Rademacher-Carlitz Polynomials

We introduce and study the \emph{Rademacher-Carlitz polynomial} $$\text{RC}(u,v,s,t,a,b):=\sum _{k=\lceil s\rceil }^{\lceil s\rceil +b-1}{u}^{\text{fl}\frac{ka+t}{b}}{v}^k$$ where $a,b\in {\mathbb{Z}}_{>0}$, $s,t\in \mathbb{R}$, and $u$ and $v$ are variables. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view $\RC(u, v, s, t, a, b)$ as a polynomial analogue (in the sense of Carlitz) of the \emph{Dedekind-Rademacher sum} \r_t(a,b) := \sum_{k=0}^{b-1}\left(\left(\frac{ka+t}{b} \right)\right) \left(\left(\frac{k}{b} \right)\right), which appears in various number-theoretic, combinatorial, geometric, and computational contexts. Our results come in three flavors: we prove a reciprocity theorem for Rademacher-Carlitz polynomials, we show how they are the only nontrivial ingredients of integer-point transforms $$\sigma (x,y):=\sum _{(j,k)\in \mathcal{P}\cap {\mathbb{Z}}^2}{x}^j{y}^k$$ of any rational polyhedron $\mathcal{P}$, and we derive a novel reciprocity theorem for Dedekind-Rademacher sums, which follows naturally from our setup.