Multiplicative Two-Step Recurrence and Complete Factorization in Generalized Pascal Triangles: Classification for m ∈ {−1,−2}

We study generalized Pascal triangles {T(m)(n, k)} defined by the recurrence T(m)(n, k) =(m(n − k) + 1)T(m)(n − 1, k − 1) + (m(k − 1) + 1)T(m)(n − 1, k). For the critical parameterm = −1, we prove a Complete Factorization Theorem: the row polynomialRn(y) = Pn k=1 T(−1)(n, k) yk−1 (degree n − 1, constant term 1) satisfies, for n ≥ 4,Rn(y) = (1 − y)dn · Qn(y),where dn = n − 2 (n even) or n − 3 (n odd), and Qn(y) = 1 + y (n even) or 1 + y2 (n odd).The key step is a Multiplicative Lemma: Rn+2(y) = (1 − y)2Rn(y). We prove thisby showing that the “excess” E(n, k) satisfies the same recurrence as T(m)(n, k), so thatE(n, k) ≡ 0 follows from E(4, k) = 0 by induction.We also prove a Classification Theorem: among integer parameters m, the multiplicativeidentity holds if and only if m ∈ {−1,−2}. While m = −2 also satisfies the multiplicativelaw, its odd rows make Qn(1) = 0, so universal cyclotomic quotients Qn ∈ {Φ2,Φ4} withconstant boundary residue Qn(1) = 2 occur only at m = −1.