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Separate Numerator and Denominator Factorizations

Abstract

At both endpoint indices corresponding to heights 4m and 4m + 1, the Narayana numerator and denominator factor into signed continuants.

Theorem 1.1 (The four continuant products).

Lean statement: D5/S3/Combinatorics/NarayanaStrip/CiglerStripProductSeries.product_factorization

Proof. Machine-checked in Lean as D5/S3/Combinatorics/NarayanaStrip/CiglerStripProductSeries.product_factorization (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Johann Cigler (2026). Some sequences and number triangles which are related to Narayana polynomials and to q-Narayana polynomials for q=-1. DOI: 10.48550/arXiv.2608.03363. URL: https://arxiv.org/abs/2608.03363v2.

Commentary.

For t and z in any commutative ring, let P^+(t, z) and Q^+(t, z) be the continuants with coefficients alternating z and tz, and let P^-(t, z) and Q^-(t, z) be the continuants with coefficients repeating z, tz, -z, -tz. Each numerator P has initial values (0, 1), and each denominator Q has initial values (1, 1). For every nonnegative integer m and each index j equal to 4m + 1 or 4m + 2, P^+_j(t^2, z^2) = P^-_j(t, z)P^-_j(-t, -z) and Q^+_j(t^2, z^2) = Q^-_j(t, z)Q^-_j(-t, -z). The endpoint formulas and addition and doubling identities for the second-order recurrence give the two numerator products and the two denominator products.

References