bibkey: krasikov2011turan authors: Ilia Krasikov year: 2011 title: Turán inequalities for three-term recurrences with monotonic coefficients doi: 10.1016/j.jat.2011.04.007 url: https://arxiv.org/html/1101.3204v1 claim: “Theorem 8 gives nonnegativity of the correctly weighted S_2 determinant for monotone recurrence coefficients on t >= b_n - 2a_n.” strata_touched:

• D5/S1/Recurrence/Turan/StrictlyIncreasingTail
• D5/S1/Recurrence/Sun/GStrict
• D5/S1/Recurrence/Sun/VTail license: citation-only triage: anchor

Krasikov’s weighted right-tail inequality

Ilia Krasikov, Turán inequalities for three-term recurrences with monotonic coefficients, Journal of Approximation Theory 163(9) (2011), 1227-1248. DOI: 10.1016/j.jat.2011.04.007. Versioned source URL: https://arxiv.org/html/1101.3204v1.

Verified locator

DOI: 10.1016/j.jat.2011.04.007. Versioned source URL: https://arxiv.org/html/1101.3204v1. The checked scope is the displayed three-term recurrence, the weighted S_2 definition, and Theorem 8’s right-tail conclusion with its monotonicity hypotheses.

Exact normalization

The source recurrence, at positive k, is

$$\frac{a_k}{c_k}{p}_k(t)=(t-b_{k-1})p_{k-1}(t)-a_{k-1}{c}_{k-1}{p}_{k-2}(t),$$

with p_{-1}=0, p_0=1, a_0=0, positive a_k,c_k for positive k, and positive c_0. Its determinant is

$$S_2(p_n)=p_n(t{)}^2-\frac{c_n}{c_{n+1}}\frac{a_{n+1}}{a_n}{p}_{n-1}(t)p_{n+1}(t).$$

Theorem 8 assumes nondecreasing a_k and b_k and gives S_2(p_n) >= 0 for t >= b_n-2a_n. It is a non-strict weighted tail result, not either full strict Sun clause. The local proof specializes to c_k=1, requires strict increase of a, and proves its own base, central, and farther-tail cases. For Sun’s g transform, a_n=n^2, b_n=2n(n+1) and the threshold is t>=2n. For the v transform, a_n=n^3/sqrt(4n^2-1), b_n=n(n+1) and the threshold is t>=n(n+1)-2n^3/sqrt(4n^2-1). The positive normalization factor d_n=1/sqrt(2n+1) connects the latter source sequence to the unit-normalized recurrence; the Lean VTail proof establishes that connection by its initials and recurrence.

Theorem 5 of the same paper cannot be applied unchanged to the g parameters: its needed discriminant condition would compare 16i^2-4 against 16i^2 in the wrong direction. Theorem 8 covers only the tail; endpoint parity, local positive quadratics, and the weighted-to-strict conversion are additional proof content.