Lecture notes on Legendre polynomials: their origin and main properties

It is well-known that separation of variables in 2nd order partial differential equations (PDEs) for physical problems with spherical symmetry usually leads to Cauchy's differential equation for the radial coordinate 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.