The Ramanujan Journal

, 20:179 | Cite as

On an asymptotic series of Ramanujan

Open Access


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.


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

Mathematics Subject Classification (2000)

34E05 60E05 


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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of CaliforniaIrvineUSA

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