Polynomials of Binomial Type and Lucas' Theorem

TL;DR

This paper explores polynomial sequences that satisfy the Binomial Theorem in finite characteristic, using additive polynomial theory, and discusses their constructions and related actions, leaving open questions about their completeness.

Contribution

It introduces new constructions of binomial-type polynomials in finite characteristic based on additive polynomial theory and examines their algebraic actions.

Findings

Multiple constructions of binomial-type polynomial sequences are presented.

The paper discusses algebraic actions on these polynomial sequences.

It raises open questions about the completeness of these constructions.

Abstract

We present various constructions of sequences of polynomials satisfying the Binomial Theorem in finite characteristic based on the theory of additive polynomials. Various actions on these constructions are also presented. It is an open question whether we then have accounted for all sequences in finite characteristic which satisfy the Binomial Theorem.