Exponential Taylor Series

TL;DR

This paper introduces a novel series expansion for differentiable complex functions using powers of $(1-e^{x})$, providing explicit formulas for remainders and applications to infinite series involving Stirling Numbers.

Contribution

It presents a new exponential Taylor series formulation with explicit remainder formulas and demonstrates its utility in calculating series involving Stirling Numbers.

Findings

Derived a series expansion for complex functions.

Provided explicit formulas for remainders in the series.

Applied the series to compute series with Stirling Numbers.

Abstract

This paper derives a way to express differentiable complex-valued functions as the sum of powers of $(1-e^{\lambda x})$, where $\lambda\in\mathbb{R}$, with an explicit formula for the remainder. This formulation is then used to associate an infinite series to $C^\infty$ functions, which is shown to recover the original function under suitable conditions on the remainder. These results are also used to calculate some infinite series involving Stirling Numbers, as well as providing a few examples.