Legendre Polynomials - Orthogonality

Orthogonality

An important property of the Legendre polynomials is that they are orthogonal with respect to the L2 inner product on the interval −1 ≤ x ≤ 1:

(where δ_mn denotes the Kronecker delta, equal to 1 if m = n and to 0 otherwise). In fact, an alternative derivation of the Legendre polynomials is by carrying out the Gram-Schmidt process on the polynomials {1, x, x2, ...} with respect to this inner product. The reason for this orthogonality property is that the Legendre differential equation can be viewed as a Sturm–Liouville problem, where the Legendre polynomials are eigenfunctions of a Hermitian differential operator:

where the eigenvalue λ corresponds to n(n + 1).