The associated Legendre differential equation is a generalization of the Legendre differential equation given by d/(d x)[(1 - x^2)(d y)/(d x)] + [l(l + 1) - m^2/(1 - x^2)] y = 0, which can be written (1 - x^2)(d^2 y)/(d x^2) - 2x(d y)/(d x) + [l(l + 1) - m^2/(1 - x^2)] y = 0 (Abramowitz and Stegun 1972; Zwillinger 1997, p. 124). The solutions P_l^m(x) to this equation are called the associated Legendre polynomials (if l is an integer), or associated Legendre functions of the first kind (if l is not an integer).
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