What is the orthogonality condition for associate legendre polynomials with different indexes? I am looking for a closed-form expression for:
$$\int_0^\pi  {P_n^m } \left( {\cos \theta } \right)P_k^l \left( {\cos \theta } \right)\,d\theta$$
where $P_n$ and $P_l$ are Legendre polynomials.

From (simple) Sturm-Liouville considerations, it follows that (x = $cos\theta$)
$$\int_{-1}^1 P_{n_1}^m (x) P_{n_2}^m (x)dx=0$$ if $n_1 \not= n_2$
Thus the associated Legendre polynomials $P_{n_1}^m(x)$ and $P_{n_2}^m(x)$ are orthogonal if $n_1 \not= n_2$.
