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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 $rge n$. In this paper, we provide an alternative approach to Beck and Stapledons conjecture by proving the following general result: if the polynomial sequence $left( h^{langle r,r-i rangle}(x)right)_{1le i le r}$ is interlacing, so is $left( U^{n}_{r,r-i}, h(x) right)_{1le i le r}$. Our result has many other interesting applications. In particular, this enables us to give a new proof of Savage and Visontais 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.
It was observed by Bump et al. that Ehrhart polynomials in a special family exhibit properties similar to the Riemann {zeta} function. The construction was generalized by Matsui et al. to a larger family of reflexive polytopes coming from graphs. We
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We prove a sharp analogue of Minkowskis inhomogeneous approximation theorem over fields of power series $mathbb{F}_q((T^{-1}))$. Furthermore, we study the approximation to a given point $underline{y}$ in $mathbb{F}_q((T^{-1}))^2$ by the $SL_2(mathbb{
We investigate a class of power series occurring in some problems in quantum optics. Their coefficients are either Gegenbauer or Laguerre polynomials multiplied by binomial coefficients. Although their sums have been known for a long time, we employ
This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree coefficients o