Periodicities in the Riordan arrays of polynomials over finite fields

What the math gives to ML

The paper gives an exact finite-state description of coefficient sequences generated by Riordan arrays over finite fields: after a transient, every column is represented by repeatedly applying a circulant matrix over \(\mathbb{F}_q\). The transferable asset is not the generating-function notation itself, but the resulting linear dynamical system with bounded state space, exact periodic orbits, and very cheap shared transition structure. This suggests a modular recurrent/state-space core whose long-horizon behavior is guaranteed to be periodic or eventually periodic, with the nonlinear neural network used only for input injection and readout. The first useful target is algorithmic tasks requiring exact modular counting or bounded symbolic memory, where ordinary floating-point recurrent models often suffer from drift.