  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.
