Some identities and an explicit formula for Bernoulli and Stirling numbers

Abstract

In the paper, the authors establish eight identities which reveal that the functions 1(1−e±t)k and the derivatives (1e±t−1)(i) can be expressed by each other by linear combinations with coefficients involving the combinatorial numbers and Stirling numbers of the second kind, find an explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind, and present two identities for Stirling numbers of the second kind.