  We present exponential generating function analogues to two classical
identities involving the ordinary generating function of the complete
homogeneous symmetric functions. After a suitable specialization the new
identities reduce to identities involving the first and second order Eulerian
polynomials. The study of these identities led us to consider a family of
symmetric functions associated with a class of permutations introduced by
Gessel and Stanley, known in the literature as Stirling permutations. In
particular, we define certain type statistics on Stirling permutations that
refine the statistics of descents, ascents and plateaux and we show that their
refined versions are equidistributed, generalizing a result of B\'ona. The
definition of this family of symmetric functions extends to the generality of
$r$-Stirling permutations. We discuss some occurrences of these symmetric
functions in the cases of $r=1$ and $r=2$.
