A simple continued fraction expansion for $e^n$

Abstract

In this paper we present a family of continued fraction expansions for $e^n$, with $n\ge 1$, with a simple expression having partial denominators given by arithmetic progressions. We give an estimate for the convergence speed showing that the convergence is faster than the corresponding regular continued fractions. Moreover, we prove that the continued fractions for $e^n$ given in this paper are special cases of continued fraction expansions, different from the standard ones, of the confluent hypergeometric function, or equivalently, of the incomplete gamma function. In addition, using the same method we give a related family of continued fraction expansions of $e^{\ell/n}$ for positive integers $1\leq\ell< n $ that contains the case of integral exponent as a limit case.