Irrationality proofs \`a la Hermite

TL;DR

This paper explores historical and mathematical proofs of irrationality, including Hermite's original work, providing new proofs and generalizations related to nd  and Bessel functions.

Contribution

It uncovers Hermite's overlooked proof of nd 's irrationality, offers a new proof for nd  r r, and generalizes to ratios of Bessel functions.

Findings

Hermite's simple proof of nd 's irrationality

New proof of irrationality of r r for rational r^2

Generalization to ratios of Bessel functions

Abstract

As rewards of reading two great papers of Hermite from 1873, we trace the historical origin of the integral Niven used in his well-known proof of the irrationality of $\pi$, uncover a rarely acknowledged simple proof by Hermite of the irrationality of $\pi^2$, give a new proof of the irrationality of $r\tan r$ for nonzero rational $r^2$, and generalize it to a proof of the irrationality of certain ratios of Bessel functions.