On the inversion of $y^{\alpha }{e}^y$ in terms of associated Stirling numbers

Abstract

The function $y={\Phi }_{\alpha }(x)$, the solution of $y^{\alpha }{e}^y=x$ for $x$ and $y$ large enough, has a series expansion in terms of $\ln x$ and $\ln \ln x$, with coefficients given in terms of Stirling cycle numbers. It is shown that this expansion converges for $x>(\alpha e{)}^{\alpha }$ for $\alpha \ge 1$. It is also shown that new expansions can be obtained for ${\Phi }_{\alpha }$ in terms of associated Stirling numbers. The new expansions converge more rapidly and on a larger domain.