Rademacher–Carlitz polynomials
Tom 163 / 2014
Streszczenie
We introduce and study the Rademacher–Carlitz polynomial \[ \operatorname{R}(u, v, s, t, a, b) := \sum_{ k = \lceil s \rceil }^{ \lceil s \rceil + b - 1 } u^{ \lfloor{ {( ka + t) \rfloor}/{ b } } } v^k \] where $a, b \in \mathbb Z_{ >0 }$, $s, t \in \mathbb R$, and $u$ and $v$ are variables. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view $\operatorname{R}(u, v, s, t, a, b)$ as a polynomial analogue (in the sense of Carlitz) of the Dedekind–Rademacher sum \[ \operatorname{r}_t(a,b) := \sum_{k=0}^{b-1}\bigg(\bigg(\frac{ka+t}{b}\bigg)\bigg) \bigg(\bigg(\frac{k}{b} \bigg)\bigg) , \] 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 the reciprocity theorem for Dedekind–Rademacher sums as a corollary which follows naturally from our setup.
