A log-concavity criterion for finite nonnegative linearization
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 $0<p_0\leq1/2$. 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.