Associated Legendre Polynomial

The associated Legendre polynomials and generalize the Legendre polynomials and are solutions to the associated Legendre differential equation, where is a positive integer and , ..., . They are implemented in the Wolfram Language as LegendreP[l, m, x]. For positive , they can be given in terms of the unassociated polynomials by

			(1)
			(2)

where are the unassociated Legendre polynomials. The associated Legendre polynomials for negative are then defined by

(3)

There are two sign conventions for associated Legendre polynomials. Some authors (e.g., Arfken 1985, pp. 668-669) omit the Condon-Shortley phase , while others include it (e.g., Abramowitz and Stegun 1972, Press et al. 1992, and the LegendreP[l, m, z] command in the Wolfram Language). Care is therefore needed in comparing polynomials obtained from different sources. One possible way to distinguish the two conventions is due to Abramowitz and Stegun (1972, p. 332), who use the notation

(4)

to distinguish the two.

Associated polynomials are sometimes called Ferrers' functions (Sansone 1991, p. 246). If , they reduce to the unassociated polynomials. The associated Legendre functions are part of the spherical harmonics, which are the solution of Laplace's equation in spherical coordinates. They are orthogonal over with the weighting function 1

(5)

and orthogonal over with respect to with the weighting function ,

(6)

The associated Legendre polynomials also obey the following recurrence relations

(7)

Letting (commonly denoted in this context),

(8)

(9)

Additional identities are

(10)

(11)

Including the factor of , the first few associated Legendre polynomials are

			(12)
			(13)
			(14)
			(15)
			(16)
			(17)
			(18)
			(19)
			(20)
			(21)
			(22)
			(23)
			(24)
			(25)
			(26)
			(27)

Written in terms (commonly written ), the first few become

			(28)
			(29)
			(30)
			(31)
			(32)
			(33)
			(34)
			(35)
			(36)
			(37)

The derivative about the origin is

(38)

(Abramowitz and Stegun 1972, p. 334), and the logarithmic derivative is

(39)

(Binney and Tremaine 1987, p. 654).