Strict Turan Tail for v

Abstract

Positive radical normalization transports the weighted recurrence bound to a strict lowercase-v tail.

Theorem 1.1 (Strictness beyond the v tail threshold).

$$\forall n:\mathbb{N},1\le n\Rightarrow \forall t:\mathbb{R},(n:\mathbb{R})\cdot \left((n:\mathbb{R})+1\right)-\frac{2\cdot (n:\mathbb{R}{)}^3}{\sqrt{4\cdot (n:\mathbb{R}{)}^2-1}}\le t\Rightarrow 0<{\left((-1{)}^n\cdot v\left(-t,n\right)\right)}^2-(-1{)}^{n-1}\cdot v\left(-t,n-1\right)\cdot (-1{)}^{n+1}\cdot v\left(-t,n+1\right)$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/Sun/VTail.v_tail_strict (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Zhi-Hong Sun (2026). Generalizations of the Christoffel-Darboux formula and congruences involving Apéry-like numbers. DOI: 10.48550/arXiv.2608.13192. URL: https://arxiv.org/html/2608.13192v1.

Acknowledgement. Ilia Krasikov (2011). Turán inequalities for three-term recurrences with monotonic coefficients. DOI: 10.1016/j.jat.2011.04.007. URL: https://arxiv.org/html/1101.3204v1.

Commentary.

Write R(j)=(-1)^j v(-t,j), a(j)=j^3/sqrt(4j^2-1) for j>0, b(j)=j(j+1), and q(j)=sqrt(2j+1)R(j). The positive factor d(j)=1/sqrt(2j+1) gives R(j)=d(j)q(j). The proof checks both initials and the exact recurrence by two-step uniqueness; it does not assume that the source sequence has the supplier’s normalization. Positivity of the radicals and monotonicity of b are proved. Strict increase of a follows by clearing positive denominators: at j=k+1 the numerator is 16k^7+168k^6+746k^5+1815k^4+2604k^3+2187k^2+982k+177, positive for k at least zero. The direct supplier call yields a weighted nonnegative determinant on the closed threshold t at least n(n+1)-2n^3/sqrt(4n^2-1). Exact positive scaling transports it to R, with neighbor weight (n+1)^3(2n-1)/(n^3(2n+1))>1. The recurrence excludes adjacent zeros, and the sign split of the neighbor product makes the unweighted determinant strict. VStrict consumes this tail.

References

• Truth anchor: D5/S1/Recurrence/Sun/VTail.v_tail_strict
• Dependency: D5/S1/Recurrence/Sun/Sequences
• Dependency: D5/S1/Recurrence/Turan/StrictlyIncreasingTail