bibkey: zacharovas2025depoissonization
authors: Vytas Zacharovas
year: 2025
title: An Elementary Approach to Depoissonization
doi: null
url: https://arxiv.org/abs/2511.14324v2
claim: Theorem 2.8 bounds arbitrary-center Poisson–Charlier truncation error using an absolute Poisson average of the next derivative's coefficients; it imposes no zero-location or Laguerre–Polya hypothesis.
strata_touched:
  - D5/S3/Zeros/Jensen/NormalizedJensenDegreeLowering
license: citation-only
triage: anchor

An arbitrary-center coefficient supplier

For a real sequence $(u_{\ell }{)}_{\ell \ge 0}$ whose exponential generating series is entire, write

$$g(z)=e^{-z}\sum _{\ell \ge 0}{u}_{\ell }{z}^{\ell }/\ell !,\mathcal{E}(g;\rho )=e^{-\rho }\sum _{\ell \ge 0}|u_{\ell }|{\rho }^{\ell }/\ell !.$$

The source uses the Charlier normalization

$${\mathsf{C}}_j(\rho ,\ell )=\sum _{k=0}^j(-1{)}^{j-k}(\genfrac{}{}{0ex}{}{j}{k})\frac{{\ell }^{\underline{k}}}{{\rho }^k}.$$

Theorem 2.8 states, for every $\rho >0$, integer $\ell \ge 2$ and $N\ge 0$,

$$\left|u_{\ell }-\sum _{j=0}^N\frac{g^{(j)}(\rho )}{j!}{\rho }^j{\mathsf{C}}_j(\rho ,\ell )\right|\le 12\frac{\ell !e^{\rho }}{{\rho }^{\ell }}{\ell }^{N/2}\mathcal{E}(g^{(N+1)};\rho ).$$

Here $\mathcal{E}(g^{(j)};\rho )$ uses the coefficients of $e^z{g}^{(j)}(z)$, equivalently the $j$th forward differences of $u_{\ell }$. It is not generally $|g^{(j)}(\rho )|$. Theorem 1.2’s center $\rho =\ell$ and constant 33 are a separate specialization; the arbitrary center above is retained for the source-zero application.

Corollary A.6 gives the displayed finite-sum normalization. In DLMF’s convention, ${\mathsf{C}}_j(\rho ,\ell )=(-1{)}^j{C}_j(\ell ;\rho )$. DLMF 18.21.9 supplies

$$\underset{a\to \infty }{\lim }(2a{)}^{j/2}{C}_j(a+\sqrt{2a}y;a)=(-1{)}^j{H}_j(y).$$

These are cited results, not new convergence or depoissonization theorems. The theorem applies to real coefficient sequences without assuming real-rootedness of $g$. The original-source application must separately map its coefficients, control this remainder and capture the actual polynomial nodes. A generic supplier is not an all-height Jensen margin or an RH/Robin proof.

The inspected source is arXiv:2511.14324v2, revised 3 September 2026. The initial preprint is from 18 November 2025.