Expanding the function $\ln(1+\operatorname{e}^x)$ into power series in terms of the Dirichlet eta function and the Stirling numbers of the second kind

Abstract

In the paper, using several approaches, the authors expand the composite function $\ln(1+\operatorname{e}^x)$ into power series around $x=0$, whose coefficients are expressed in terms of the Dirichlet eta function $\eta(1-n)$ and the Stirling numbers of the second kind $S(n,k)$.