On the Eulerian polynomials of type D

We define and study sub-Eulerian polynomials of type D, which count the elements of the group of even-signed permutations Dn with respect to the number of descents in a refined sense. The recurrence relations and exponential generating functions of the sub-Eulerian polynomials are determined, by which the solution to a problem of Brenti, concerning the recurrence relation for the Eulerian polynomials of type D, is also obtained.