Positivity and recurrence formulas for oscillatory Gegenbauer Hankel determinants

Unrefereed research candidate. In one sentence: for the complex weight (1-t2)αeixt on [-1,1], every even-size Hankel determinant is strictly positive for every exponent α > -1 and every real frequency x, so the even-degree orthogonal polynomials used in complex Gaussian quadrature for oscillatory integrals always exist. Gaussian-type rules for highly oscillatory integrals can be built from polynomials orthogonal with respect to a complex weight. Such a polynomial of degree N exists only when the N-by-N Hankel determinant of the moments is nonzero, and with a complex weight this can fail. Celsus, Deaño, Huybrechs and Iserles (2022) proved even-size positivity for the flat weight (α = 0). This paper extends it to every α > -1 by continuation in α, using a Pearson identity (adjacent determinants never vanish together, and every complex zero is simple) and a large-frequency expansion with constants uniform on compact exponent intervals. Odd-size determinants have infinitely many real zeros. The same expansion proves the formal degree and leading-coefficient conjectures of Milovanović, Cvetković and Marjanović (2008) and gives complete proofs of the recurrence-coefficient formulas in Conjectures 2.5 and 2.7 of Lyu and Zhou (2026), in their stated ranges. An exact Bessel-Jacobi determinant identity, combined with an unrefereed preprint of Kounchev and Render, gives the critical length j7/2,1 for the span of xjcos x, xjsin x, 0 ≤ j ≤ 3. Prior claim. A ResearchGate preprint by J. M. Ashfaque dated 24 July 2026 (https://www.researchgate.net/publication/410769119) claims that the Lyu-Zhou recurrence formulas follow from a discrete Painlevé IV identification. Its rewriting of the Lyu-Zhou difference equations is correct. Its Proposition 3.1 fails as printed, at odd and even sizes, and it gives no construction or degree computation for the Lyu-Zhou polynomials (competing_claim_audit.md, prior_claim_check.py). This release claims the first complete proofs known to it of Conjectures 2.5 and 2.7, not the first claim; the preprint does not state the positivity theorem or the 2008 degree results. Version 0.2.0-candidate corrects the description of that preprint in version 0.1.0 (10.5281/zenodo.23933929) from its full text: it credits the correct parts, adds that Proposition 3.1 also fails at even sizes and that the preprint's proof contains the correct difference formula, and adds prior_claim_check.py. The mathematics and verify.py are unchanged. Limits: existence only (no claim about distinct quadrature nodes, nonzero weights or conditioning); the general Carnicer-Mainar-Peña critical-length conjecture remains open; the n = 3 corollary depends on an unrefereed preprint. The archive contains the manuscript (PDF and LaTeX source), verify.py (mpmath 1.3.0 and SymPy 1.14.0 numerical and exact checks with a negative control, valid under python -O) and its retained output, the novelty report, the competing-claim audit, a public review record and SHA-256 sums. Assurance: written proofs; the finite checks test formulas and conventions, not the statements for all sizes; internal AI-assisted review only (not human peer review, independent reproduction or formal verification). Creator: Anonymous. Text CC0-1.0; code MIT.