On the inversion of $y^αe^y$ in terms of associated Stirling numbers

Abstract

The function $y=Φ_α(x)$, the solution of $y^αe^y=x$ 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 $x>(αe)^α$ for $α\ge 1$. It is also shown that new expansions can be obtained for $Φ_α$ in terms of associated Stirling numbers. The new expansions converge more rapidly and on a larger domain.