  We introduce and study the \emph{Rademacher-Carlitz polynomial} \[
  \RC(u, v, s, t, a, b) := \sum_{k = \lceil s \rceil}^{\lceil s \rceil + b - 1}
u^{\fl{\frac{ka + t}{b}}} v^k \] where $a, b \in \Z_{>0}$, $s, t \in \R$, 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 \[
  \sigma(x,y):=\sum_{(j,k) \in \mathcal{P}\cap \Z^2} x^j y^k \] of any rational
polyhedron $\mathcal{P}$, and we derive a novel reciprocity theorem for
Dedekind-Rademacher sums, which follows naturally from our setup.
