Eight interesting identities involving the exponential function, derivatives, and Stirling numbers of the second kind

TL;DR

This paper derives new identities linking exponential functions, their derivatives, and Stirling numbers of the second kind, revealing intricate combinatorial relationships and providing explicit linear combinations between these mathematical entities.

Contribution

The paper introduces novel identities connecting exponential functions, derivatives, and Stirling numbers, expanding the understanding of their combinatorial and analytical relationships.

Findings

Identities expressing functions and derivatives via Stirling numbers

Explicit linear combinations between exponential functions and derivatives

Enhanced understanding of combinatorial structures in exponential functions

Abstract

In the paper, the author establishes some identities which show that the functions $\frac1{(1-e^{\pm t})^k}$ and the derivatives $\bigl(\frac1{e^{\pm t}-1}\bigr)^{(i)}$ can be expressed each other by linear combinations with coefficients involving the combinatorial numbers and the Stirling numbers of the second kind, where $t\ne0$ and $i,k\in\mathbb{N}$.