Noncommutative binomial theorem, shuffle type polynomials and Bell polynomials

TL;DR

This paper develops new noncommutative binomial theorems using Lyndon-Shirshov bases and shuffle polynomials, connecting them to Bell polynomials and applications in algebraic structures like bialgebras.

Contribution

It introduces novel noncommutative binomial theorems based on shuffle polynomials and explores their applications to Bell polynomials and algebraic structures.

Findings

Established a free noncommutative binomial theorem using Lyndon-Shirshov basis.

Derived Bell and q-Bell differential polynomials from the second binomial theorem.

Applied shuffle type polynomials to bialgebras and Hopf algebras.

Abstract

In this paper we use the Lyndon-Shirshov basis to study the shuffle type polynomials. We give a free noncommutative binomial (or multinomial) theorem in terms of the Lyndon-Shirshov basis. Another noncommutative binomial theorem given by the shuffle type polynomials with respect to an adjoint derivation is established. As a result, the Bell differential polynomials and the $q$-Bell differential polynomials can be derived from the second binomial theorem. The relation between the shuffle type polynomials and the Bell differential polynomials is established. Finally, we give some applications of the free noncommutative binomial theorem including application of the shuffle type polynomials to bialgebras and Hopf algebras.