Legendre Polynomial

Legendre Polynomials like sine cosines are complete and orthogonal. And come up in solutions of ODE where general solution is experessed as linear combination of legendre polynomials.

It is the solution of the following differential equation:

\begin{align*} (1-x^2) P''_n(x) - 2x P'_n(x) + n (n+1)P_n(x) = 0 \end{align*}

In Physics Legendre Polynomial come up naturally in the solution of Laplace Equation by separation of variables in spherical coordinates.