  We introduce and study an alternative form of the chaotic expansion for
counting processes using the Poisson imbedding representation; we name this
alternative form \textit{pseudo-chaotic expansion}. As an application, we prove
that the coefficients of this pseudo-chaotic expansion for any linear Hawkes
process are obtained in closed form, whereas those of the usual chaotic
expansion cannot be derived explicitly. Finally, we study further the structure
of linear Hawkes processes by constructing an example of a process in a
pseudo-chaotic form that satisfies the stochastic self-exciting intensity
equation which determines a Hawkes process (in particular its expectation
equals the one of a Hawkes process) but which fails to be a counting process.
