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On Two Conjectures Related to the Boros-Moll Sequences

  • Heshan Aravinda
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Abstract

The Boros-Moll sequences {di(m)}0≤ i≤ m are defined as di(m)=2−2m∑k=im 2k binom2m−2km−kbinomm+kkbinomki. Consider the ratio sequence ui(m)=(di−1(m)di+1(m))/di(m)2. Chen and Gu conjectured that {ui(m)}2≤ i ≤ m−2 is both reverse ultra log-concave and log-concave. In this paper, we prove the reverse ultra log-concavity conjecture using bounds of Chen-Gu and Zhao, and prove the log-concavity conjecture asymptotically by showing that {ui(m)}2≤ i ≤ m−2 is strictly log-concave for all sufficiently large m. The key ingredient in the latter result is a recurrence of Kauers and Paule, which we interpret as a nonlinear discrete dynamical system through a backward map. We construct an approximation to the ratio sequence using its stable limiting fixed point and combine localization and contraction arguments with finite-difference estimates and separate interior and edge analyses to obtain the desired strict log-concavity.

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Cite

BibTeXRIS
BibTeX
@misc{aravinda2026conjectures,
  title = {{On Two Conjectures Related to the Boros-Moll Sequences}},
  author = {Heshan Aravinda},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10234},
  eprint = {2609.10234},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10234},
}
RIS
TY  - UNPB
TI  - On Two Conjectures Related to the Boros-Moll Sequences
AU  - Heshan Aravinda
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.10234
UR  - https://arxiv.org/abs/2609.10234
AB  - The Boros-Moll sequences {d_i(m)}_0≤ i≤ m are defined as d_i(m)=2^−2m∑_k=i^m 2^k binom2m−2km−kbinomm+kkbinomki. Consider the ratio sequence u_i(m)=(d_i−1(m)d_i+1(m))/d_i(m)^2. Chen and Gu conjectured that {u_i(m)}_2≤ i ≤ m−2 is both reverse ultra log-concave and log-concave. In this paper, we prove the reverse ultra log-concavity conjecture using bounds of Chen-Gu and Zhao, and prove the log-concavity conjecture asymptotically by showing that {u_i(m)}_2≤ i ≤ m−2 is strictly log-concave for all sufficiently large m. The key ingredient in the latter result is a recurrence of Kauers and Paule, which we interpret as a nonlinear discrete dynamical system through a backward map. We construct an approximation to the ratio sequence using its stable limiting fixed point and combine localization and contraction arguments with finite-difference estimates and separate interior and edge analyses to obtain the desired strict log-concavity.
ER  -