Orthogonal polynomials, Legendre

The Legendre polynomials are the solutions of an important differential equation in Science, namely $$\begin{equation*} C(1-x^2)P-m_l^2P+(1-x^2)\frac{d}{dx}\left((1-x^2)\frac{dP}{dx}\right)=0. \end{equation*}$$ Here $C$ is a constant. For $m_l=0$ we obtain the Legendre polynomials as solutions, whereas $m_l \ne 0$ yields the so-called associated Legendre polynomials. This differential equation arises in for example the solution of the angular dependence of Schroedinger's equation with spherically symmetric potentials such as the Coulomb potential.

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