"Associated Legendre polynomials" P(l,m) = test(m < 0, (-1)^m (l + m)! / (l - m)! P(l,-m), 1/(2^l l!) (1 - x^2)^(m/2) d((x^2 - 1)^l, x, l + m)) "Verify equation (1)" for(l, 0, 2, for(m, -l, l, f = P(l,m), check((1 - x^2) d(f,x,2) - 2 x d(f,x) + (l (l + 1) - m^2 / (1 - x^2)) f == 0) )) "ok" Run