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arXiv · 2511.14324

An Elementary Approach to Depoissonization

Vytas Zacharovas
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Abstract

We study depoissonization for sequences with entire exponential generating functions. We establish explicit finite-order remainder bounds for Poisson--Charlier approximations, controlled by Poisson-weighted averages of the absolute values of higher-order forward differences of the coefficient sequence. This complements classical analytic depoissonization, which instead relies on complex-plane growth estimates: classical methods are natural when such estimates are available, whereas the present approach is useful when the relevant finite differences can be bounded, without requiring estimates at nonreal arguments. We also study the inverse problem of deriving asymptotic expansions of the Poisson transform at large positive arguments from the coefficient sequence. This leads to finite-difference analogues of Ramanujan's expansion. Applications to combinatorics and probability are presented.

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Vytas Zacharovas. 2025-11-18. An Elementary Approach to Depoissonization. https://arxiv.org/abs/2511.14324

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