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.
Ferraz et al. (Tue,) studied this question.