  Fixing a positive integer $r$ and $0 \le k \le r-1$, define $f^{\langle r,k
\rangle}$ for every formal power series $f$ as $ f(x) = f^{\langle r,0
\rangle}(x^r)+xf^{\langle r,1 \rangle}(x^r)+ \cdots +x^{r-1}f^{\langle r,r-1
\rangle}(x^r).$ Jochemko recently showed that the polynomial $U^{n}_{r,k}\,
h(x) := \left( (1+x+\cdots+x^{r-1})^{n} h(x) \right)^{\langle r,k \rangle}$ has
only nonpositive zeros for any $r \ge \deg h(x) -k$ and any positive integer
$n$. As a consequence, Jochemko confirmed a conjecture of Beck and Stapledon on
the Ehrhart polynomial $h(x)$ of a lattice polytope of dimension $n$, which
states that $U^{n}_{r,0}\,h(x)$ has only negative, real zeros whenever $r\ge
n$. In this paper, we provide an alternative approach to Beck and Stapledon's
conjecture by proving the following general result: if the polynomial sequence
$\left( h^{\langle r,r-i \rangle}(x)\right)_{1\le i \le r}$ is interlacing, so
is $\left( U^{n}_{r,r-i}\, h(x) \right)_{1\le i \le r}$. Our result has many
other interesting applications. In particular, this enables us to give a new
proof of Savage and Visontai's result on the interlacing property of some
refinements of the descent generating functions for colored permutations.
Besides, we derive a Carlitz identity for refined colored permutations.
