A unified approach to infinite log-concavity of combinatorial sequences
Abstract
We establish a sufficient condition for infinite log-concavity and apply it to five families of combinatorial sequences. We give a new proof of infinite log-concavity for the Boros--Moll coefficient sequences (dk(n))k=0n for fixed n. For the transposed sequences (dℓ(ℓ+k))k≥0, we prove Zhao's conjecture on infinite log-concavity for every fixed integer ℓ≥3. For the normalized sequences in Euler's difference table, we confirm the infinite log-concavity conjecture of Chen, Gu, Ma and Wang under essential iteration, in which both endpoints are discarded after each step. We also confirm a conjecture of Medina, Moll and Rowland on the infinite log-concavity of the coefficient sequences of polynomials arising from iterated primitives of log(1+x). Finally, we answer a question of Brändén and Chasse by showing that, for positive integers d, the sequence (kd)k≥0 is infinitely log-concave if and only if d≠2. For every real d≥3, all its iterates are positive at every index k≥1. The proofs combine holomorphic estimates with finite computer-assisted verification.
Related
- Infinite log-concavity of the Boros--Moll sequences
- Infinite log-concavity of the Taylor coefficients of the Riemann xi-function
- Jacobian layers and higher Milnor sequences: rigidity, stabilization, and log-concavity
- A Unified Framework for Wasserstein Convergence of ULMC Methods beyond Log-Concavity: Old and New
- Two Infinite Families of Regular Sequences of Power Sums in Three Variables
BibTeX
@misc{xie2026unified,
title = {{A unified approach to infinite log-concavity of combinatorial sequences}},
author = {Matthew H. Y. Xie and Candice X. T. Zhang and Philip B. Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12237},
eprint = {2610.12237},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12237},
}
RIS
TY - UNPB
TI - A unified approach to infinite log-concavity of combinatorial sequences
AU - Matthew H. Y. Xie
AU - Candice X. T. Zhang
AU - Philip B. Zhang
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12237
UR - https://arxiv.org/abs/2610.12237
AB - We establish a sufficient condition for infinite log-concavity and apply it to five families of combinatorial sequences. We give a new proof of infinite log-concavity for the Boros--Moll coefficient sequences (d_k(n))_k=0^n for fixed n. For the transposed sequences (d_ℓ(ℓ+k))_k≥0, we prove Zhao's conjecture on infinite log-concavity for every fixed integer ℓ≥3. For the normalized sequences in Euler's difference table, we confirm the infinite log-concavity conjecture of Chen, Gu, Ma and Wang under essential iteration, in which both endpoints are discarded after each step. We also confirm a conjecture of Medina, Moll and Rowland on the infinite log-concavity of the coefficient sequences of polynomials arising from iterated primitives of log(1+x). Finally, we answer a question of Brändén and Chasse by showing that, for positive integers d, the sequence (k^d)_k≥0 is infinitely log-concave if and only if d≠2. For every real d≥3, all its iterates are positive at every index k≥1. The proofs combine holomorphic estimates with finite computer-assisted verification.
ER -