslash sum\_^n\; a\_\backslash ell\; P\_\backslash ell(x)$$converges in the mean to f(x) as n \to inf ty , provided we take$$a\_\backslash ell\; =\; \backslash frac\; \backslash int\_^1\; f(x)\; P\_\backslash ell(x)\backslash ,dx.$$

This completeness property underlies all the expansions discussed in this article, and is often stated in the form$$\backslash sum\_^\backslash infty\; \backslash frac\; P\_\backslash ell(x)P\_\backslash ell(y)\; =\; \backslash delta(x-y),$$with and .

Applications

Expanding an inverse distance potential

See main article: Laplace expansion (potential).

The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre^[2] as the coefficients in the expansion of the Newtonian potential$$\backslash frac\; =\; \backslash frac\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; P\_\backslash ell(\backslash cos\; \backslash gamma),$$where and are the lengths of the vectors and respectively and is the angle between those two vectors. The series converges when . 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,, 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 is the axis of symmetry and is the angle between the position of the observer and the axis (the zenith angle), the solution for the potential will be$$\backslash Phi(r,\backslash theta)\; =\; \backslash sum\_^\backslash infty\; \backslash left(A\_\backslash ell\; r^\backslash ell\; +\; B\_\backslash ell\; r^\; \backslash right)\; P\_\backslash ell(\backslash cos\backslash theta)\; \backslash ,.$$

and are to be determined according to the boundary condition of each problem.^[3]

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

In multipole expansions

Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently):$$\backslash frac\; =\; \backslash sum\_^\backslash infty\; \backslash eta^k\; P\_k(x),$$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 (in spherical coordinates) due to a point charge located on the -axis at (see diagram right) varies as$$\backslash Phi\; (r,\; \backslash theta)\; \backslash propto\; \backslash frac\; =\; \backslash frac.$$

If the radius of the observation point is greater than, the potential may be expanded in the Legendre polynomials$$\backslash Phi(r,\; \backslash theta)\; \backslash propto\; \backslash frac\; \backslash sum\_^\backslash infty\; \backslash left(\backslash frac\; \backslash right)^k\; P\_k(\backslash cos\; \backslash theta),$$where we have defined and . This expansion is used to develop the normal multipole expansion.

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

In trigonometry

The trigonometric functions, also denoted as the Chebyshev polynomials, can also be multipole expanded by the Legendre polynomials . The first several orders are as follows:

Another property is the expression for, which is$$\backslash frac=\backslash sum\_^n\; P\_\backslash ell(\backslash cos\backslash theta)\; P\_(\backslash cos\backslash theta).$$

In recurrent neural networks

A recurrent neural network that contains a -dimensional memory vector,

m \in \R ^{d} , can be optimized such that its neural activities obey the linear time-invariant system given by the following state-space representation:$$\backslash theta\; \backslash dot(t)\; =\; A\backslash mathbf(t)\; +\; Bu(t),$$

In this case, the sliding window of

u across the past \theta units of time is best approximated by a linear combination of the first d shifted Legendre polynomials, weighted together by the elements of m at time t :$$u(t\; -\; \backslash theta\text{'})\; \backslash approx\; \backslash sum\_^\; \backslash widetilde\_\backslash ell\; \backslash left(\backslash frac\; \backslash right)\; \backslash ,\; m\_(t),\; \backslash quad\; 0\; \backslash le\; \backslash theta\text{'}\; \backslash le\; \backslash theta\; .$$

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.^[4]

Additional properties

Legendre polynomials have definite parity. That is, they are even or odd, according to$$P\_n(-x)\; =\; (-1)^n\; P\_n(x)\; \backslash ,.$$

Another useful property is$$\backslash int\_^1\; P\_n(x)\backslash ,dx\; =\; 0\; \backslash text\; n\backslash ge1,$$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 is simply given by the leading expansion coefficient a _{0} .

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

The derivative at the end point is given by$$P\_n\text{'}(1)\; =\; \backslash frac\; \backslash ,.$$

The Askey–Gasper inequality for Legendre polynomials reads$$\backslash sum\_^n\; P\_j(x)\; \backslash ge\; 0\; \backslash quad\; \backslash text\backslash quad\; x\backslash ge\; -1\; \backslash ,.$$

The Legendre polynomials of a scalar product of unit vectors can be expanded with spherical harmonics using$$P\_\backslash ell\; \backslash left(r\; \backslash cdot\; r\text{'}\backslash right)\; =\; \backslash frac\; \backslash sum\_^\backslash ell\; Y\_(\backslash theta,\backslash varphi)\; Y\_^*(\backslash theta\text{'},\backslash varphi\text{'})\backslash ,,$$where the unit vectors and have spherical coordinates and, respectively.

The product of two Legendre polynomials ^[5] $$\backslash sum\_^\backslash infty\; t^P\_p(\backslash cos\backslash theta\_1)P\_p(\backslash cos\backslash theta\_2)=\backslash frac2\backslash pi\backslash frac\backslash ,,$$where

K( ⋅ ) is the complete elliptic integral 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$$(n+1)\; P\_(x)\; =\; (2n+1)\; x\; P\_n(x)\; -\; n\; P\_(x)$$and$$\backslash frac\; \backslash frac\; P\_n(x)\; =\; xP\_n(x)\; -\; P\_(x)$$or, with the alternative expression, which also holds at the endpoints$$\backslash frac\; P\_(x)\; =\; (n+1)P\_n(x)\; +\; x\; \backslash fracP\_(x)\; \backslash ,.$$

Useful for the integration of Legendre polynomials is$$(2n+1)\; P\_n(x)\; =\; \backslash frac\; \backslash bigl(P\_(x)\; -\; P\_(x)\; \backslash bigr)\; \backslash ,.$$

From the above one can see also that$$\backslash frac\; P\_(x)\; =\; (2n+1)\; P\_n(x)\; +\; \backslash bigl(2(n-2)+1\backslash bigr)\; P\_(x)\; +\; \backslash bigl(2(n-4)+1\backslash bigr)\; P\_(x)\; +\; \backslash cdots$$or equivalently$$\backslash frac\; P\_(x)\; =\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots$$where is the norm over the interval $$\backslash |\; P\_n\; \backslash |\; =\; \backslash sqrt\; =\; \backslash sqrt\; \backslash ,.$$

Asymptotics

Asymptotically, for

\ell \to inf ty , the Legendre polynomials can be written as ^[6] and for arguments of magnitude greater than 1^[7] where,, and are Bessel functions.

Zeros

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.

From this property and the facts that

P _{n(\pm} 1 ) \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 ) .

Pointwise evaluations

The parity and normalization implicate the values at the boundaries

x = \pm 1 to be$$P\_n(1)\; =\; 1\; \backslash ,,\; \backslash quad\; P\_n(-1)\; =\; (-1)^n$$At the origin x =0 one can show that the values are given by$$P\_(0)\; =\; \backslash frac\; \backslash binom\; =\; \backslash frac\; \backslash frac=\; (-1)^n\backslash frac$$$$P\_(0)\; =\; 0$$

Variants with transformed argument

Shifted Legendre polynomials

The shifted Legendre polynomials are defined as$$\backslash widetilde\_n(x)\; =\; P\_n(2x-1)\; \backslash ,.$$Here the "shifting" function is an affine transformation that bijectively maps the interval to the interval, implying that the polynomials are orthogonal on :$$\backslash int\_0^1\; \backslash widetilde\_m(x)\; \backslash widetilde\_n(x)\backslash ,dx\; =\; \backslash frac\; \backslash delta\_\; \backslash ,.$$

An explicit expression for the shifted Legendre polynomials is given by$$\backslash widetilde\_n(x)\; =\; (-1)^n\; \backslash sum\_^n\; \backslash binom\; \backslash binom\; (-x)^k\; \backslash ,.$$

The analogue of Rodrigues' formula for the shifted Legendre polynomials is$$\backslash widetilde\_n(x)\; =\; \backslash frac\; \backslash frac\; \backslash left(x^2\; -x\; \backslash right)^n\; \backslash ,.$$

The first few shifted Legendre polynomials are:

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

Legendre rational functions

See main article: 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:$$R\_n(x)\; =\; \backslash frac\backslash ,P\_n\backslash left(\backslash frac\backslash right)\backslash ,.$$

They are eigenfunctions of the singular Sturm–Liouville problem:$$\backslash left(x+1\backslash right)\; \backslash frac\; \backslash left(x\; \backslash frac\; \backslash left[\backslash left(x+1\backslash right)\; v(x)\backslash right]\backslash right)\; +\; \backslash lambda\; v(x)\; =\; 0$$with eigenvalues $$\backslash lambda\_n=n(n+1)\backslash ,.$$

See also

• Gaussian quadrature
• Gegenbauer polynomials
• Turán's inequalities
• Legendre wavelet
• Legendre function
• Jacobi polynomials
• Romanovski polynomials
• Laplace expansion (potential)

References

• • Book: George B.. Arfken. George B. Arfken. Hans J.. Weber. 2005. Mathematical Methods for Physicists. Elsevier Academic Press. 0-12-059876-0.
• Book: Bayin, S. S.. 2006. Mathematical Methods in Science and Engineering. Wiley. 978-0-470-04142-0. ch. 2.
• Book: Belousov, S. L.. 1962. Tables of Normalized Associated Legendre Polynomials. Mathematical Tables. 18. Pergamon Press. 978-0-08-009723-7.
• Book: Richard. Courant. Richard Courant. David. Hilbert. David Hilbert. 1953. Methods of Mathematical Physics. 1. Interscience . New York, NY. 978-0-471-50447-4.
  • Book: El Attar, Refaat . Legendre Polynomials and Functions . CreateSpace . 2009 . 978-1-4414-9012-4.

External links

• A quick informal derivation of the Legendre polynomial in the context of the quantum mechanics of hydrogen
  • Wolfram MathWorld entry on Legendre polynomials
• Dr James B. Calvert's article on Legendre polynomials from his personal collection of mathematics
• The Legendre Polynomials by Carlyle E. Moore
• Legendre Polynomials from Hyperphysics

Notes and References

1. Book: Boas, Mary L. . Mathematical methods in the physical sciences . 2006 . Wiley . 978-0-471-19826-0 . 3rd . Hoboken, NJ.
2. Book: A.-M. . Legendre . Recherches sur l'attraction des sphéroïdes homogènes . Mémoires de Mathématiques et de Physique, présentés à l'Académie Royale des Sciences, par divers savans, et lus dans ses Assemblées . X . 411–435 . Paris . 1785 . 1782 . fr . http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf . dead . https://web.archive.org/web/20090920070434/http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf . 2009-09-20 .
3. Book: Jackson, J. D. . Classical Electrodynamics . limited . 3rd . Wiley & Sons . 1999 . 103 . 978-0-471-30932-1.
4. Voelker . Aaron R. . Kajić . Ivana . Eliasmith . Chris . Legendre Memory Units: Continuous-Time Representation in Recurrent Neural Networks . Advances in Neural Information Processing Systems . https://neurips.cc . 2019 .
5. Leonard C. Maximon. A generating function for the product of two Legendre polynomials. Norske Videnskabers Selskab Forhandlinger . 29 . 1957 . 82–86 .
6. Book: Szegő, Gábor . Orthogonal polynomials. 1975. American Mathematical Society. 0821810235. 4th. Providence. 194 (Theorem 8.21.2) . 1683237.
7. Web site: DLMF: 14.15 Uniform Asymptotic Approximations.