The Lowest Term of the Shifted Hankel Determinant

Abstract

The reverse permutation gives the unique lowest-degree nonzero term of the backward-shifted Hankel determinant.

Theorem 1.1 (The first nonzero coefficient).

Lean statement: D5/S3/Combinatorics/PartialTheta/PartialThetaHankelLowest.lowest_term

Proof. Machine-checked in Lean as D5/S3/Combinatorics/PartialTheta/PartialThetaHankelLowest.lowest_term (✓ 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 pair of nonnegative integers m and n, every coefficient of D_{-m,n+m+1}(q) below degree (m + n + 1) binom(n, 2) vanishes, and the coefficient at that degree is (-1)^{binom(m + n + 1, 2)}. A determinant term can be nonzero only when its permutation sends each index i to an index j with i + j at least m. Among these permutations, the full reversal uniquely minimizes the sum of binom(i + j - m, 2). Its sign is the displayed coefficient.

References

• Truth anchor: D5/S3/Combinatorics/PartialTheta/PartialThetaHankelLowest.lowest_term
• Dependency: D5/S3/Combinatorics/PartialTheta/PartialThetaHankelVandermonde