As is well-known, the separation of variables in second order partial

differential equations (PDEs) for physical problems with spherical symmetry

usually leads to Cauchy’s differential equation for the radial coordinate $r$

and Legendre’s differential equation for the polar angle $\theta$. For

eigenvalues of the form $n\,(n+1)$, $n \ge 0\,$ being an integer, Legendre’s

equation admits certain polynomials $P_n(\cos{\theta})$ as solutions, which

form a complete set of continuous orthogonal functions for all $\theta \in

[0,\pi]$. This allows us to take the polynomials $P_n(x)$, where $x =

\cos{\theta}$, as a basis for the Fourier-Legendre series expansion of any

function $f(x)$ continuous by parts over $\,x \in [-1,1]$. These lecture notes

correspond to the end of my course on Mathematical Methods for Physics, when I

did derive the differential equations and solutions for physical problems with

spherical symmetry. For those interested in Number Theory, I have included an

application of shifted Legendre polynomials in \emph{irrationality proofs},

following a method introduced by Beukers to show that $\zeta{(2)}$ and

$\zeta{(3)}$ are irrational numbers.