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Two algebraic proofs of the transcendence of $\mathrm{e}$ based on formal power series

Martin Klazar

Abstract

We first remind the classical analytical proof of the transcendence of $\mathrm{e}$ due to Hilbert. Then, using formal power series, we give two algebraic semiformal proofs of the same result. The first proof is a specialization of a proof of the Lindemann-Weierstrass theorem given by Beukers, Bézivin and Roba [2]. The second proof uses improper integrals of formal power series and is due to this author. At the beginning, we explain what ``semiformal'' means and indicate yet another analytical proof of the transcendence of $\mathrm{e}$. Like the two algebraic proofs, it avoids uncountable sets. It is due to this author in [4].

Two algebraic proofs of the transcendence of $\mathrm{e}$ based on formal power series

Abstract

We first remind the classical analytical proof of the transcendence of due to Hilbert. Then, using formal power series, we give two algebraic semiformal proofs of the same result. The first proof is a specialization of a proof of the Lindemann-Weierstrass theorem given by Beukers, Bézivin and Roba [2]. The second proof uses improper integrals of formal power series and is due to this author. At the beginning, we explain what ``semiformal'' means and indicate yet another analytical proof of the transcendence of . Like the two algebraic proofs, it avoids uncountable sets. It is due to this author in [4].
Paper Structure (3 sections, 14 theorems, 93 equations)

This paper contains 3 sections, 14 theorems, 93 equations.

Table of Contents

  1. Hilbert's proof that $\mathrm{e}$ is transcendental. Why this proof is not completely satisfactory. How to fix it
  2. Specializing the proof of Beukers, Bézivin and Robba
  3. The proof by formal integration

Key Result

Proposition 2.1

If $p(x)\in\mathbb{C}[x]$, $m\in\mathbb{N}_0$, $c\in\mathbb{C}^*$ ($=\mathbb{C}\setminus\{0\}$) and if $A(x)$ in $\mathbb{C}[[x]]$ is rational, then the rational formal power series either has no pole (is a polynomial) or all its poles have orders at least $2$.

Theorems & Definitions (18)

  • Proposition 2.1
  • Theorem 2.2
  • Proposition 3.1
  • Definition 3.2
  • Definition 3.3: semiformal shifts
  • Proposition 3.4
  • Proposition 3.5
  • Theorem 3.6
  • Definition 3.7: Newton integral of fps
  • Proposition 3.8
  • ...and 8 more