Year: 2026
Type: preprint
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 $(d_k(n))_{k=0}^n$ 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.
Source: arXiv (Cornell University)
Institutions: Tianjin University of Technology, Tianjin Normal University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G6545402480, G6545595343, G7610362479
OpenAlex ID: W7221100655