Associated Legendre Polynomial

The associated Legendre polynomials P_l^m(x) and P_l^(-m)(x) generalize the Legendre polynomials P_l(x) and are solutions to the associated Legendre differential equation, where l is a positive integer and m=0, ..., l. They are implemented in the Wolfram Language as LegendreP[l, m, x]. For positive m, they can be given in terms of the unassociated polynomials by


where P_l(x) are the unassociated Legendre polynomials. The associated Legendre polynomials for negative m are then defined by


There are two sign conventions for associated Legendre polynomials. Some authors (e.g., Arfken 1985, pp. 668-669) omit the Condon-Shortley phase (-1)^m, 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


to distinguish the two.

Associated polynomials are sometimes called Ferrers' functions (Sansone 1991, p. 246). If m=0, 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 [-1,1] with the weighting function 1


and orthogonal over [-1,1] with respect to m with the weighting function (1-x^2)^(-1),


The associated Legendre polynomials also obey the following recurrence relations


Letting x=costheta (commonly denoted mu in this context),


Additional identities are


Including the factor of (-1)^m, the first few associated Legendre polynomials are


Written in terms x=costheta (commonly written mu=costheta), the first few become


The derivative about the origin is


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


(Binney and Tremaine 1987, p. 654).

See also

Associated Legendre Differential Equation, Condon-Shortley Phase, Gegenbauer Polynomial, Legendre Function of the First Kind, Legendre Function of the Second Kind, Legendre Polynomial, Spherical Harmonic, Toroidal Function

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Associated Legendre Polynomial

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

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