  We transformed the generalized exponential power series to another functional
form suitable for further analysis. By applying the Cauchy-Euler differential
operator in the form of an exponential operator, the series became a sum of
exponential differential operators acting on a simple exponential (exp(-x). In
the process we found new relations for the operator and a new polynomial with
some interesting properties. Another form of the exponential power series
became a nested sum of the new polynomial, thus isolating the main variable to
a different functional dependence. We studied shortly the asymptotic behavior
by using the dominant terms of the transformed series. New series expressions
were created for common functions, like the trigonometric and exponential
functions, in terms of the polynomial.
