On the inversion of yαey in terms of associated Stirling numbers

The function y = Φα(x), the solution of y e = x for x and y large enough, has a series expansion in terms of lnx and ln lnx, with coefficients given in terms of Stirling cycle numbers. It is shown that this expansion converges for x > (αe) for α ≥ 1. It is also shown that new expansions can be obtained for Φα in terms of associated Stirling numbers. The new expansions converge more rapidly and on a larger domain. 1. Stirling numbers — Stirling cycle numbers [ n m ]