Fixing a positive integer r and
$0 \les k \les r-1$
, 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
$U^{n}_{r,k}\, h(x) := ( (1+x+\cdots +x^{r-1})^{n} h(x) )^{\langle r,k \rangle }$
has only non-positive zeros for any
$r \ges \deg h(x) -k$
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
$r\ges n$
. In this paper, we provide an alternative approach to Beck and Stapledon's conjecture by proving the following general result: if the polynomial sequence
$( h^{\langle r,r-i \rangle }(x))_{1\les i \les r}$
is interlacing, so is
$( U^{n}_{r,r-i}\, h(x) )_{1\les i \les r}$
. 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 coloured permutations. Besides, we derive a Carlitz identity for refined coloured permutations.