A unified approach to infinite log-concavity of combinatorial sequences

Source 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.