An Elementary Approach to Depoissonization

TL;DR

The paper addresses depoissonization: recovering asymptotics of coefficients $A_n$ from the exponential generating function via an elementary real-variable framework that avoids complex-analytic growth assumptions. It introduces the Poissonized function $f(x)=e^{-x}\sum_{m\ge0}\frac{A_m}{m!}x^m$ and derives a Charlier–Poisson expansion $A_n=\sum_{m=0}^N \frac{f^{(m)}(n)}{m!}\tau_m(n)+O\big(17\,n^{(N+1)/2}E(f^{(N+1)};n)\big)$, linking $A_n$ to local derivatives of $f$ via the polynomials $\tau_m$. The inverse problem is treated via Ramanujan-type expansions, expressing $f(R)$ in terms of finite differences $\Delta^s A_n$ and appropriate polynomials (e.g., $\tau_s(-R)$, $C_m$), with remainder terms and connections to Mahler polynomials. The framework yields concrete first- and higher-order depoissonization results, monotone-case corollaries, and operator-bound techniques, offering a unified, self-contained approach with broad applicability in analytic combinatorics and probability.

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.

Theorems & Definitions (62)

• Theorem 1.1: jacquet_szpanwowski_1998
• Theorem 1.2
• Corollary 1.3
• Proposition 1.4
• Theorem 1.5: "Entry 10" of berndt_1985
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 2.1
• proof