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 $\frac{1}{(1-e^{\pm t}{)}^k}$ and the derivatives $(\frac{1}{e^{\pm t}-1}{)}^{(i)}$ can be expressed each other by linear combinations with coefficients involving the combinatorial numbers and the Stirling numbers of the second kind, where $t\ne 0$ and $i,k\in \mathbb{N}$.

9 pages

References in corpus (2)

• Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions
• Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions

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