Binary shuffle bases for quasi-symmetric functions

We construct bases of quasi-symmetric functions whose product rule is given by the shuffle of binary words, as for multiple zeta values in their integral representations, and then extend the construction to the algebra of free quasi-symmetric functions colored by positive integers. As a consequence, we show that the fractions introduced in Guo and Xie (Ramanujan J 25:307–317, 2011) provide a realization of this algebra by rational moulds extending that of free quasi-symmetric functions given in Chapoton et al. (Int Math Res Not IMRN 2008, no. 9, Art. ID rnn018, 2008).