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