Polynomial continued fractions for exp(π)

Abstract

ABSTRACT We present two (inequivalent) polynomial continued fraction representations of the number with all their elements in ; no such representation was seemingly known before. More generally, a similar result for is obtained for every such that . The proof uses a classical polynomial continued fraction representation of , for and , of which we present a new proof that enables us to obtain the exact rate of convergence of the convergents of the continued fraction for . We also deduce some consequences of arithmetic interest concerning the elements of certain polynomial continued fraction representations of the (transcendental) Gel'fond-Schneider numbers , where and , where is the field of algebraic numbers, embedded into .