About the Cramér Large Deviation Property for Bell Polynomials
Abstract
If w = (w1,w2,…) is a sequence in ℕ={1,2,…}, the partial Bell polynomials based on w are Bn,k for k ∈ ℕ and n∈{k,k+1,…}. Let F(z) = ∑n=1∞ (wn/n!)zn be the exponential generating function for w, and assume the radius of convegence is positive R>0. Then F(z)k = ∑n=k∞ (k!/n!) zn Bn,k for |z|<R. Alternatively, defining Qk,n = (k!/n!)Bn,k, we have Q1,n = wn/n!, and Qk+1,n=∑m=1n−k Q1,m Qk,n−m for k≥ 1. Let us say that the Cramér-type large deviation property holds if limsubstackn → ∞ k/n → κ 1/n ln(Qk,n) = 𝒢(κ) , for every κ∈ (0,1), where 𝒢(κ)=infr ∈ (0,R) (κln(F(r))−ln(r)). The (Hardy-Ramanujan) Erdös induction argument suggests this should generally be true as long as two technical conditions are true: one an initial step, and the other a condition for small densities κ.
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BibTeX
@misc{li2026about,
title = {{About the Cramér Large Deviation Property for Bell Polynomials}},
author = {Sophia Li and Shannon Starr},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06281},
eprint = {2609.06281},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06281},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - About the Cramér Large Deviation Property for Bell Polynomials
AU - Sophia Li
AU - Shannon Starr
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.06281
UR - https://arxiv.org/abs/2609.06281
AB - If w = (w_1,w_2,…) is a sequence in ℕ={1,2,…}, the partial Bell polynomials based on w are B_n,k for k ∈ ℕ and n∈{k,k+1,…}. Let F(z) = ∑_n=1^∞ (w_n/n!)z^n be the exponential generating function for w, and assume the radius of convegence is positive R>0. Then F(z)^k = ∑_n=k^∞ (k!/n!) z^n B_n,k for |z|<R. Alternatively, defining Q_k,n = (k!/n!)B_n,k, we have Q_1,n = w_n/n!, and Q_k+1,n=∑_m=1^n−k Q_1,m Q_k,n−m for k≥ 1. Let us say that the Cramér-type large deviation property holds if lim_substackn → ∞ k/n → κ 1/n ln(Q_k,n) = 𝒢(κ) , for every κ∈ (0,1), where 𝒢(κ)=inf_r ∈ (0,R) (κln(F(r))−ln(r)). The (Hardy-Ramanujan) Erdös induction argument suggests this should generally be true as long as two technical conditions are true: one an initial step, and the other a condition for small densities κ.
ER -