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Rademacher-Carlitz Polynomials | | Matthias Beck
; Florian Kohl
; | | Date: |
1 Oct 2013 | | Abstract: | We introduce and study the emph{Rademacher-Carlitz polynomial} [
RC(u, v, s, t, a, b) := sum_{k = lceil s
ceil}^{lceil s
ceil + b - 1}
u^{fl{frac{ka + t}{b}}} v^k ] where $a, b in _{>0}$, $s, t in R$, 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} [
_t(a,b) := sum_{k=0}^{b-1}left(left(frac{ka+t}{b}
ight)
ight)
left(left(frac{k}{b}
ight)
ight), ] 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 ^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. | | Source: | arXiv, 1310.0380 | | Services: | Forum | Review | PDF | Favorites | |
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