bibkey: sun2026generalizations authors: Zhi-Hong Sun year: 2026 title: 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 claim: “Conjecture 5.2 states both strict lowercase g and v Turan inequalities for every positive index on the complete real domains x <= -1 and x <= -1/8.” strata_touched:

• D5/S1/Recurrence/Sun/Sequences
• D5/S1/Recurrence/Sun/GEndpoint
• D5/S1/Recurrence/Sun/GStrict
• D5/S1/Recurrence/Sun/VTail
• D5/S1/Recurrence/Sun/VStrict license: citation-only triage: anchor

Sun’s two lowercase Turan conjectures

Zhi-Hong Sun, Generalizations of the Christoffel-Darboux formula and congruences involving Apéry-like numbers, arXiv:2608.13192v1, 13 August 2026. Source URL: https://arxiv.org/html/2608.13192v1. DOI: 10.48550/arXiv.2608.13192. The source’s Conjecture 5.2 is a single two-clause question; neither clause alone settles it.

Verified locator

DOI: 10.48550/arXiv.2608.13192. Versioned source URL: https://arxiv.org/html/2608.13192v1. The displayed lowercase definitions and Conjecture 5.2 are the source scope checked here; no theorem about the older uppercase sequences is substituted for either negative-domain clause.

Source statement and definitions

The paper defines lowercase sequences over real arguments by

$$g_0(x)=1,g_1(x)=\frac{x+1}{2},(n+1{)}^2{g}_{n+1}(x)=\left(2n(n+1)+\frac{x+1}{2}\right)g_n(x)-n^2{g}_{n-1}(x),$$

and

$$v_0(x)=1,v_1(x)=x,(n+1{)}^3{v}_{n+1}(x)=(2n+1)(n(n+1)+x)v_n(x)-n^3{v}_{n-1}(x),$$

with both recurrences for every integer n >= 1. Conjecture 5.2 states, for every such n,

$$g_n(x{)}^2>g_{n-1}(x)g_{n+1}(x)(x\le -1),v_n(x{)}^2>v_{n+1}(x)v_{n-1}(x)(x\le -1/8).$$

Both endpoints belong to the source domains. Lean Sequences.g and Sequences.v use the quotient form of these same multiplied recurrences. GStrict.g_strict_turan and VStrict.v_strict_turan prove the two literal lowercase conclusions for arbitrary positive natural index and every real argument in the respective closed domain. Reversing the two factors in the second product uses commutativity of real multiplication.

Prior art and limits

The paper relates uppercase G_n,V_n to lowercase g_n,v_n at argument 2x^2+2x+1. For real x, that argument is at least 1/2; the uppercase results in Jianxi Mao and Qiqi Xiao, Second order determinants for three-term recurrence polynomial sequences, Filomat 40(9) (2026), 3169-3178, Theorems 3.2 and 3.6, therefore do not cover either negative lowercase domain. Y.-T. Li and Z.-H. Sun, arXiv:2609.28098v1, Theorem 3.3 and Remark 3.1 address p-adic material around Conjecture 4.2, not these real inequalities. Krasikov’s weighted tail result is recorded separately in D5/L/Recurrence/krasikov2011turan; it requires additional local and endpoint arguments for this target.

The bounded source/version/formula searches reported in the source assessment did not locate an exact later proof through 25 September 2026. This is a search limit, not a claim of worldwide absence or priority. The bodies of Abreu–Bustoz and Bustoz–Ismail were unread in that assessment; they supply neither a verified match nor negative evidence. No external Lean code was ported.