Associated Legendre functions P^k_n(x)_?are needed, e.g., in quantum physics. They are defined by

and are solutions of the ODE

where q(x) = n(n + 1) - k²/(1 - x²). Find P¹₁(x), P¹₂(x), P²₂(x), and P²₄(x) and verify that they satisfy (16).

P(x) = (1 - x) x/2 d Pn(x) dxk (1-x)y" - 2xy + q(x)y = 0