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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...