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The Negative-Index Unit Endpoint

Abstract

Uniform horizontal weights give a negative-index vanishing interval and a unit determinant at its next endpoint.

Theorem 1.1 (Vanishing and the first unit determinant).

Lean statement: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinColumnNegative.negative_endpoint

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinColumnNegative.negative_endpoint (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Johann Cigler (2022). Some remarks and conjectures about Hankel determinants of polynomials which are related to Motzkin paths. DOI: 10.48550/arXiv.2204.09910. URL: https://arxiv.org/abs/2204.09910v4.

Commentary.

Let h be nonnegative and let g(y) be monic of degree h over the integer polynomial ring in t. Specialize s to t in the orthogonal polynomials p_r and the backward polynomials b_r, where b_0 = s - t, b_1 = (y - t)b_0 - 1 and b_{r+2} = (y - t)b_{r+1} - b_r. For each gap from one through h, form an h by h coefficient matrix with row polynomial b_{gap-1-i} when i is below gap and p_{i-gap} otherwise, taking remainders modulo g and coefficients of y^j in column j. Its determinant is zero. The matrix with row polynomial b_{h-i} modulo g has determinant (-1)^binom(h+1,2). After specialization b_0 vanishes and b_{r+1} = -p_r, so a zero row gives the gap and reversed monicity gives the endpoint.

References