Rademacher-Carlitz polynomials

Matthias Beck ; Florian Kohl

Acta Arithmetica, Tome 166 (2014), p. 379-393 / Harvested from The Polish Digital Mathematics Library

Résumé

We introduce and study the Rademacher-Carlitz polynomial $R(u,v,s,t,a,b) := ∑_{k=⌈s⌉}^{⌈s⌉+b-1} u^{⌊(ka+t)⌋}/b_{v^k}$ where $a,b ∈ ℤ_{>0}$, 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 $r_t(a,b) := ∑_{k=0}^{b-1} (((ka+t)/b)) ((k/b))$, 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 $σ(x,y) := ∑_{(j,k)∈ ∩ ℤ²} x^jy^k$ of any rational polyhedron , and we derive the reciprocity theorem for Dedekind-Rademacher sums as a corollary which follows naturally from our setup.

Détail

Publié le : 2014-01-01

Zbl 1318.11057

EUDML-ID : urn:eudml:doc:279109