Legendre polynomials

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  is an integer, they are polynomials. The Legendre polynomials  are illustrated above for  and , 2, ..., 5. They are implemented in the Wolfram Language as LegendreP[n, x].

The associated Legendre polynomials  and  are solutions to the associated Legendre differential equation, where  is a positive integer and , ..., .

The Legendre polynomial  can be defined by the contour integral

(1)

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

The first few Legendre polynomials are

(2)

(3)

(4)

(5)

(6)

(7)

(8)

When ordered from smallest to largest powers and with the denominators factored out, the triangle of nonzero coefficients is 1, 1

 , 3, , 5, 3, , ... (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

(9)

(10)

(11)

(12)

(13)

(14)

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

(15)

(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  with the weighting function 1.

(16)

(17)

(18)

(19)

(20)

(21)

(22)

Normalizing so that  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

(23)

The first few are

(24)

(25)

(26)

(27)

The Legendre polynomials are 

orthogonal over  with weighting function 1 and satisfy

(28)

where  is the Kronecker delta.

The Legendre polynomials are a special case of the Gegenbauer polynomials with , a special case of the Jacobi polynomials  with , and can be written as a hypergeometric function using Murphy's formula

(29)

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

The Rodrigues representation provides the formula

(30)

which yields upon expansion

(31)

(32)

where  is the floor function. Additional sum formulas include

(33)

(34)

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

(35)

(36)

(37)

(Koepf 1998, p. 3).

A generating function for  is given by

(38)

Take ,

(39)

Multiply (39) by ,

(40)

and add (38) and (40),

(41)

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

(42)

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

The Legendre polynomials satisfy the recurrence relation

(43)

(Koepf 1998, p. 2). In addition,

(44)

(correcting Hildebrand 1956, p. 324).

A complex generating function is

(45)

and the Schläfli integral is

(46)

Integrals over the interval  include the general formula

(47)

for  (Byerly 1959, p. 172), from which the special case

(48)

(49)

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

(50)

for  (Byerly 1959, p. 172), which gives the special case

(51)

where

(52)

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

(53)

where

(54)

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

Integrals over  with weighting functions  and  are given by

(55)

(56)

(Arfken 1985, p. 700).

The Laplace transform is given by

(57)

where  is a modified Bessel function of the first kind.

A sum identity is given by

(58)

where  is the th root of  (Szegö 1975, p. 348). A similar identity is

(59)