Advertisement

Stirling Numbers, Lambert W and the Gamma Function

  • David J. Jeffrey
  • Nick Murdoch
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10693)

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.

References

  1. 1.
    Borwein, J.M., Corless, R.M.: The Gamma function in the Monthly, American Math Monthly, in press. arXiv:1703.05349 [math.HO]
  2. 2.
    Coppersmith, D.: Personal communicationGoogle Scholar
  3. 3.
    Copson, E.T.: An Introduction to the Theory of Functions of a Complex Variable. The Clarendon Press, Oxford (1935)MATHGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Jeffrey, D.J., Hare, D.E.G., Corless, R.M.: Unwinding the branches of the Lambert W function. Math. Sci. 21, 1–7 (1996)MathSciNetMATHGoogle Scholar
  6. 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
  7. 7.
    Marsaglia, G., Marsaglia, J.C.: A new derivation of Stirling’s approximation to \(n!\). Am. Math. Monthly 97, 826–829 (1990)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsThe University of Western OntarioLondonCanada

Personalised recommendations