Riordan array representation of recursive polynomial sequences, orthogonal polynomial sequences, and $d$-orthogonal polynomial sequences

What the math gives to ML

The paper supplies a structured change-of-basis framework in which polynomial coefficient matrices are replaced by production matrices, with orthogonal-polynomial recurrences appearing as tridiagonal operators and d-orthogonal recurrences as (d+2)-banded lower-Hessenberg operators. This suggests neural layers that compute learned polynomial features through short recurrences rather than dense Vandermonde-like matrix multiplication, reducing parameter count and improving numerical conditioning. The most transferable asset is the equivalence between a basis transform and a sparse recurrence operator, together with the inverse-matrix moment interpretation that can initialize or regularize the learned basis.