Euler and the Legendre Polynomials

TL;DR

This paper explores Euler's historical contributions to Legendre polynomials, revealing that many properties and representations were first documented by Euler, including generating functions, difference equations, and integral formulas.

Contribution

It uncovers Euler's original work on Legendre polynomials, highlighting his early derivations of key properties and lesser-known integral representations.

Findings

Euler's papers contain the generating function for Legendre polynomials.

Euler's difference equations for Legendre polynomials are identified.

An integral representation derived from Euler's continued fractions is presented.

Abstract

In this note we will present how Euler's investigations on various different subjects lead to certain properties of the Legendre polynomials. More precisely, we will show that the generating function and the difference equation for the Legendre polynomials was already written down by Euler in at least two different papers. Furthermore, we will demonstrate that some familiar expressions for the Legendre polynomials are corollaries of the before-mentioned works. Finally, we will show that Euler's ideas on continued fractions lead to an integral representation for the Legendre polynomials that seems to be less generally known.