Polynomial continued fractions for $\exp(\pi)$
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 $\beta \in \overline{\mathbb{Q}} \setminus \mathbb Q$.
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