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Cigler’s Partial Theta Hankel Determinant Conjecture

Abstract

Every normalized backward-shifted partial theta Hankel quotient is a monic integer polynomial with the conjectured degree and values at zero and one.

Theorem 1.1 (The normalized quotient formula for all shifts).

Lean statement: D5/S3/Combinatorics/PartialTheta/PartialThetaHankel.result

Proof. Machine-checked in Lean as D5/S3/Combinatorics/PartialTheta/PartialThetaHankel.result (✓ std3). ∎

Resolves. Problems/cigler-partial-theta-hankel (proved) by D5/S3/Combinatorics/PartialTheta/PartialThetaHankel.result.

Source. Repository-derived.

Acknowledgement. Johann Cigler (2024). Hankel determinants of backward shifts of the coefficients of a partial theta function. DOI: 10.48550/arXiv.2407.05768. URL: https://arxiv.org/abs/2407.05768v2.

Commentary.

For every pair of nonnegative integers m and n, D_{0,n+1}(q) is nonzero, and there exists a polynomial r_{m,n}(q) with integer coefficients satisfying 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 polynomial r_{m,n} is monic, has degree mn(n + m + 2)/2, and satisfies r_{m,n}(1) = 1 and r_{m,n}(0) = (-1)^{mn}. Here D_{-m,N}(q) is the determinant with entries a(-m + i + j, q), where a(s, q) is q^{binom(s, 2)} for nonnegative s and zero otherwise. Integral alternant division gives the polynomial quotient; the extremal determinant terms give monicity, degree, and the value at zero; coalescence and the staircase Pascal determinant give the value at one. These identities prove the conjecture following equation (2) in Section 1 of Cigler’s paper.

References