An Elementary Approach to Depoissonization

TL;DR

This paper introduces a simple real-variable method for depoissonization, enabling the recovery of sequence asymptotics from generating functions without complex-analytic assumptions, and extends Ramanujan's expansion.

Contribution

It develops an elementary real-variable approach to depoissonization and its inverse, broadening the tools available for asymptotic analysis in combinatorics and probability.

Findings

Provides a unified framework for depoissonization and its inverse.

Avoids complex-analytic growth conditions in asymptotic analysis.

Extends Ramanujan's original expansion to a broader context.

Abstract

We investigate depoissonization, the problem of recovering asymptotics of sequence coefficients from their exponential generating function. Classical approaches rely on complex-analytic growth conditions, but here we develop real-variable methods that avoid such assumptions. We also address the inverse problem, deriving asymptotic expansions of the generating function itself in terms of its coefficients, thereby extending Ramanujan's original expansion. Taken together, these results offer a unified and elementary framework for depoissonization and its reverse, with applications to analytic combinatorics and probability.