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

David J. Jeffrey, Robert M. Corless, David E. G. Hare +1

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 Sti…