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

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 $\mathcal{L}(a_n)=b_n{a}_n^2-a_{n+1}{a}_{n-1}$ is a quadratic form in the state vector ${\mathbf{v}}_n$, and identify the matrix $Q_n=Q^{(0)}+n{Q}^{(1)}$ 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 $\mathrm{\infty }$-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.