Legendre Polynomial


The Legendre polynomials, sometimes called Legendre functions of the first kind, Legendre coefficients, or zonal harmonics (Whittaker and Watson 1990, p. 302), are solutions to the Legendre differential equation. If l is an integer, they are polynomials. The Legendre polynomials P_n(x) are illustrated above for x in [-1,1] and n=1, 2, ..., 5. They are implemented in the Wolfram Language as LegendreP[n, x].

The associated Legendre polynomials P_l^m(x) and P_l^(-m) are solutions to the associated Legendre differential equation, where l is a positive integer and m=0, ..., l.

The Legendre polynomial P_n(z) can be defined by the contour integral


where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).

The first few Legendre polynomials are


When ordered from smallest to largest powers and with the denominators factored out, the triangle of nonzero coefficients is 1, 1, -1, 3, -3, 5, 3, -30, ... (OEIS A008316). The leading denominators are 1, 1, 2, 2, 8, 8, 16, 16, 128, 128, 256, 256, ... (OEIS A060818).

The first few powers in terms of Legendre polynomials are


(OEIS A008317 and A001790). A closed form for these is given by


(R. Schmied, pers. comm., Feb. 27, 2005). For Legendre polynomials and powers up to exponent 12, see Abramowitz and Stegun (1972, p. 798).

The Legendre polynomials can also be generated using Gram-Schmidt orthonormalization in the open interval (-1,1) with the weighting function 1.


Normalizing so that P_n(1)=1 gives the expected Legendre polynomials.

The "shifted" Legendre polynomials are a set of functions analogous to the Legendre polynomials, but defined on the interval (0, 1). They obey the orthogonality relationship


The first few are


The Legendre polynomials are orthogonal over (-1,1) with weighting function 1 and satisfy


where delta_(mn) is the Kronecker delta.

The Legendre polynomials are a special case of the Gegenbauer polynomials with alpha=1/2, a special case of the Jacobi polynomials P_n^((alpha,beta)) with alpha=beta=0, and can be written as a hypergeometric function using Murphy's formula


(Bailey 1933; 1935, p. 101; Koekoek and Swarttouw 1998).

The Rodrigues representation provides the formula


which yields upon expansion

=1/(2^l)sum_(k=0)^(|_l/2_|)(-1)^k(l; k)(2l-2k; l)x^(l-2k)

where |_r_| is the floor function. Additional sum formulas include

P_l(x)=1/(2^l)sum_(k=0)^(l)(l; k)^2(x-1)^(l-k)(x+1)^k
=sum_(k=0)^(l)(l; k)(-l-1; k)((1-x)/2)^k

(Koepf 1998, p. 1). In terms of hypergeometric functions, these can be written

P_n(x)=(2n; n)(x^n)/(2^n)_2F_1(-n/2,(1-n)/2;1/2-n;x^(-2))

(Koepf 1998, p. 3).

A generating function for P_n(x) is given by


Take partialg/partialt,


Multiply (39) by 2t,


and add (38) and (40),


This expansion is useful in some physical problems, including expanding the Heyney-Greenstein phase function and computing the charge distribution on a sphere. Another generating function is given by


where J_0(x) is a zeroth order Bessel function of the first kind (Koepf 1998, p. 2).

The Legendre polynomials satisfy the recurrence relation


(Koepf 1998, p. 2). In addition,


(correcting Hildebrand 1956, p. 324).

A complex generating function is


and the Schläfli integral is


Integrals over the interval [x,1] include the general formula


for m!=0 (Byerly 1959, p. 172), from which the special case

={1 m=0; 0 m even !=0; (-1)^((m-1)/2)(m!!)/(m(m+1)(m-1)!!) m odd

follows (OEIS A002596 and A046161; Byerly 1959, p. 172). For the integral over a product of Legendre functions,


for m!=n (Byerly 1959, p. 172), which gives the special case

 int_0^1P_m(x)P_n(x)dx={1/(2n+1)   m=n; 0   m!=n, m,n both even or odd; f_(m,n)   m even, n odd; f_(n,m)   m odd, n even



(OEIS A078297 and A078298; Byerly 1959, p. 172). The latter is a special case of




and Gamma(z) is a gamma function (Gradshteyn and Ryzhik 2000, p. 762, eqn. 7.113.1)

Integrals over [-1,1] with weighting functions x and x^2 are given by

int_(-1)^1xP_L(x)P_N(x)dx={(2(L+1))/((2L+1)(2L+3)) for N=L+1; (2L)/((2L-1)(2L+1)) for N=L-1
int_(-1)^1x^2P_L(x)P_N(x)dx={(2(L+1)(L+2))/((2L+1)(2L+3)(2L+5)) for N=L+2; (2(2L^2+2L-1))/((2L-1)(2L+1)(2L+3)) for N=L; (2L(L-1))/((2L-3)(2L-1)(2L+1)) for N=L-2

(Arfken 1985, p. 700).

The Laplace transform is given by

 L[P_n(t)](s)={1/2sqrt(pi)[sqrt(2/s)I_(-n-1/2)(s)-1/2s_1F_2(1;2+1/2n,1/2(3-n);1/4s^2)]   for n even; 1/2sqrt(pi)[sqrt(2/s)I_(-n-1/2)(s)+_1F_2(1;1/2(3+n),1-1/2n;1/4s^2)]   for n odd,

where I_n(s) is a modified Bessel function of the first kind.

A sum identity is given by


where x_nu is the nuth root of P_n(x) (Szegö 1975, p. 348). A similar identity is


which is responsible for the fact that the sum of weights in Legendre-Gauss quadrature is always equal to 2.

See also

Associated Legendre Polynomial, Conical Function, Laplace's Integral, Laplace-Mehler Integral, Legendre Function of the First Kind, Legendre Function of the Second Kind, Mehler-Dirichlet Integral, Spherical Harmonic, Super Catalan Number, Zonal Harmonic

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Legendre Polynomial

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Weisstein, Eric W. "Legendre Polynomial." From MathWorld--A Wolfram Web Resource.

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