An Integral Quotient for the Shifted Determinant

Abstract

Integral division of an augmented alternant constructs the normalized shifted Hankel quotient.

Theorem 1.1 (Integral alternant division).

Lean statement: D5/S3/Combinatorics/PartialTheta/PartialThetaHankelQuotient.integral_quotient

Proof. Machine-checked in Lean as D5/S3/Combinatorics/PartialTheta/PartialThetaHankelQuotient.integral_quotient (✓ 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 nonnegative integers m and n, set N = n + m + 1. There exists r(q) in the integer polynomial ring such that D_{-m,N}(q) = (-1)^{binom(m + 1, 2)} r(q) q^{m binom(n, 2)} D_{0,n+1}(q). More precisely, form an N by N matrix A over the polynomial ring in x_0 through x_n with coefficients in the integer polynomial ring in q. For row i below m, its entry in column j is zero if j is below m - i and is q^{binom(m - i + 1, 2) + (m - i)(N - j)} otherwise. For row i at least m, its entry is x_{i-m}^j. There exists an integral polynomial B with det A = product over 0 at most u less than v at most n of (x_v - x_u), multiplied by B. Set t_i = (m - i)N for i below m and t_i = 0 otherwise. The same r and B satisfy q^{sum_i t_i + N binom(n, 2)} r(q) = (-1)^{binom(m + 1, 2)} q^{sum_j binom(j, 2) + 2 binom(n + 1, 3)} B(1, q, through q^n), where both sums range from zero through N - 1.

References

• Truth anchor: D5/S3/Combinatorics/PartialTheta/PartialThetaHankelQuotient.integral_quotient
• Dependency: D5/S3/Combinatorics/PartialTheta/PartialThetaHankelLowest