The Unshifted Partial Theta Hankel Determinant

Abstract

A Vandermonde product evaluates the unshifted Hankel determinant and establishes its nonvanishing.

Theorem 1.1 (The unshifted determinant formula).

Lean statement: D5/S3/Combinatorics/PartialTheta/PartialThetaHankelVandermonde.unshifted

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

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 nonnegative integer n, put V_n(q) = product over d from one through n of (q^d - 1)^{n + 1 - d}. Then D_{0,n+1}(q) = q^{(n + 1) binom(n, 2)} V_n(q), and this determinant is nonzero. The polynomial V_n is monic, has degree binom(n + 2, 3), and satisfies V_n(0) = (-1)^{binom(n + 1, 2)}. Empty products are one. Extracting the row and column monomials leaves the Vandermonde matrix at 1, q, through q^n; grouping its factors by the difference of the two indices gives the displayed product.

References

• Truth anchor: D5/S3/Combinatorics/PartialTheta/PartialThetaHankelVandermonde.unshifted
• Dependency: D5/S3/Combinatorics/PartialTheta/PartialThetaHankelDefs