The associated Legendre polynomials, named after Adrien-Marie Legendre, are defined by:

<math>P_\ell ^m (x) = \frac{(-1)^m}{2^\ell \ell!} (1 - x^2) ^ \frac{m}{2} \left( \frac{d}{dx} \right ) ^{\ell+m} (x^2 - 1)^\ell.<math>

These differ from the Legendre polynomials.

They satisfy the orthogonality condition

<math>\int_{-1}^{1} P_k ^m P_\ell ^m dx = \delta _{k,\ell} \frac{2 (\ell+m)!}{(2\ell+1)(\ell-m)!}<math>

Where <math>\delta _{k,\ell}<math> is the Kronecker delta.

In many occusation in physics, Legendre polynomials occurs where spherical symmetry is involved. Then, the Legendre polynomials represents the dependence in azimuth as <math>\ x = \cos\theta<math>. The Associated Legendre polynomials are expensions of the above, where <math>\sqrt{1-x^2}<math> terms are <math>\ \sin\theta<math>. In other words, while the Legendre polynomials are polynomials of <math>\ \cos\theta<math>, the Associated Legendre polynomials are polynomials of <math>\cos\theta \ \ , \ \ \sin\theta<math>.

The Associated Legendre polynomials are an important part of spherical harmonics.

See also:

• Spherical harmonics.
• Angular momentum.
• Legendre polynomials.