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PreprintarXiv licencearXiv

Infinite log-concavity of the Boros--Moll sequences

  • Matthew H. Y. Xie
  • Philip B. Zhang
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Abstract

Let (di(n))i=0n be the Boros--Moll coefficient sequence. We prove that, for every integer n≥1, the polynomial Mn(x)=∑i=0n (di(n)2−di−1(n)di+1(n))xi has only simple negative zeros, which strictly interlace those of the Narayana polynomial of the same degree. This proves a conjecture of Chen, Yang, and Zhang and, by Brändén's preservation theorem, settles the infinite log-concavity conjecture of Boros and Moll. The proof uses an expansion of the reversed and normalized form of Mn(x) in derivatives of the Narayana polynomial, together with estimates for the weights and partial sums of the normalized derivatives.

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Cite

BibTeXRIS
BibTeX
@misc{xie2026infinite,
  title = {{Infinite log-concavity of the Boros--Moll sequences}},
  author = {Matthew H. Y. Xie and Philip B. Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20653},
  eprint = {2609.20653},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20653},
}
RIS
TY  - UNPB
TI  - Infinite log-concavity of the Boros--Moll sequences
AU  - Matthew H. Y. Xie
AU  - Philip B. Zhang
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20653
UR  - https://arxiv.org/abs/2609.20653
AB  - Let (d_i(n))_i=0^n be the Boros--Moll coefficient sequence. We prove that, for every integer n≥1, the polynomial M_n(x)=∑_i=0^n (d_i(n)^2−d_i−1(n)d_i+1(n))x^i has only simple negative zeros, which strictly interlace those of the Narayana polynomial of the same degree. This proves a conjecture of Chen, Yang, and Zhang and, by Brändén's preservation theorem, settles the infinite log-concavity conjecture of Boros and Moll. The proof uses an expansion of the reversed and normalized form of M_n(x) in derivatives of the Narayana polynomial, together with estimates for the weights and partial sums of the normalized derivatives.
ER  -