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.
