Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Borwein, J.M., Corless, R.M.: The Gamma function in the Monthly, American Math Monthly, in press. arXiv:1703.05349 [math.HO]
Coppersmith, D.: Personal communication
Copson, E.T.: An Introduction to the Theory of Functions of a Complex Variable. The Clarendon Press, Oxford (1935)
Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the lambert W function. Adv. Comput. Math. 5, 329–359 (1996)
Jeffrey, D.J., Hare, D.E.G., Corless, R.M.: Unwinding the branches of the Lambert W function. Math. Sci. 21, 1–7 (1996)
Jeffrey, D.J., Kalugin, G.A., Murdoch, N.: Lagrange inversion and Lambert W. In: SYNASC 2015 Proceedings, pp 42–46. IEEE Computer Society (2015)
Marsaglia, G., Marsaglia, J.C.: A new derivation of Stirling’s approximation to \(n!\). Am. Math. Monthly 97, 826–829 (1990)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Jeffrey, D.J., Murdoch, N. (2017). Stirling Numbers, Lambert W and the Gamma Function. In: Blömer, J., Kotsireas, I., Kutsia, T., Simos, D. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2017. Lecture Notes in Computer Science(), vol 10693. Springer, Cham. https://doi.org/10.1007/978-3-319-72453-9_21
Download citation
DOI: https://doi.org/10.1007/978-3-319-72453-9_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-72452-2
Online ISBN: 978-3-319-72453-9
eBook Packages: Computer ScienceComputer Science (R0)