发表机构
Tianjin University of Technology; Tianjin Normal University（天津工业大学; 天津师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对组合序列无限对数凹性，本文建立了统一的充分条件，将其应用于五类组合序列，证明了相关猜想并回答了问题，结合全纯估计与有限计算机辅助验证完成证明。
AI 中文摘要
我们建立了无限对数凹性的一个充分条件，并将其应用于五类组合序列。对于固定$n$的Boros–Moll系数序列$(d_k(n))_{k=0}^n$，我们给出了其无限对数凹性的新证明。对于转置序列$(d_\ell(\ell+k))_{k\ge0}$，我们证明了对于每个固定整数$\ell\ge3$的无限对数凹性的赵猜想。对于欧拉差分表中的归一化序列，我们在每次迭代后丢弃两个端点的本质迭代下，证实了Chen、Gu、Ma和Wang的无限对数凹性猜想。我们还证实了Medina、Moll和Rowland关于$\log(1+x)$迭代原函数产生的多项式系数序列的无限对数凹性的猜想。最后，我们通过证明对于正整数$d$，序列$(k^d)_{k\ge0}$是无限对数凹的当且仅当$d\ne2$，回答了Brändén和Chasse的一个问题。对于每个实数$d\ge3$，其所有迭代在每个索引$k\ge1$处均为正。证明结合了全纯估计与有限计算机辅助验证。
英文摘要
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 $(d_\ell(\ell+k))_{k\ge0}$, we prove Zhao's conjecture on infinite log-concavity for every fixed integer $\ell\ge3$. 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 $(k^d)_{k\ge0}$ is infinitely log-concave if and only if $d\ne2$. For every real $d\ge3$, all its iterates are positive at every index $k\ge1$. The proofs combine holomorphic estimates with finite computer-assisted verification.
