TITLE: Quasi-symmetric generalizations of classical symmetric functions
QUESTION [5 upvotes]: I am looking for quasi-symmetric versions of the classical $e_\lambda$ and $h_\lambda$ (the elementary and complete homogeneous symmetric functions). Is there some reference for this?
I am aware of at least four quasi-symmetric generalizations of Schur polynomials:

Extended Schur (Assaf, Searles)
Quasi-symmetric Schur (Haglund et al)
Young quasi-symmetric Schur
Dual Immaculate functions

There are also two quasi-symmetric power-sums which I know of.
I have defined a non-symmetric extension of the elementary symmetric functions, (as a certain specialization of non-symmetric Macdonald polynomials), and this can perhaps be turned into some quasi-symmetric version. However, I have not yet managed to do this.

REPLY [2 votes]: I sent an email to Stephanie van Willigenburg,
and she suggested the following reference, which contains such generalizations.