  As a natural basis of the Hopf algebra of quasisymmetric functions, monomial
quasisymmetric functions are formal power series defined from compositions. The
same definition applies to left weak compositions, while leads to divergence
for other weak compositions. We adapt the method of renormalization in quantum
field theory, in the framework of Connes and Kreimer, to deal with such
divergency. This approach defines monomial quasisymmetric functions for any
weak composition as power series while extending the quasi-shuffle (stuffle)
relation satisfied by the usual quasisymmetric functions. The algebra of
renormalized quasisymmetric functions thus obtained turns out to be isomorphic
to the quasi-shuffle algebra of weak compositions, giving the former a natural
Hopf algebra structure and the latter a power series realization. This
isomorphism also gives the free commutative Rota-Baxter algebra a power series
realization, in support of a suggestion of Rota that Rota-Baxter algebra should
provide a broad context for generalizations of symmetric functions.
