Rademacher-Carlitz polynomials

Matthias Beck, and Florian Kohl. "Rademacher-Carlitz polynomials." Acta Arithmetica 163.4 (2014): 379-393. <http://eudml.org/doc/279109>.

@article{MatthiasBeck2014,
abstract = {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.},
author = {Matthias Beck, Florian Kohl},
journal = {Acta Arithmetica},
keywords = {Rademacher-Carlitz polynomials; Dedekind sums; reciprocity theorem; lattice points; generating functions; rational cones},
language = {eng},
number = {4},
pages = {379-393},
title = {Rademacher-Carlitz polynomials},
url = {http://eudml.org/doc/279109},
volume = {163},
year = {2014},
}

TY - JOUR
AU - Matthias Beck
AU - Florian Kohl
TI - Rademacher-Carlitz polynomials
JO - Acta Arithmetica
PY - 2014
VL - 163
IS - 4
SP - 379
EP - 393
AB - 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.
LA - eng
KW - Rademacher-Carlitz polynomials; Dedekind sums; reciprocity theorem; lattice points; generating functions; rational cones
UR - http://eudml.org/doc/279109
ER -