$r-$Bell polynomials in combinatorial Hopf algebras

TL;DR

This paper introduces partial r-Bell polynomials within three combinatorial Hopf algebras and establishes a generating function factorization based on the Zassenhauss formula.

Contribution

It extends the theory of Bell polynomials to the setting of combinatorial Hopf algebras, providing new algebraic tools and formulas.

Findings

Derived a factorization formula for generating functions

Connected partial r-Bell polynomials to the Zassenhauss formula

Expanded Bell polynomial theory in Hopf algebra context

Abstract

We introduce partial $r$-Bell polynomials in three combinatorial Hopf algebras. We prove a factorization formula for the generating functions which is a consequence of the Zassenhauss formula.