Associated Legendre Polynomials - Orthogonality

Orthogonality

Assuming 0 ≤ m ≤ ℓ, they satisfy the orthogonality condition for fixed m:

Where δk, ℓ is the Kronecker delta.

Also, they satisfy the orthogonality condition for fixed ℓ:

Read more about this topic:  Associated Legendre Polynomials