Binomial Root Transforms, Geometric Scalings, and Irregular Hadamard Products of Signatures of Arbitrary Order

We study binomial root transforms and their geometric scalings in the framework of signatures of linear recurrent sequences of arbitrary order. If a characteristic polynomial of degree m is given, the ordinary binomial transform is governed by the root shift λ -> λ + 1, and more generally its n-th iterate is obtained through the substitution P(t-n). We show that the corresponding k-binomial transform is exactly the geometric scaling of the transformed signature, placing binomial transforms inside the general theory of scaling-induced irregular Hadamard products. In the generic branch, the Hadamard product between a transformed signature and its scaled companion collapses systematically from degree m² to the triangular number m(m+1)/2. We also discuss non-generic branches arising from repeated roots and additional product collisions. Explicit formulas are developed for orders 2, 3, and 4.