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.
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Yu, Y. On an asymptotic series of Ramanujan. Ramanujan J 20, 179–188 (2009). https://doi.org/10.1007/s11139-009-9169-x
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DOI: https://doi.org/10.1007/s11139-009-9169-x
Keywords
- Asymptotic expansion
- Binomial distribution
- Central moments
- Cumulants
- Gamma distribution
- Negative binomial distribution
- Poisson distribution
- Ramanujan’s notebooks