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The Ramanujan Journal

, Volume 16, Issue 3, pp 235–245 | Cite as

Factorial series connected with the Lambert function, and a problem posed by Ramanujan

  • Hans Volkmer
Article

Abstract

Ramanujan’s sequence {y n } n=0 defined by \(\sum _{j=0}^{n-1}\frac{n^{j}}{j!}+\frac{n^{n}}{n!}y_{n}=\frac{e^{n}}{2}\) is expanded in factorial series derived from a series representing the Lambert W function. As a corollary, it is shown that the sequence {y n } is completely monotonic.

Keywords

Lambert function Ramanujan’s conjecture Factorial series Gamma function 

Mathematics Subject Classification (2000)

33E20 33A15 

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References

  1. 1.
    Corless, R.M., Gonnet, G.H., Hare, D.E.G., Knuth, D.E.: On the Lambert W function. Adv. Comput. Math. 5, 329–359 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Corless, R.M., Jeffrey, D.J., Knuth, D.E.: A sequence of series for the Lambert W function. In: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, Kihei, HI, pp. 197–204 (electronic). ACM, New York (1997) CrossRefGoogle Scholar
  3. 3.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2, 2nd edn. Wiley, New York (1971) zbMATHGoogle Scholar
  4. 4.
    Hardy, G.H., Seshu Aiyar, P.V., Wilson, B.M.: Collected Papers of Srinivasa Ramanujan. Cambridge University Press, Cambridge (1927), reprinted Chelsea, New York, 1962 Google Scholar
  5. 5.
    Huntington, E.V.: Stirling’s formula with remainder. Biometrika 31, 390 (1940) MathSciNetGoogle Scholar
  6. 6.
    Karamata, J.: Sur quelques problèmes posés par Ramanujan. J. Indian Math. Soc. 24, 343–365 (1960) MathSciNetGoogle Scholar
  7. 7.
    Key, E.: A probabilistic approach to a conjecture of Ramanujan. J. Ramanujan Math. Soc. 4(2), 109–119 (1989) zbMATHMathSciNetGoogle Scholar
  8. 8.
    Knopp, K.: Theory and Application of Infinite Series. Dover, New York (1990) Google Scholar
  9. 9.
    Marsaglia, G., Marsaglia, J.: A new derivation of Stirling’s approximation to n! Am. Math. Mon. 97, 826–829 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Olver, F.W.J.: Asymptotics and Special Functions. A.K. Peters, Natick (1997) zbMATHGoogle Scholar
  11. 11.
    Olver, F.W.J.: Error bounds for the Laplace approximation for definite integrals. J. Approx. Theory 1, 293–313 (1968) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Szegö, G.: Über einige von S. Ramanujan gestellte Aufaben. J. Lond. Math. Soc. 3, 225–232 (1928) CrossRefGoogle Scholar
  13. 13.
    Watson, G.N.: Theorems stated by Ramanujan (V): approximations connected with e x. Proc. Lond. Math. Soc. (2) 28, 239–308 (1928) CrossRefGoogle Scholar
  14. 14.
    Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1941) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of Wisconsin–MilwaukeeMilwaukeeUSA

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