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

David J. Jeffrey·Robert M. Corless·David E. G. Hare·Donald E. Knuth
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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.

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David J. Jeffrey, Robert M. Corless, David E. G. Hare, Donald E. Knuth. 1995-12-01. On the inversion of $y^αe^y$ in terms of associated Stirling numbers. https://arxiv.org/abs/math/9512230

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