Applications of Some Formulae by Hermite to the Approximation of Exponentials and Logarithms

Abstract

While Liouville gave the first examples of transcendental numbers, the modern theory of proofs of transcendency started with Hermite’s beautiful paper “Sur la fonction exponentielle” (Hermite, 1873). In this paper, for a given system of distinct complex numbers ω ₀ , ω ₁ ...,ω _m and of positive integers ϱ ₀ , ϱ ₁ ..., ϱ _m with the sum σ, Hermite constructed a set of m + 1 polynomials

$$ \mathfrak{A}o(z),{\mathfrak{A}_1}(z),...,{\mathfrak{A}_m}(z) $$

of degrees not exceeding σ-ϱ₀, σ-ϱ₁,..., σ-ϱ_m respectively, such that all the functions

$$ {\mathfrak{A}_k}(z){e^{{\omega _1}z}},{\mathfrak{A}_l}(z){e^{{\omega _k}z}}\;(0\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } k < l\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } m) $$

vanish at z = 0 at least to the order σ + 1. On putting z = 1, these formulae produce simultaneous rational approximations of the numbers 1, e, e ² ,..., e ^m that are so good that they imply the linear independence of these numbers and hence the transcendency of e.