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The Catalan Power Moment Dictionary

Abstract

The Jacobi recurrence expresses odd Catalan power coefficients as moments of a monic orthonormal polynomial family.

Theorem 1.1 (Normalized orthogonality and Catalan moments).

Lean statement: D5/S3/Combinatorics/CatalanPowerHankel/CiglerElevenMoments.moment_dictionary

Proof. Machine-checked in Lean as D5/S3/Combinatorics/CatalanPowerHankel/CiglerElevenMoments.moment_dictionary (✓ 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.

Specialize the boundary-weighted Motzkin orthogonal polynomials to interior weight two and boundary weight one. The resulting rational polynomials satisfy p_0 = 1, p_1 = X-1 and p_{j+2} = (X-2)p_{j+1}-p_j. There exists a rational linear functional L on rational polynomials such that L(p_a p_b) is one when a equals b and zero otherwise, and L(X^t p_k) = C_{2k+1,t-k} for all nonnegative integers t and k. This includes the zero moments when t is less than k. The Jacobi operator acts on sequences by (Jv)0 = v_0+v_1 and (Jv){j+1} = v_j+2v_{j+1}+v_{j+2}; L(f) is the zeroth coordinate of f(J) applied to the zeroth unit vector. Its powers have coordinates binom(2t,t+k)-binom(2t,t+k+1), which satisfy the two-step Pascal recurrence and identify the Catalan coefficients.

References