Orthogonal Moments for Motzkin Columns
Abstract
One orthogonal moment functional recovers every column of the boundary-weighted Motzkin triangle.
Theorem 1.1 (The column moment functional).
Lean statement: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinColumnTransfer.column_moments
Proof. Machine-checked in Lean as D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinColumnTransfer.column_moments (✓ 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.
Over the integer polynomial ring in t and s, let p_0(y) = 1, p_1(y) = y - s and p_{r+2}(y) = (y - t)p_{r+1}(y) - p_r(y). There is a linear functional ell on polynomials in y such that ell(y^n p_k(y)) = M_{n,k}(t,s) for all nonnegative n and k, and ell(p_i p_j) is one when i equals j and zero otherwise. Here M_{n,k}(t,s) counts Motzkin paths with horizontal weight s on the axis and t above it. Moving the three-term recurrence across the pairing yields the column identities, and triangularity of the Motzkin array yields orthogonality.
References
- Truth anchor:
D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinColumnTransfer.column_moments - Dependency: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinColumnDefs
- Dependency: D5/S3/Combinatorics/MotzkinHankel/CiglerMotzkinHankelOrthogonal