Polynomial Reflection and Numerator Reciprocity
Abstract
Polynomial reflection gives exact-degree palindromic generating-series numerators.
Theorem 1.1 (Reflection gives a palindromic numerator).
Lean statement: D5/S3/Analytic/PolynomialReflectionNumerator.palindromic_numerator_of_reflection
Proof. Machine-checked in Lean as D5/S3/Analytic/PolynomialReflectionNumerator.palindromic_numerator_of_reflection (✓ std3). ∎
Source. Repository-derived.
Commentary.
A real polynomial p of degree at most n, normalized by p(0)=1 and satisfying p(-1-x)=(-1)^n p(x), has a generating-series numerator of degree exactly n over (1-z)^(n+1). Its coefficients at j and n-j coincide. Expansion in the binomial polynomial basis converts reflection of p into coefficient reversal of the numerator.