On Transforming the Generalized Exponential Power Series

TL;DR

This paper transforms the generalized exponential power series into a new functional form using exponential operators, revealing new relations and polynomials, and explores their properties and asymptotic behavior.

Contribution

It introduces a novel transformation of the exponential power series employing the Cauchy-Euler operator, leading to new polynomial relations and series representations for common functions.

Findings

Derived new relations for the exponential operator.

Introduced a new polynomial with interesting properties.

Created series expressions for trigonometric and exponential functions.

Abstract

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.