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

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 $i,k\in\mathbb{N}$.