Explicit antipode formulas are proved for commutative and non-commutative q-quasi-symmetric functions, plus a partial antipode on a new fundamental basis of NCQSym, via a cancelation argument that recovers the classical QSym antipode.
Math., vol 142 issue 1, 1--30 (2006)
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The Antipodes of $q$-Quasi-Symmetric Functions and Non-Commutative Quasi-Symmetric Functions
Explicit antipode formulas are proved for commutative and non-commutative q-quasi-symmetric functions, plus a partial antipode on a new fundamental basis of NCQSym, via a cancelation argument that recovers the classical QSym antipode.