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arXiv · 0904.1226

On an Asymptotic Series of Ramanujan

Yaming Yu
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Abstract

An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 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 problem of inverse moments for distribution families such as the binomial or the negative binomial.

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Yaming Yu. 2009-04-07. On an Asymptotic Series of Ramanujan. https://doi.org/10.1007/s11139-009-9169-x

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