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 P_l^m(x) | = | (-1)^m (1 - x^2)^(m/2) d^m/(d x^m) P_l(x) | = | (-1)^m/(2^l l!) (1 - x^2)^(m/2) d^(l + m)/(d x^(l + m)) (x^2 - 1)^l, where P_l(x) are the unassociated Legendre polynomials.

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