Noncommutative binomial theorem, shuffle-type polynomials, and Bell polynomials

Abstract

This paper studies shuffle-type polynomials and their associated binomial identities. First, we establish an explicit formula for the expansion coefficients of shuffle-type polynomials with respect to the Lyndon–Shirshov basis, yielding a general noncommutative binomial (and multinomial) theorem valid in arbitrary free algebras. Second, by extending the Bell polynomial framework, we derive an alternative binomial theorem based on shuffle-type polynomials; this construction naturally produces the q-Bell differential polynomials. Furthermore, we elucidate the precise relationship between shuffle-type polynomials and Bell differential polynomials. Finally, we illustrate the effectiveness of our free noncommutative binomial theorem and present applications of shuffle-type polynomials to bialgebras and Hopf algebras.