Stirling Numbers, Lambert W and the Gamma Function
Stirling’s asymptotic expansion for the Gamma function can be derived from an expansion of the Lambert W function about one of its branch points. Although the series expansions around this branch point have been known for some time, the coefficients in the series were only known as solutions of nonlinear recurrence relations. Here we show that the coefficients can be expressed using associated Stirling numbers.
- 1.Borwein, J.M., Corless, R.M.: The Gamma function in the Monthly, American Math Monthly, in press. arXiv:1703.05349 [math.HO]
- 2.Coppersmith, D.: Personal communicationGoogle Scholar
- 6.Jeffrey, D.J., Kalugin, G.A., Murdoch, N.: Lagrange inversion and Lambert W. In: SYNASC 2015 Proceedings, pp 42–46. IEEE Computer Society (2015)Google Scholar