The crown auxiliary Chebyshev identity

Abstract

The weighted auxiliary polynomial is a shifted Chebyshev quotient.

This polynomial identity is a repository-derived ingredient for the log-concavity argument. It concerns the auxiliary polynomial, not the actual f-polynomial discussed in source Remark 3.8.

Theorem 1.1 (The exact identity for every n).

Lean statement: D5/S3/Combinatorics/Geometry/CrownOrderPolytopeChebyshev.crownAuxiliaryPolynomial_chebyshev

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Geometry/CrownOrderPolytopeChebyshev.crownAuxiliaryPolynomial_chebyshev (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Teemu Lundström and Leonardo Saud Maia Leite (2025). Order polytopes of crown posets. DOI: 10.48550/arXiv.2504.05123. URL: https://arxiv.org/abs/2504.05123v3.

Commentary.

For every natural n, X times Q_n equals twice T_n((X+2)/2) minus two, as an identity of rational polynomials. Q_n is the sum of A(n,m)X^(m-1) over 1 <= m <= n. The identity is proved through its coefficients and the Chebyshev recurrence, including n equals zero.

References

• Truth anchor: D5/S3/Combinatorics/Geometry/CrownOrderPolytopeChebyshev.crownAuxiliaryPolynomial_chebyshev
• Dependency: D5/S3/Combinatorics/Geometry/CrownOrderPolytopeScalar