Hankel Determinants of Backward Shifts of Partial Theta Coefficients

Johann Cigler, Hankel determinants of backward shifts of the coefficients of a partial theta function, arXiv:2407.05768v2, Section 1, the Conjecture after equation (2): with a(s, q) = q^{binom(s,2)} for s ≥ 0 and a(s, q) = 0 for s < 0, D_{−m,N}(q) = det(a(−m + i + j, q)){0≤i,j<N}, and r{m,n}(q) defined by D_{−m,n+m+1}(q) = (−1)^{binom(m+1,2)} r_{m,n}(q) q^{m binom(n,2)} D_{0,n+1}(q),

The functions r_{m,n}(q) are monic polynomials with integer coefficients with deg r_{m,n}(q) = mn(n + m + 2)/2 which satisfy r_{m,n}(1) = 1 and r_{m,n}(0) = (−1)^{mn}.

The paper proves the cases m = 1 and m = 2.

The theorem D5/S3/Combinatorics/PartialTheta/PartialThetaHankel.result establishes all four assertions for every m, n ≥ 0, together with D_{0,n+1}(q) ≠ 0, so that r_{m,n} is determined by (2).

Pre-registration issue 12580 records the literature screen: web, GitHub and repository searches found no proof for general m; Cigler's later work on the same sequences settles a different conjecture quoted in the paper's Appendix. This is a bounded negative finding.

1. D_{0,n+1}(q) is a Vandermonde determinant in q⁰, q¹, …, qⁿ times a monomial, hence a monomial times ∏_{d=1}^{n} (q^d − 1)^{n+1−d}, which is nonzero.
2. After removing row and column monomials, D_{−m,n+m+1}(q) is a generalized Vandermonde determinant: an alternant in independent variables x_r evaluated at x_r = q^r.
3. Every such alternant is divisible by the Vandermonde product with an integral symmetric quotient, so r_{m,n} is an integer polynomial in q.
4. Among the terms of the quotient a unique one has the highest q-degree, with coefficient 1, giving monicity and the degree mn(n + m + 2)/2; the unique lowest term gives r_{m,n}(0) = (−1)^{mn}.
5. At q = 1 the Vandermonde factors vanish; letting the variables coalesce turns the quotient into a ratio of confluent determinants, which evaluates to 1.

The statement would fail if the normalizing monomial in (2) were misread, if two terms of the quotient shared the highest degree, or if the coalescence at q = 1 changed the sign.

An independent referee implementation checked all four assertions and the intermediate determinant identities exactly for m ≤ 7 and n ≤ 6, and agreement with the paper's printed examples and its theorems for m = 1, 2.

theorem; the statement is the Section 1 Conjecture of arXiv:2407.05768v2, quantified over all m, n ≥ 0.

• Proved (formalized): r_{m,n}(q) is a monic integer polynomial of degree mn(n + m + 2)/2 with r_{m,n}(1) = 1 and r_{m,n}(0) = (−1)^{mn}, for every m, n ≥ 0.
• Proved (formalized): D_{0,n+1}(q) ≠ 0, through the Vandermonde evaluation.
• Open: the paper's recurrences for r_{m,n} in the general case and a combinatorial interpretation of the coefficients of r_{m,n}.

The literature screen is limited to the web, GitHub and repository searches recorded above.