Associated Legendre polynomials

Associated Legendre polynomials are a generalization of Legendre polynomials, which arise in the context of solving problems in physics, particularly in potential theory, quantum mechanics, and in the theory of spherical harmonics. The associated Legendre polynomials, denoted as \( P_\ell^m(x) \), are defined for non-negative integers \( \ell \) and \( m \), where \( m \) can take on values from \( 0 \) to \( \ell \).