The Ramanujan Journal

, 20:179 | Cite as

On an asymptotic series of Ramanujan

Open Access
Article

Abstract

An asymptotic series in Ramanujan’s second notebook (Entry 10, Chap. 3) is concerned with the behavior of the expected value of φ(X) for large λ where X is a Poisson random variable with mean λ and φ is a function satisfying certain growth conditions. We generalize this by studying the asymptotics of the expected value of φ(X) when the distribution of X belongs to a suitable family indexed by a convolution parameter. Examples include the binomial, negative binomial, and gamma families. Some formulas associated with the negative binomial appear new.

Keywords

Asymptotic expansion Binomial distribution Central moments Cumulants Gamma distribution Negative binomial distribution Poisson distribution Ramanujan’s notebooks 

Mathematics Subject Classification (2000)

34E05 60E05 

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1964) MATHGoogle Scholar
  2. 2.
    Berndt, B.C.: Ramanujan’s Notebooks, Part I. Springer, New York (1985) MATHGoogle Scholar
  3. 3.
    Berndt, B.C.: Ramanujan’s Notebooks, Part II. Springer, New York (1989) MATHGoogle Scholar
  4. 4.
    David, F.N., Johnson, N.L.: Reciprocal Bernoulli and Poisson variables. Metron 18, 77–81 (1956) MATHMathSciNetGoogle Scholar
  5. 5.
    Evans, R.J.: Ramanujan’s second notebook: asymptotic expansions for hypergeometric series and related functions. In: Proc. Ramanujan Centennial Conference. Academic Press, New York (1988) Google Scholar
  6. 6.
    Evans, R.J., Boersma, J., Blachman, N.M., Jagers, A.A.: The entropy of a Poisson distribution: problem 87-6. SIAM Rev. 30, 314–317 (1988) CrossRefGoogle Scholar
  7. 7.
    Flajolet, P.: Singularity analysis and asymptotics of Bernoulli sums. Theor. Comput. Sci. 215, 371–381 (1999) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Grab, E.L., Savage, I.R.: Tables for the expected value of 1/x for positive Bernoulli and Poisson variables. J. Am. Stat. Assoc. 49, 169–177 (1954) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gut, A.: Probability: A Graduate Course. Springer, New York (2005) MATHGoogle Scholar
  10. 10.
    Jacquet, P., Szpankowski, W.: Analytical depoissonization and its applications. Theor. Comput. Sci. 201, 1–62 (1998) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Jacquet, P., Szpankowski, W.: Entropy computations via analytic depoissonization. IEEE Trans. Inf. Theory 45, 1072–1081 (1999) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Johnson, N.L., Kemp, A.W., Kotz, S.: Univariate Discrete Distributions, 3rd edn. Wiley, Hoboken (2005) MATHGoogle Scholar
  13. 13.
    Krichevskiy, R.E.: Laplace’s law of succession and universal encoding. IEEE Trans. Inf. Theory 44, 296–303 (1998) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Marciniak, E., Wesolowski, J.: Asymptotic Eulerian expansions for binomial and negative binomial reciprocals. Proc. Am. Math. Soc. 127, 3329–3338 (1999) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Rempala, G.A.: Asymptotic factorial powers expansions for binomial and negative binomial reciprocals. Proc. Am. Math. Soc. 132, 261–272 (2004) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Stephan, F.F.: The expected value and variance of the reciprocal and other negative powers of a positive Bernoullian variate. Ann. Math. Stat. 16, 50–61 (1945) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Willink, R.: Relationships between central moments and cumulants, with formulae for the central moments of gamma distributions. Commun. Stat. Theory Methods 32, 701–704 (2003) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Wuyungaowa and Wang, T.: Asymptotic expansions for inverse moments of binomial and negative binomial. Stat. Probab. Lett. 78, 3018–3022 (2008) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Žnidarič, M.: Asymptotic expansion for inverse moments of binomial and Poisson distributions. Open Stat. Probab. J. 1, 7–10 (2009) CrossRefGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of CaliforniaIrvineUSA

Personalised recommendations