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arxiv: 2604.14391 · v1 · submitted 2026-04-15 · 🧮 math.CO · math.NT

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

Piero Giacomelli This is my paper

Pith reviewed 2026-05-10 12:34 UTC · model grok-4.3

classification 🧮 math.CO math.NT
keywords log-concavityP-recursive sequencesholonomic sequencescompanion matrixquadratic forminfinite log-concavitylinear recurrence
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The pith

Log-concavity of P-recursive sequences follows from positive semi-definiteness of a matrix linear in the index n.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to decide log-concavity for sequences obeying linear recurrences with coefficients linear in n by reducing the question to the positive semi-definiteness of a certain matrix. Using the companion-matrix formulation of the recurrence, the log-concavity defect is rewritten as a quadratic form on the current state vector whose matrix is affine in n. This immediately yields a sufficient condition for the entire sequence to be log-concave. In the important special case of second-order recurrences with constant coefficients the same construction produces an exact if-and-only-if criterion, because the defect sequence is itself geometric and vanishes after one step when the initial defect is non-negative. The authors extend the exact criterion to sequences fixed by the log-concavity operator and to those with dominant-root growth.

Core claim

Writing the recurrence in companion-matrix form v_{n+1} = (nA + B) v_n, the log-concavity quantity b_n = a_n^2 - a_{n+1}a_{n-1} equals the quadratic form v_n^T Q_n v_n with Q_n = Q^{(0)} + n Q^{(1)}. Positive semi-definiteness of Q_n therefore suffices to guarantee b_n >= 0 for all n. For second-order constant-coefficient recurrences, b_n itself obeys a geometric recurrence, so that the sequence is infinitely log-concave if and only if b_1 >= 0.

What carries the argument

The matrix Q_n = Q^{(0)} + n Q^{(1)} that makes the log-concavity defect b_n a quadratic form in the companion state vector v_n.

Load-bearing premise

The sequence exactly satisfies the given linear recurrence with real coefficients linear in n, and the matrix condition is claimed only as a sufficient test except in the low-order constant-coefficient cases.

What would settle it

A sequence satisfying a higher-order recurrence with linear coefficients where the associated matrix Q_n is indefinite for some n yet b_n remains non-negative for every n would show that positive semi-definiteness is not always necessary.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript presents a companion-matrix approach to log-concavity for P-recursive sequences, i.e., sequences satisfying linear recurrences with coefficients linear in the index n. It expresses the log-concavity defect b_n = a_n² − a_{n+1}a_{n−1} as the quadratic form v_n^T Q_n v_n, where v_n is the companion state vector and Q_n = Q^{(0)} + n Q^{(1)}. Positive semi-definiteness of Q_n is shown to be sufficient for log-concavity. For second-order constant-coefficient recurrences, a tight necessary-and-sufficient criterion for infinite log-concavity is proved by showing that b_n itself satisfies a geometric recurrence, implying that iterated applications of the log-concave operator vanish identically. Analogous tight criteria are derived for sequences fixed by the operator and for sequences with dominant-root asymptotics.

Significance. If the algebraic derivations hold, the work supplies a systematic matrix-theoretic tool for establishing log-concavity in a broad class of holonomic sequences that arise in combinatorics. The reduction of infinite log-concavity to the geometric property of b_n for the constant-coefficient second-order case is particularly clean and yields a necessary-and-sufficient condition. The paper correctly labels its PSD criterion as sufficient only and restricts the tight results to the indicated subclasses; it also ships self-contained algebraic identities derived directly from the companion recurrence without additional parameters or fitted quantities.

minor comments (2)
  1. The identification of the matrix Q_n = Q^{(0)} + n Q^{(1)} is central to the sufficient condition, yet the manuscript does not exhibit the explicit entries of Q^{(0)} and Q^{(1)} in terms of the recurrence matrices A and B (or even for the d=2 case). This makes direct verification and application of the PSD test difficult for readers.
  2. No concrete worked examples or counter-examples are supplied to illustrate either the sufficient PSD condition or the tight criteria for infinite log-concavity. Adding at least one explicit sequence (e.g., a known P-recursive sequence such as the central binomial coefficients or a simple hypergeometric term) would allow readers to check the matrix construction and the geometric property of b_n.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and encouraging report, which accurately summarizes our companion-matrix approach to log-concavity for P-recursive sequences and highlights the clean necessary-and-sufficient criterion for infinite log-concavity in the second-order constant-coefficient case. We are pleased that the work is viewed as supplying a systematic matrix-theoretic tool.

Circularity Check

0 steps flagged

No significant circularity; derivations are direct algebraic identities

full rationale

The central results follow from expressing the log-concavity operator L(a_n) = b_n as the quadratic form v_n^T Q_n v_n where Q_n = Q^{(0)} + n Q^{(1)} is obtained by direct matrix multiplication from the companion recurrence v_{n+1} = (nA + B) v_n. For second-order constant-coefficient cases the proof that L(a_n) is geometric (hence L^2(a_n) = 0) is an explicit computation on the recurrence coefficients. All steps are self-contained algebraic verifications using only the given linear recurrence and real-valued assumption; the PSD condition is correctly labeled sufficient and no fitted parameters, self-citations, or imported uniqueness theorems appear in the load-bearing derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the definition of P-recursive sequences and the standard companion-matrix representation of linear recurrences; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The sequence satisfies a d-th order linear recurrence whose coefficients are linear functions of n.
    This is the class of sequences under study, stated at the opening of the abstract.

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