Article

The Ramanujan Journal

, Volume 20, Issue 2, pp 179-188

Open Access This content is freely available online to anyone, anywhere at any time.

On an asymptotic series of Ramanujan

  • Yaming YuAffiliated withDepartment of Statistics, University of California Email author 

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