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Negative-Index Motzkin Determinants

Abstract

Backward orthogonal polynomials determine the vanishing interval and first nonzero determinant at negative indices.

Definition 1.1 (Backward orthogonal polynomials).

Lean statement: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelNegative.backward

Formalization. D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelNegative.backward (✓ 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.

The polynomials b_r(y) over the integer polynomial ring in t and s satisfy b_0 = s - t, b_1 = (y - t)b_0 - 1 and b_{r+2} = (y - t)b_{r+1} - b_r for every nonnegative integer r. They continue the orthogonal recurrence with b_r corresponding to p_{-r-1}.

Theorem 1.2 (Vanishing and the first nonzero backward determinant).

Lean statement: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelNegative.negative_determinants

Proof. Machine-checked in Lean as D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelNegative.negative_determinants (✓ 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.

For every nonnegative integer m and every integer a with 1 at most a and a less than m, the m by m coefficient determinant with row polynomial b_{a-1-i} when i is less than a and p_{i-a} otherwise is zero. Its column j consists of coefficients of y^j. At a = m, (-1)^m times the determinant with row polynomial b_{m-1-i} equals (-1)^binom(m+1,2) times (s - t)^m, and this polynomial is nonzero in the integer polynomial ring in t and s. This nonvanishing concerns independent indeterminates; specialization to s = t can make it zero when m is positive.

References