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Palindromic Numerators of OEIS A398690

Abstract

Every simplified Verlinde row has a palindromic numerator of its expected degree.

For natural r and q, the source function is the real signed sine sum A(r,q)=(2q+1)^(-1) sum from j=0 to 2q of (-1)^(rj) sin((2j+1)pi/(4q+2))^(2-r). The exponent is an integer exponent. The conjectured rows use r=n+3, so r is at least three.

Definition 1.1 (Simplified Verlinde source sum).

Lean statement: D5/S3/Analytic/OeisA398690Palindromic.simplifiedVerlinde

Formalization. D5/S3/Analytic/OeisA398690Palindromic.simplifiedVerlinde (✓ std3).

Citation. F. Chapoton (2026). OEIS A398690, triangle of polynomials related to Verlinde numbers. URL: https://oeis.org/A398690.

Commentary.

This is the A398690 comment’s signed finite sum. Its angle 2pi/(8q+4) simplifies to pi/(4q+2). The real power uses the same integer exponent 2-r.

Theorem 1.2 (All row numerators are palindromic).

Lean statement: D5/S3/Analytic/OeisA398690Palindromic.result

Proof. Machine-checked in Lean as D5/S3/Analytic/OeisA398690Palindromic.result (✓ std3). ∎

Resolves. Problems/oeis-a398690-palindromic (proved) by D5/S3/Analytic/OeisA398690Palindromic.result.

Source. Repository-derived.

Acknowledgement. F. Chapoton (2026). OEIS A398690, triangle of polynomials related to Verlinde numbers. URL: https://oeis.org/A398690.

Commentary.

For every natural n there is a real polynomial R of degree exactly n such that its formal power series equals (1-z)^(n+1) times the series whose coefficient at q is A(n+3,q). For every j at most n, the coefficients of R at j and n-j are equal. Chebyshev node moments identify every source coefficient with a polynomial model; the model’s reflection gives numerator reciprocity.

References