Bourlet's Theorem for the product of differential operators, an application of the operator method and a proof for $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$, that Euler missed, derived from difference equations

TL;DR

This paper presents a new proof of the Basel problem sum using difference equations, extending Bourlet's theorem for differential operators and offering insights into operator methods.

Contribution

It introduces a novel proof of the Basel problem sum based on difference equations, connecting operator theory with classical analysis.

Findings

New proof of alculus sum using difference equations

Application of Bourlet's theorem to operator methods

Insight into the connection between difference equations and classical sums

Abstract

We give another proof for \[ \sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6} \] that basically follows from the theory of difference equations.