Some interlacing sequences related to Eulerian and derangement polynomials

The Eulerian polynomials and derangement polynomials arise often in combinatorics, algebra and geometry. It is well known that the Eulerian polynomials and derangement polynomials form Sturm sequences, respectively. In this paper, we give new sufficient conditions for the interlacing property of recurrence sequences of polynomials. As applications, we show some interesting interlacing sequences among the Eulerian polynomials, the derangement polynomials and those generalized polynomials for colored permutations.