A log-concavity criterion for finite nonnegative linearization
Abstract
In a 2005 survey, R. Szwarc asked for criteria, stated in the recurrence coefficients, for nonnegative linearization (property (P)) of finite orthogonal polynomial systems such as the Krawtchouk polynomials. For a finite three-term recurrence normalized to have row sums one, whose diagonal and downward coefficients have a constant nonnegative ratio, we prove that property (P) holds for every product, including products beyond the terminal degree, whenever the Newton coefficients of the profile di=bi+ci are log-concave without internal zeros. The criterion contains the full positive Krawtchouk range and the dual q-Krawtchouk arrays, and through a quadratic decomposition of symmetric finite systems it covers Szwarc's quadratic system Qn in its sharp range 0<p0≤1/2. As a limit to any such criterion, property (P) fails at coefficient distance O((1−r2)N) from the Krawtchouk array of size N with fixed ratio ci/di=r<1. All theorems, corollaries and propositions are formalized in Lean 4.
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BibTeX
@misc{polzer2026logconcavity,
title = {{A log-concavity criterion for finite nonnegative linearization}},
author = {Leslie P. Polzer},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01561},
eprint = {2611.01561},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01561},
}
RIS
TY - UNPB
TI - A log-concavity criterion for finite nonnegative linearization
AU - Leslie P. Polzer
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01561
UR - https://arxiv.org/abs/2611.01561
AB - In a 2005 survey, R. Szwarc asked for criteria, stated in the recurrence coefficients, for nonnegative linearization (property (P)) of finite orthogonal polynomial systems such as the Krawtchouk polynomials. For a finite three-term recurrence normalized to have row sums one, whose diagonal and downward coefficients have a constant nonnegative ratio, we prove that property (P) holds for every product, including products beyond the terminal degree, whenever the Newton coefficients of the profile d_i=b_i+c_i are log-concave without internal zeros. The criterion contains the full positive Krawtchouk range and the dual q-Krawtchouk arrays, and through a quadratic decomposition of symmetric finite systems it covers Szwarc's quadratic system Q_n in its sharp range 0<p_0≤1/2. As a limit to any such criterion, property (P) fails at coefficient distance O((1−r^2)^N) from the Krawtchouk array of size N with fixed ratio c_i/d_i=r<1. All theorems, corollaries and propositions are formalized in Lean 4.
ER -