Log-Concavity and Infinite Log-Concavity of Linear Recurrent Sequences with Linear Coefficients via Companion Matrix Methods

Abstract

We study log-concavity properties of real sequences $(a_n)_{n \ge 0}$ satisfying a $d$-th order linear recurrence whose coefficients are linear functions of $n$; the so-called P-recursive (or holonomic) sequences. Writing the recurrence in companion-matrix form $\mathbf{v}_{n+1} = M_n\,\mathbf{v}_n$ with $M_n = nA + B$, we show that the log-concave operator value $\mathcal{L}(a_n) = b_n \coloneqq a_n^2 - a_{n+1}a_{n-1}$ is a quadratic form in the state vector $\mathbf{v}_n$, and identify the matrix $Q_n = Q^{(0)} + nQ^{(1)}$ whose positive semi-definiteness gives a sufficient condition for log-concavity. For the class of second-order recurrences with constant coefficients, we prove a tight (necessary and sufficient) criterion for the sequence to be $\infty$-log-concave, a consequence of the fact that $\mathcal{L}(a_n)$ is itself a geometric sequence so that $\mathcal{L}^2(a_n) = 0$ identically. We obtain analogous tight criteria for sequences fixed by $\mathcal{L}$, and for P-recursive sequences satisfying a dominant-root asymptotic behaviour. We leave some further insight in case this criteria break down in full generality.