Lecture notes on Legendre polynomials: their origin and main properties

TL;DR

The notes show how spherical symmetry in PDEs leads to Legendre's equation and a Legendre polynomial basis on $[-1,1]$, enabling Fourier-Legendre expansions of functions on that interval. They derive Rodrigues' formula, establish orthogonality with $\int_{-1}^1 P_m(x)P_n(x)dx = 2/(2n+1)$ when $m=n$ and zero otherwise, and discuss the generating function that generates the full sequence of polynomials. They introduce shifted polynomials tilde P_n on $[0,1]$, including Rodrigues-type and explicit representations, and highlight their orthogonality on $[0,1]$ and their role in Beukers-style irrationality proofs via integration by parts that transfer derivatives to the test function. The framework connects Legendre theory to numerical quadrature on general intervals and provides a bridge between classical physics applications and number-theoretic methods using Beukers-type integrals.

Abstract

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 $θ$. For eigenvalues of the form $\,n\,(n+1)$, $n \ge 0\,$ being an integer, Legendre's equation admits certain polynomials $P_n(\cosθ)$ as solutions, which form a complete set of continuous orthogonal functions for all $θ\in [0,π]$. This allows us to take the polynomials $P_n(x)$, where $x = \cosθ$, 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 $ζ{(2)}$ and $ζ{(3)}$ are irrational numbers.

Figures (3)

• Figure 1: Cartesian and spherical coordinates. As usual, $r$ is the distance of (arbitrary) point $P=(x,y,z)$ to the origin $O=(0,0,0)$, $\theta$ is the polar angle (between $\mathbf{k}$ and the vector $\,\vec{r} := x\,\mathbf{i} +y\,\mathbf{j} +z\,\mathbf{k}$), and $\phi$ is the azimuthal angle (between $\mathbf{i}$ and the projection of vector $\vec{r}$ onto the $x y$-plane).
• Figure 2: The first few Legendre polynomials $P_n(x)$, as given by Eq. \ref{['eq:PnExplicito']}. Note that they all obey the 'practical normalization' $\,P_n(1)=1$.
• Figure 3: The first few shifted Legendre polynomials $\,\tilde{P}_n(x)$, as found from Eq. \ref{['eq:BkTildePn']}, which obey the 'practical normalization' $\,\tilde{P}_n(0)=1$.

Theorems & Definitions (20)

• Lemma 1: Rodrigues' formula for $P_n(x)\,$
• proof
• Lemma 2: Orthogonality of Legendre polynomials
• proof
• Lemma 3: Orthonormality of Legendre polynomials
• proof
• Theorem 1: Orthonormality relation for $P_n(x)\,$
• Theorem 2: Simpler explicit representation of $\tilde{P}_n(x)\,$
• proof
• Theorem 3: Rodrigues' formula for $\tilde{P}_n(x)\,$