  We develop recursion relations, in particle number, for all (unprojected)
Jain composite fermion (CF) wave functions. These recursions generalize a
similar recursion originally written down by Read for Laughlin states, in mixed
first-second quantized notation. In contrast, our approach is purely
second-quantized, giving rise to an algebraic, `pure guiding center' definition
of CF states that de-emphasizes first quantized many-body wave functions. Key
to the construction is a second-quantized representation of the flux attachment
operator that maps any given fermion state to its CF counterpart. An algebra of
generators of edge excitations is identified. In particular, in those cases
where a well-studied parent Hamiltonian exists, its properties can be entirely
understood in the present framework, and the identification of edge state
generators can be understood as an instance of `microscopic bosonization'. The
intimate connection of Read's original recursion with `non-local order
parameters' generalizes to the present situation, and we are able to give
explicit second quantized formulas for non-local order parameters associated
with CF states.
