The transcendence of $\mathrm{e}$ via formal power series

TL;DR

This paper explores algebraic proofs of the transcendence of e using formal power series, including two distinct methods based on classical and modern approaches.

Contribution

It presents two novel proofs of e's transcendence via formal power series, extending classical results with algebraic and FPS-based techniques.

Findings

Two proofs of e's transcendence using formal power series

Extension of Hilbert's classical proof to algebraic FPS methods

Connection between FPS techniques and Lindemann-Weierstrass theorem

Abstract

We review Hilbert's classical analytical proof of the transcendence of the number $\mathrm{e}$. Then, we show how this result can be obtained algebraically by means of formal power series (FPS). We give two proofs of the transcendence of $\mathrm{e}$ based on FPS. The first of them is a specialization of the 1990 proof by Beukers, B\'ezivin and Robba of the Lindemann-Weierstrass theorem. The second proof is due to this author and is an adaptation of Hilbert's argument to FPS.