On multiple and infinite log-concavity

TL;DR

This paper explores the properties of sequences that are infinitely log-concave, characterizing those fixed by certain operators and conjecturing about their behavior under convolution, linking operator fixed points to linear recurrences.

Contribution

It characterizes sequences fixed by powers of the log-concavity operator using linear recurrences and proposes a conjecture on their infinite log-concavity through convolution.

Findings

Sequences fixed by the operators are characterized by a linear 4-term recurrence.

Sequences with only negative real roots are included in the class of infinitely log-concave sequences.

Positive sequences may become infinitely log-concave after finite convolutions.

Abstract

Following Boros--Moll, a sequence $(a_n)$ is $m$-log-concave if $\mathcal{L}^j (a_n) \geq 0$ for all $j = 0, 1, \ldots, m$. Here, $\mathcal{L}$ is the operator defined by $\mathcal{L} (a_n) = a_n^2 - a_{n - 1} a_{n + 1}$. By a criterion of Craven--Csordas and McNamara--Sagan it is known that a sequence is $\infty$-log-concave if it satisfies the stronger inequality $a_k^2 \geq r a_{k - 1} a_{k + 1}$ for large enough $r$. On the other hand, a recent result of Br\"and\'en shows that $\infty$-log-concave sequences include sequences whose generating polynomial has only negative real roots. In this paper, we investigate sequences which are fixed by a power of the operator $\mathcal{L}$ and are therefore $\infty$-log-concave for a very different reason. Surprisingly, we find that sequences fixed by the non-linear operators $\mathcal{L}$ and $\mathcal{L}^2$ are, in fact, characterized by a linear 4-term recurrence. In a final conjectural part, we observe that positive sequences appear to become $\infty$-log-concave if convoluted with themselves a finite number of times.