Staircase palindromic polynomials

The paper proves Theorem 3.3, which states exactly the cyclotomic factorization of the staircase palindromic polynomial S(x,n,h) for h>1 and the h=1 special case. The proof first shows S(x,n,h) = (1 + ⋯ + x^{h−1})(1 + ⋯ + x^{n−h+1}) via a convolution identity (Lemma 3.2 and Equation (3.3)), then applies the standard factorization x^m − 1 = ∏_{d|m} Ψ_d(x) to obtain the claimed product, cancelling Ψ_1(x)=x−1 appropriately. This matches the candidate solution’s steps in substance: writing S as a product of two geometric sums, converting to (x^m − 1)/(x − 1), invoking the cyclotomic factorization, and cancelling the Ψ_1 factors. The candidate provides slightly more explicit coefficient bookkeeping, but the argument is the same. See the paper’s definitions and statement (Definition 3.1, Theorem 3.3) and derivation (Lemma 3.2, Eq. (3.3), and the use of Theorem 2.3 on cyclotomic factorization) for confirmation . The paper’s abstract and framing are consistent with this result .