The eigenfunctions of distinct eigenvalues for a Hermitian operator are proved to be orthogonal. Why does the same not apply to Legendre polynomials and functions that have different eigenvalues ?
http://en.wikipedia.org/wiki/Legendre_polynomials
Just for clarity, I am talking of orthogonality of non-polynomial solution and polynomial solution here.
An eigenfunction/eigenvalue problem involves boundary conditions, not just the ODE. The proof of orthogonality of eigenfunctions for different eigenvalues essentially relies on the boundary conditions.
Legendre's differential equation being singular at $\pm 1$, the boundary conditions are somewhat implicit: they amount to the statement that the solution $y$ must stay bounded at both endpoints. This could be recast as the Dirichlet boundary condition for the function $(1-x^2)y(x)$.
Legendre functions of the second kind do not obey the aforementioned boundary condition: they blow up at the endpoints, as the linked article shows.