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Infinitely many negative terms in two weighted partition sequences
Abstract
Recently, Andrews and El Bachraoui introduced partition-counting functions a2(n) and b2(n), together with their signed analogues a'2(n) and b'2(n), and proved that a'2(n), b'2(n)>0 for all n≥ 3. For m≥ 2, the latter functions extend naturally to a'm(n) and b'm(n), defined by ∑n≥ 0 a'm(n)qn &= −q/(1−q) (qm;qm)∞/(q;qm)∞ +q/((1−q)2), ∑n≥ 0 b'm(n)qn &= −(q(1−qm−1))/(1−q) (q2m;qm)∞/(q;qm)∞ +q/(1−q) ( 1/(1−q) −(qm−1)/(1−qm+1) ). They asked whether there exists an integer m>2 for which a'm(n) is eventually positive, and posed the same question for b'm(n). In this paper, we resolve these two questions.
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BibTeX
@misc{li2026infinitely,
title = {{Infinitely many negative terms in two weighted partition sequences}},
author = {Meng-Yao Li and Su-Ping Cui},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02389},
eprint = {2611.02389},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02389},
}
RIS
TY - UNPB
TI - Infinitely many negative terms in two weighted partition sequences
AU - Meng-Yao Li
AU - Su-Ping Cui
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02389
UR - https://arxiv.org/abs/2611.02389
AB - Recently, Andrews and El Bachraoui introduced partition-counting functions a_2(n) and b_2(n), together with their signed analogues a'_2(n) and b'_2(n), and proved that a'_2(n), b'_2(n)>0 for all n≥ 3. For m≥ 2, the latter functions extend naturally to a'_m(n) and b'_m(n), defined by ∑_n≥ 0 a'_m(n)q^n &= −q/(1−q) (q^m;q^m)_∞/(q;q^m)_∞ +q/((1−q)^2), ∑_n≥ 0 b'_m(n)q^n &= −(q(1−q^m−1))/(1−q) (q^2m;q^m)_∞/(q;q^m)_∞ +q/(1−q) ( 1/(1−q) −(q^m−1)/(1−q^m+1) ). They asked whether there exists an integer m>2 for which a'_m(n) is eventually positive, and posed the same question for b'_m(n). In this paper, we resolve these two questions.
ER -