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The Ramanujan Journal

, Volume 16, Issue 1, pp 1–5 | Cite as

On the complete monotonicity of a Ramanujan sequence connected with e n

  • José A. Adell
  • P. Jodrá
Article

Abstract

We show that the Ramanujan sequence (θ n ) n≥0 defined as the solution to the equation
$$\frac{e^{n}}{2}=\sum_{k=0}^{n-1}\frac{n^{k}}{k!}+\frac{n^{n}}{n!}\theta_{n}$$
is completely monotone. Our proof uses the fact that (θ n ) n≥0 coincides, up to translation and renorming, with the moment sequence of a probability distribution function on [0,1] involving the two real branches of the Lambert W function.

Keywords

Ramanujan sequence Complete monotonicity Moment sequence Lambert W function 

Mathematics Subject Classification (2000)

41A60 33B30 

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References

  1. 1.
    Alm, S.E.: Monotonicity of the difference between median and mean of gamma distributions and of a related Ramanujan sequence. Bernoulli 9, 351–371 (2003) CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Alzer, H.: On Ramanujan’s inequalities for exp (k). J. Lond. Math. Soc. (2) 69, 639–656 (2004) CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Barry, D.A., Parlange, J.-Y., Li, L., Prommer, H., Cunningham, C.J., Stagnitti, F.: Analytical approximations for real values of the Lambert W-function. Math. Comput. Simul. 53, 95–103 (2000) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Berndt, B.C.: Ramanujan’s Notebooks: Part II. Springer, New York (1989) zbMATHGoogle Scholar
  5. 5.
    Berndt, B.C., Choi, Y.-S., Kang, S.-Y.: The problems submitted by Ramanujan to the Journal of the Indian Mathematical Society. In: Continued Fractions: From Analytic Number Theory to Constructive Approximation, Columbia, MO, 1998. Contemporary Mathematics, vol. 236, pp. 15–56. American Mathematical Society, Providence (1999) Google Scholar
  6. 6.
    Bracken, P.: A function related to the central limit theorem. Comment. Math. Univ. Carol. 39, 765–775 (1998) MathSciNetzbMATHGoogle Scholar
  7. 7.
    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) CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2. Wiley, New York (1966) zbMATHGoogle Scholar
  9. 9.
    Flajolet, P., Grabner, P.J., Kirschenhofer, P., Prodinger, H.: On Ramanujan’s Q-function. J. Comput. Appl. Math. 58, 103–116 (1995) CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Karamata, J.: Sur quelques problèmes posés par Ramanujan. J. Indian Math. Soc. 24, 343–365 (1960) MathSciNetGoogle Scholar
  11. 11.
    Ramanujan, S.: Question 294. J. Indian Math. Soc. 3, 128 (1911) Google Scholar
  12. 12.
    Ramanujan, S.: On question 294. J. Indian Math. Soc. 4, 151–152 (1912) Google Scholar
  13. 13.
    Ramanujan, S.: Collected Papers. Chelsea, New York (1962) Google Scholar
  14. 14.
    Szegö, G.: Über einige von S. Ramanujan gestelle Aufgaben. J. Lond. Math. Soc. 3, 225–232 (1928) CrossRefGoogle Scholar
  15. 15.
    Watson, G.N.: Theorems stated by Ramanujan (V): approximations connected with e x. Proc. Lond. Math. Soc. (2) 29, 293–308 (1929) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Departamento de Métodos EstadísticosUniversidad de ZaragozaZaragozaSpain

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