A family of symmetric functions associated with Stirling permutations

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 $r=2$.