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Polynomial Identities for the Residue Reduction

Abstract

Chebyshev product identities control periodic remainders, reflection and paired rows of the Catalan moment polynomials.

Theorem 1.1 (Periodicity, reflection and monic quotients).

Lean statement: D5/S3/Combinatorics/CatalanPowerHankel/CiglerElevenPolynomials.polynomial_structure

Proof. Machine-checked in Lean as D5/S3/Combinatorics/CatalanPowerHankel/CiglerElevenPolynomials.polynomial_structure (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Johann Cigler (2023). Some experimental observations about Hankel determinants of convolution powers of Catalan numbers. DOI: 10.48550/arXiv.2308.07642. URL: https://arxiv.org/abs/2308.07642v2.

Commentary.

Let p_j and h_j be the boundary-weighted Motzkin orthogonal polynomials specialized to interior weight two and boundary weights one and three respectively. Thus p_0 = h_0 = 1, p_1 = X-1, h_1 = X-3, and each family satisfies q_{j+2} = (X-2)q_{j+1}-q_j. Write C_0 = 2, C_1 = X and S_0 = 1, S_1 = X for the Chebyshev families satisfying q_{j+2} = Xq_{j+1}-q_j. For all nonnegative k and j, p_{2k+1+j}-p_j = p_k C_{k+j+1}(X-2). For every nonnegative s at most k, p_{k+s}+p_{k-s} = p_k C_s(X-2). For every positive B and every nonnegative t less than B, p_{B+t}+p_{B-1-t} = X S_{B-1}(X-2) h_t. The recurrence gives p_j = S_j(X-2)+S_{j-1}(X-2) and h_j = S_j(X-2)-S_{j-1}(X-2), with S_{-1} = 0. The product identity C_m S_r = S_{r+m}+S_{r-m} and the identity C_{t+1}(X-2)+C_t(X-2) = Xh_t yield the three formulas.

References