LAUSR

We present two (inequivalent) polynomial continued fraction representations of the number eπ with all their elements in Q; no such representation was seemingly known before. More generally, a similar result for erπ is obtained for every r ∈ Q. The proof uses a classical polynomial continued fraction representation of αβ, for | arg(α)| < π and β ∈ C \ 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 αβ, where α ∈ Q \ {0, 1} and β ∈ Q \Q.