Polynomial continued fractions for $\exp(\pi)$ - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Journal of Difference Equations and Applications Année : 2023

Polynomial continued fractions for $\exp(\pi)$

Tanguy Rivoal

Résumé

We present two (inequivalent) polynomial continued fraction representations of the number $e^\pi$ with all their elements in~$\mathbb{Q}$; no such representation was seemingly known before. More generally, a similar result for $e^{r\pi}$ is obtained for every $r\in \mathbb Q$. The proof uses a classical polynomial continued fraction representation of $\alpha^\beta$, for $\vert \arg(\alpha)\vert <\pi$ and $\beta \in \mathbb{C}\setminus \mathbb{Z}$, of which we offer a proof using a complex contour integral originating from interpolation theory. We also deduce some consequences of arithmetic interest concerning the elements of certain polynomial continued fraction representations of the (transcendental) Gel'fond-Schneider numbers $\alpha^\beta$, where $\alpha\in \overline{\mathbb{Q}}\setminus \{0,1\}$ and $\beta \in \overline{\mathbb{Q}} \setminus \mathbb Q$.
Fichier principal
Vignette du fichier
fracontexpi.pdf (284.89 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03269677 , version 1 (24-06-2021)
hal-03269677 , version 2 (26-06-2021)
hal-03269677 , version 3 (05-09-2022)

Identifiants

Citer

Tanguy Rivoal. Polynomial continued fractions for $\exp(\pi)$. Journal of Difference Equations and Applications, 2023, 29 (4), pp.377-392. ⟨10.1080/10236198.2023.2204979⟩. ⟨hal-03269677v3⟩
67 Consultations
221 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More