On multiple and infinite log-concavity

Abstract

Following Boros--Moll, a sequence $(a_n)$ is $m$-log-concave if ${\mathcal{L}}^j(a_n)\ge 0$ for all $j=0,1,\dots ,m$. 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 $a_k^2\ge r{a}_{k-1}{a}_{k+1}$ for large enough $r$. On the other hand, a recent result of Brändén 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.

13 pages