Binomial Transforms, Geometric Scalings, and Irregular Hadamard Products of Order-2 Signatures

We study ordinary binomial transforms and k-binomial transforms from the viewpoint of signatures of second-order linear recurrences. The main observation is that the k-binomial transform is precisely the geometric scaling of the ordinary binomial transform. This places k-binomial transforms inside the general theory of scaling-induced irregular Hadamard products of signatures. As a consequence, products of the form BT(a, b) ∗ kBT(a, b) collapse systematically from the generic fourth order to a cubic signature, and more generally multifactor products of scaled binomial transforms collapse from the generic order to order r + 1. We derive explicit formulas for the binary and trifactor cases, extend them to the n-th ordinary binomial transform, and interpret the results structurally: a signature identifies a class of sequences through the denominator of the generating function, while the Hadamard product acts on such classes by predicting the denominator of the resulting class.