We introduce and study the Rademacher-Carlitz polynomial $R(u,v,s,t,a,b):={\sum }_{k=\lceil s\rceil }^{\lceil s\rceil +b-1}{u}^{\lfloor (ka+t)\rfloor }/b_{v^k}$ where $a,b\in {\mathbb{Z}}_{>0}$, s,t ∈ ℝ, and u and v are variables. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view R(u,v,s,t,a,b) as a polynomial analogue (in the sense of Carlitz) of the Dedekind-Rademacher sum $r_t(a,b):={\sum }_{k=0}^{b-1}\left(((ka+t)/b)\right)\left((k/b)\right)$, 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...

It is well-known that the $N$-th Riemann sum of a compactly supported function on the real line converges to the Riemann integral at a much faster rate than the standard $O(1/N)$ rate of convergence if the sum is over the lattice, $Z/N$. In this paper we prove an n-dimensional version of this result for Riemann sums over polytopes.

All 3-dimensional convex polytopes are known to be rigid. Still their Minkowski differences (virtual polytopes) can be flexible with any finite freedom degree. We derive some sufficient rigidity conditions for virtual polytopes and present some examples of flexible ones. For example, Bricard's first and second flexible octahedra can be supplied by the structure of a virtual polytope.