Some interlacing properties related to the Eulerian and derangement polynomials

Abstract

In this paper, we consider two matrices of polynomials ${\left[h_{n,k}\right(t\left)\right]}_{n\ge 0}$ and ${\left[l_{n,k}\right(t\left)\right]}_{n\ge 0}$, which are defined by a recurrence relation from a sequence of polynomials with real coefficients. These matrices play an important role in the study of the Eulerian and derangement transformations by Athanasiadis, who asked when their rows form interlacing sequences of real-rooted polynomials. In this paper, we give an answer to this question. In our setup, all columns of these matrices are shown to be generalized Sturm sequences. As applications, we show that the derangement transformation, its type B analogue and the r-colored derangement transformation of a class of polynomials with nonnegative coefficients, and all roots in the interval $[-1,0]$, have only real roots in a unified manner. The question about the type B derangement transformation was also raised by Athanasiadis. Furthermore, we show that the diagonal line of ${\left[h_{n,k}\right(t\left)\right]}_{n\ge 0}$ and ${\left[l_{n,k}\right(t\left)\right]}_{n\ge 0}$ forms a generalized Sturm sequence respectively, i.e., we give sufficient conditions for the binomial transformation to preserve the interlacing property.