The Antipodes of $q$-Quasi-Symmetric Functions and Non-Commutative Quasi-Symmetric Functions

We prove the antipode formula for the $q$-deformations of quasi-symmetric functions.

We also define a fundamental basis for non-commutative quasi-symmetric functions, and establish a partial antipode formula there.

Our method also reduce to a new proof for the known antipode formula on the usual quasi-symmetric functions.